Introduction
Bernoulli percolation is one of the simplest and most fundamental models of statistical mechanics. In two dimensions, it has been central to the study of critical phenomena through conformal invariance. For critical site percolation on the triangular lattice, Smirnov [31 ] rigorously proved the conformal invariance of crossing probabilities and Cardy’s formula. Together with the introduction of Schramm–Loewner evolution (SLE6 _6 6 [28 ] , this establishes the convergence of the critical percolation interface to SLE6 _6 6 [13 ] , described by the nested conformal loop ensemble (CLE) CLE6 _6 6 [29 ] . The arm exponents are a family of basic scaling exponents in critical percolation that describe the power-law decay rate of the probabilities of disjoint paths crossing annuli with specific colors. An arm event is called polychromatic if its prescribed colors include both black and white, and monochromatic if all arms have the same color. The exact evaluation of these exponents was an early application of SLE. The one-arm exponent is 5 / 48 5/48 5/48 [18 ] , and the polychromatic (n ≥ 2 n \ge2 n ≥ 2 α n = n 2 − 1 12 \alpha_n = \frac{n^2-1}{12} α n = 12 n 2 − 1 [32 ] . These results confirmed predictions from statistical physics [1 ] .
On the other hand, for the monochromatic n n n n ≥ 2 n \ge2 n ≥ 2 [9 ] , it is shown that ξ n \xi_n ξ n α n < ξ n < α n + 1 \alpha_n < \xi_n < \alpha_{n+1} α n < ξ n < α n + 1 n ≥ 2 n \ge2 n ≥ 2 [25 ] that the monochromatic two-arm exponent ξ 2 = 0.3566668 ⋯ \xi_2 = 0.3566668\cdots ξ 2 = 0.3566668 ⋯
However, for three or more monochromatic arms, distinct white clusters may jointly obstruct the crossings even when none of them does so alone. In terms of the exploration process in [25 ] , the previously discovered loops become boundary arcs, and a newly discovered loop can separate the target together with one of these arcs. Therefore, the stopping rule depends on past connections, which makes essential difficulties in extending their derivation to general ξ n \xi_n ξ n [25 Remark 2.3] .
In this paper, we determine the monochromatic three-arm exponent ξ 3 \xi_3 ξ 3 ξ 3 = 0.76928375 ⋯ \xi_3 = 0.76928375\cdots ξ 3 = 0.76928375 ⋯ ξ 3 \xi_3 ξ 3 n n n n ≥ 4 n \ge4 n ≥ 4 § 1.3 .
The exponent ξ 3 \xi_3 ξ 3
Let δ T \delta\mathbb{T} δ T δ > 0 \delta> 0 δ > 0 1 / 2 1/2 1/2 0 < r < R 0 < r < R 0 < r < R p 3 ( r , R ) p_3(r,R) p 3 ( r , R ) { z ∈ C : r < ∣ z ∣ < R } \{z \in\mathbb{C} : r < |z| < R\} { z ∈ C : r < ∣ z ∣ < R } δ ↓ 0 \delta\downarrow0 δ ↓ 0 τ = ( 2 π ) − 1 log ( R / r ) > 0 \tau= (2\pi)^{-1}\log(R/r) > 0 τ = ( 2 π ) − 1 log ( R / r ) > 0 p 3 ( τ ) = p 3 ( e − 2 π τ , 1 ) p_3(\tau) = p_3(e^{-2\pi\tau},1) p 3 ( τ ) = p 3 ( e − 2 π τ , 1 ) [9 ] give
p 3 ( τ ) = exp { − 2 π ξ 3 τ + o ( τ ) } , 2 3 < ξ 3 < 5 4 . (1) p_3(\tau) = \exp\{-2\pi\xi_3\tau+ o(\tau)\}, \qquad\frac{2}{3} < \xi_3 < \frac{5}{4}. \tag*{(1)} p 3 ( τ ) = exp { − 2 π ξ 3 τ + o ( τ )} , 3 2 < ξ 3 < 4 5 . (1) We express ξ 3 \xi_3 ξ 3 R \mathbb{R} R t ∈ R t \in\mathbb{R} t ∈ R
x ( t ) = e t 1 + e t , a ( t ) = 1 − x ( t ) , L 0 = − d 2 d t 2 + 1 4 , x(t) = \frac{e^t}{1+e^t}, \qquad a(t) = 1-x(t), \qquad L_0 = -\frac{d^2}{dt^2} + \frac{1}{4}, x ( t ) = 1 + e t e t , a ( t ) = 1 − x ( t ) , L 0 = − d t 2 d 2 + 4 1 , H ( t ) = 9 3 16 π 2 cosh ( t / 2 ) + 3 ( 2 cosh ( t / 2 ) + 1 ) 3 . (2) H(t) = \frac{9\sqrt{3}}{16\pi}\frac{2\cosh(t/2)+3}{(2\cosh(t/2)+1)^3}. \tag*{(2)} H ( t ) = 16 π 9 3 ( 2 cosh ( t /2 ) + 1 ) 3 2 cosh ( t /2 ) + 3 . (2) For s ∈ C s \in\mathbb{C} s ∈ C ∣ Re s ∣ < 2 |\operatorname{Re}s| < 2 ∣ Re s ∣ < 2 f : R → C f:\mathbb{R}\to\mathbb{C} f : R → C
( T s f ) ( t ) = ∫ R H ( t − u ) ( [ x ( t ) ] 1 + s [ x ( u ) ] 1 − s + [ a ( t ) ] 1 + s [ a ( u ) ] 1 − s ) f ( u ) d u . (3) (T_s f)(t) = \int_{\mathbb{R}} H(t-u)\left([x(t)]^{1+s}[x(u)]^{1-s} + [a(t)]^{1+s}[a(u)]^{1-s}\right)f(u)\,du. \tag*{(3)} ( T s f ) ( t ) = ∫ R H ( t − u ) ( [ x ( t ) ] 1 + s [ x ( u ) ] 1 − s + [ a ( t ) ] 1 + s [ a ( u ) ] 1 − s ) f ( u ) d u . (3) The powers use the real logarithms of x x x a a a
Theorem 1.1. There is a unique positive parameter s ∗ ∈ ( 0 , 2 ) s_*\in(0,2) s ∗ ∈ ( 0 , 2 )
L 0 r = T s ∗ r (4) L_0r=T_{s_*}r \tag*{(4)} L 0 r = T s ∗ r (4) has a strictly positive bounded even solution r ∈ C 2 ( R ) r\in C^2(\mathbb{R}) r ∈ C 2 ( R ) 3 / 2 < s ∗ < 2 3/2<s_*<2 3/2 < s ∗ < 2
ξ 3 = s ∗ 2 − 1 / 4 3 . (5) \xi_3=\frac{s_*^2-1/4}{3}. \tag*{(5)} ξ 3 = 3 s ∗ 2 − 1/4 . (5) The solution is unique up to multiplication by a positive constant and has a positive common limit at both − ∞ -\infty − ∞ + ∞ +\infty + ∞
All coefficients and the boundary condition in (4) are explicit. Numerical evaluation gives
s ∗ ≈ 1.59932838 , ξ 3 ≈ 0.76928375. s_*\approx1.59932838,\qquad\xi_3\approx0.76928375. s ∗ ≈ 1.59932838 , ξ 3 ≈ 0.76928375. See Appendix § 8 .§ 3 . This agrees with the estimate 0.77 ± 0.02 0.77\pm0.02 0.77 ± 0.02 [17 ] .
The parameter s ∗ s_* s ∗
g ( t ) = e − ∣ t ∣ / 2 , q 0 ( t ) = 3 3 8 π ( cosh ( t / 2 ) + 1 / 2 ) , A ( t ) = 27 32 π 2 t coth 3 t 4 , g(t)=e^{-|t|/2},\qquad q_0(t)=\frac{3\sqrt{3}}{8\pi(\cosh(t/2)+1/2)},\qquad A(t)=\frac{27}{32\pi^2}t\coth\frac{3t}{4}, g ( t ) = e − ∣ t ∣/2 , q 0 ( t ) = 8 π ( cosh ( t /2 ) + 1/2 ) 3 3 , A ( t ) = 32 π 2 27 t coth 4 3 t , where A ( 0 ) = 9 / ( 8 π 2 ) A(0)=9/(8\pi^2) A ( 0 ) = 9/ ( 8 π 2 ) ( f ∗ h ) ( t ) = ∫ R f ( t − u ) h ( u ) d u (f*h)(t)=\int_{\mathbb{R}}f(t-u)h(u)\,\mathrm{d}u ( f ∗ h ) ( t ) = ∫ R f ( t − u ) h ( u ) d u
R s f = g ∗ ( T s f ) − q 0 ∗ f , S f = f − A ∗ f + A ∫ R f . R_sf=g*(T_sf)-q_0*f,\qquad Sf=f-A*f+A\int_{\mathbb{R}}f. R s f = g ∗ ( T s f ) − q 0 ∗ f , S f = f − A ∗ f + A ∫ R f . The latter formula is applied to exponentially decaying even functions. Let L e v e n 2 ( R ) L^2_{\mathrm{even}}(\mathbb{R}) L even 2 ( R ) I I I 1 1 1
Proposition 1.2. The block operator below is the identity plus a trace-class operator on C ⊕ L e v e n 2 ( R ) \mathbb{C}\oplus L^2_{\mathrm{even}}(\mathbb{R}) C ⊕ L even 2 ( R )
D ( s ) = det ( ∫ R R s 1 f ⟼ ∫ R R s f − S R s 1 I − S R s ) , ∣ Re s ∣ < 2 , (6) \mathcal{D}(s)=\det\begin{pmatrix}
\int_{\mathbb{R}}R_s1 & f\longmapsto\int_{\mathbb{R}}R_sf\\
-SR_s1 & I-SR_s
\end{pmatrix},\qquad|\operatorname{Re}s|<2, \tag*{(6)} D ( s ) = det ( ∫ R R s 1 − S R s 1 f ⟼ ∫ R R s f I − S R s ) , ∣ Re s ∣ < 2 , (6) is holomorphic in this strip. The parameter s ∗ s_* s ∗ Theorem 1.1 is its smallest positive zero:
s ∗ = min { s ∈ ( 0 , 2 ) : D ( s ) = 0 } . s_*=\min\{s\in(0,2):\mathcal{D}(s)=0\}. s ∗ = min { s ∈ ( 0 , 2 ) : D ( s ) = 0 } . This zero is simple.
The proof of Proposition 1.2 is given in Appendix § 8 .
Proof strategy
The main difficulty is to construct an observable for three-arm connectivity that can be evaluated after cutting a complete boundary-touching CLE loop. Two white clusters may jointly obstruct three disjoint black crossings, so an exploration must retain their past connections. In § 2 , we introduce two outer boundary marks and a boundary two-point observable recording these connections. For a boundary-touching loop whose white cluster avoids the inner set, we identify the observable in the retained exterior component, marked at the endpoints of its inherited boundary arc, with the indicator that the original three-arm event holds and the discarded boundary arc is maximal; see Lemma 2.5. Averaging over the original marks then expresses the averaged observable as a weighted sum of the same observable in the retained components, together with a contribution from white clusters meeting the inner set. This is the formula of Proposition 2.3 .
To evaluate this sum, we need a tractable description of the retained component together with its inherited boundary arc and the newly exposed arc along the CLE interface. We couple CLE with an independent LQG field and equip both arcs with their quantum boundary lengths. The coupling of SLE and CLE with LQG was developed in [30 , 14 , 22 , 21 ] and has since been widely used to derive exact formulas in statistical mechanics; see e.g. [5 , 4 SXZ24, LSYZ26, 12 ] . In § 3 , we use this coupling to identify the quantum surface associated with the retained component as a CLE-decorated quantum disk with one interior mark and two boundary marks. It also gives an explicit kernel for the quantum boundary lengths of the newly exposed and the discarded boundary arcs. Integrating the sum in Proposition 2.3 against the LQG field therefore gives a closed equation in terms of quantum boundary lengths; see Proposition 3.2 .
We then transfer this equation to the annulus setup, and obtain an equation for the annulus partition function in § 4 . This follows from the construction of the Brownian annulus [7 ] and its application to percolation in [SXZ24]. After Mellin transformation, this yields an integro-differential equation in Proposition 4.1 with an explicit kernel.
The remaining task is to analyze the integro-differential equation of Proposition 4.1 and derive an exact spectral formula for ξ 3 \xi_3 ξ 3 § 5 , we combine arm event estimates with moment bounds for Gaussian multiplicative chaos (GMC) [26 , 27 ] to identify the boundary condition satisfied by the transformed observable. The general spectral threshold theorem, Theorem 6.1 , then identifies the convergence endpoint of the Mellin transform with the spectral threshold, proving Theorem 1.1 . Appendix § 8 gives an alternative expression via Fredholm determinant and proves Proposition 1.2 .
The above approach extends to the monochromatic three-arm exponents of C L E κ \mathrm{CLE}_{\kappa} CLE κ 4 < κ < 8 4<\kappa<8 4 < κ < 8 § 3 is valid for general 4 < κ < 8 4<\kappa<8 4 < κ < 8 § 5 for general κ \kappa κ
Remark 1.3 (SLE-martingale approach). In [18 Appendix B] , based on the SLE exploration, the backbone exponent ξ 2 \xi_2 ξ 2 [25 ] . For ξ 3 \xi_3 ξ 3 Proposition 2.3 to develop an SLE-martingale approach that tracks both the conformal radius and the harmonic measure of the inherited boundary arc in the component retained after cutting a complete boundary-touching loop. It would be interesting to determine whether this approach yields an explicit characterization of ξ 3 \xi_3 ξ 3 Theorem 1.1 .
Monochromatic n n n n ≥ 4 n \ge4 n ≥ 4
The approach above provides a robust method for determining all of the monochromatic n n n ξ n \xi_n ξ n § 2 extends to n ≥ 4 n \ge4 n ≥ 4
Here we briefly present the spectral characterization for every ξ n \xi_n ξ n n ≥ 4 n \ge4 n ≥ 4 r ( A ) r(A) r ( A ) A A A
Δ n − 1 = { ( x 1 , … , x n − 1 ) ∈ ( 0 , ∞ ) n − 1 : ∑ i = 1 n − 1 x i = 1 } , d x = d x 1 ⋯ d x n − 2 , \Delta_{n-1}=\left\{(x_1,\ldots,x_{n-1})\in(0,\infty)^{n-1}:\sum_{i=1}^{n-1}x_i=1\right\},\qquad dx=dx_1\cdots dx_{n-2}, Δ n − 1 = { ( x 1 , … , x n − 1 ) ∈ ( 0 , ∞ ) n − 1 : i = 1 ∑ n − 1 x i = 1 } , d x = d x 1 ⋯ d x n − 2 , where x n − 1 = 1 − ∑ i = 1 n − 2 x i x_{n-1}=1-\sum_{i=1}^{n-2}x_i x n − 1 = 1 − ∑ i = 1 n − 2 x i x i x_i x i i i i L 1 ( Δ n − 1 , d x ) L^1(\Delta_{n-1},dx) L 1 ( Δ n − 1 , d x )
Theorem 1.4 (Higher monochromatic arm exponents). For each n ≥ 4 n \ge4 n ≥ 4 0 < β < 1 0<\beta<1 0 < β < 1 L n , β \mathcal{L}_{n,\beta} L n , β L 1 ( Δ n − 1 , d x ) L^1(\Delta_{n-1},dx) L 1 ( Δ n − 1 , d x ) [Cai26, (1.1)]. There is a unique β n ∈ ( 0 , 1 ) \beta_n\in(0,1) β n ∈ ( 0 , 1 )
r ( L n , β n ) = 1 , ξ n = ( n + β n ) 2 − 1 12 . (7) r(\mathcal{L}_{n,\beta_n})=1,\qquad\xi_n=\frac{(n+\beta_n)^2-1}{12}. \tag*{(7)} r ( L n , β n ) = 1 , ξ n = 12 ( n + β n ) 2 − 1 . (7) The spectral radius is less than one for 0 < β < β n 0<\beta<\beta_n 0 < β < β n β n < β < 1 \beta_n<\beta<1 β n < β < 1 β = β n \beta=\beta_n β = β n
In [Cai26], we explicitly define the operator L n , β \mathcal{L}_{n,\beta} L n , β ξ 3 \xi_3 ξ 3 ξ n \xi_n ξ n n ≥ 4 n \ge4 n ≥ 4 L n , β \mathcal{L}_{n,\beta} L n , β
The main new geometric step is the construction of a connection event requiring, for each choice of a marked point, two black paths from that point and one from every other mark to distinct points of the inner boundary, disjoint except at their common starting point; see [Cai26, Section 2]. The additional analytic work consists of estimates for simultaneous collisions of marks and control of the essential spectral radius, including contributions from cutting sequences that leave some original arc lengths unchanged. This is sketched in [Cai26, Sections 4 and 5].
Finite-dimensional approximations of L n , β \mathcal{L}_{n,\beta} L n , β ξ n \xi_n ξ n
Table 2.
For n = 4 , 5 , 6 n=4,5,6 n = 4 , 5 , 6 [17 ] . We discuss the difficulties in improving numerical precision for n ≥ 4 n \ge4 n ≥ 4
Acknowledgment. We thank Zhuoyan Xie and Zijie Zhuang for helpful discussions. This work was supported by the National Natural Science Foundation of China (Grant No.12526204).
AI statement. The author developed the proof strategy, including the geometric exploration, the LQG derivation, and the reduction to a renewal equation and a spectral threshold problem. Within this framework, AI tools were used in carrying out calculations, technical estimates, and proofs in functional analysis, especially for n ≥ 4 n \ge4 n ≥ 4
The annular event and a two-point boundary observable
Throughout this section we use the nested CLE6 _6 6 [31 , 13 , 15 ] . The black and white sides of each interface are inherited from percolation. Let D D D K ⋐ D K \Subset D K ⋐ D D ∖ K D \setminus K D ∖ K D D D K K K (10) below.
White clusters and necklaces
Consider critical site percolation on the triangular lattice of mesh size δ \delta δ H 3 δ ( D , K ) H_3^\delta(D,K) H 3 δ ( D , K ) quasi-white circuit is a simple circuit separating the two boundary components whose vertices are white except for a specified finite set of black vertices. Menger’s theorem and planar duality give
( H 3 δ ( D , K ) ) c = { there is a separating circuit with at most two black vertices } . (8) \left(H_3^\delta(D,K)\right)^c = \{\text{there is a separating circuit with at most two black vertices}\}. \tag*{(8)} ( H 3 δ ( D , K ) ) c = { there is a separating circuit with at most two black vertices } . (8) If both white portions of a quasi-white circuit with two black vertices belong to the same white cluster in D ∖ K D \setminus K D ∖ K
We use the external frontier of each white cluster, obtained at finite mesh by closing passages of width one in its black external vertex boundary, as in [25 Section 2.2] . At each passage, keep the side specified by the quasi-white circuit. In the limit, this frontier lies in the complete interface trace. A separating circuit with at most two black vertices becomes a necklace of one or two complete interfaces, each with winding number one in the complementary regions on its white side and with the prescribed side at every contact.
Let H 3 ( D , K ) H_3(D,K) H 3 ( D , K ) K K K [9 Section 4] identifies this measurable event in the full percolation scaling limit and gives
P [ H 3 δ ( D , K ) ] ⟶ P [ H 3 ( D , K ) ] ( δ ↓ 0 ) . (9) \mathbb{P}[H_3^\delta(D,K)] \longrightarrow\mathbb{P}[H_3(D,K)] \qquad(\delta\downarrow0). \tag*{(9)} P [ H 3 δ ( D , K )] ⟶ P [ H 3 ( D , K )] ( δ ↓ 0 ) . (9) For a rough inner set, such as the filled metric ball used below, first uniformize D D D D ∖ K D \setminus K D ∖ K { r < ∣ w ∣ < 1 } \{r < |w| < 1\} { r < ∣ w ∣ < 1 } r r r r + ε r+\varepsilon r + ε ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0 O ( ε − 2 ) O(\varepsilon^{-2}) O ( ε − 2 ) ε \varepsilon ε O ( ε − 1 ) O(\varepsilon^{-1}) O ( ε − 1 ) [32 Theorems 3 and 4] .
Apply these estimates to the finitely many interfaces before removing the boundary approximations, as in the frontier comparison of [25 Lemma 2.3] . Then decrease the diameter cutoff. A vanishing interface in a macroscopic obstruction involving two white clusters either leaves a one-cluster obstruction or forces the same six-arm event at the two black vertices, as in [9 Section 4] .
To define white connections after removing K K K D D D [24 Section 1.2] . Orient its outer interface η \eta η C η C_\eta C η Γ η \Gamma_\eta Γ η u , v u, v u , v V V V Σ \Sigma Σ P ( u , v ; V ; Σ ) P(u, v; V; \Sigma) P ( u , v ; V ; Σ ) V V V u u u v v v Σ \Sigma Σ C η C_\eta C η Γ η \Gamma_\eta Γ η
d p a t h , η ( u , v ) = inf { diam E ( γ ) : γ ∈ P ( u , v ; C η ‾ ; { η } ∪ Γ η ) } , d_{\mathrm{path},\eta}(u,v)=\inf\{\operatorname{diam}_{\mathrm{E}}(\gamma):\gamma\in P(u,v;\overline{C_\eta};\{\eta\}\cup\Gamma_\eta)\}, d path , η ( u , v ) = inf { diam E ( γ ) : γ ∈ P ( u , v ; C η ; { η } ∪ Γ η )} , where diam E \operatorname{diam}_{\mathrm{E}} diam E X η X_\eta X η Π η : X η → D ‾ \Pi_\eta:X_\eta\to\overline{D} Π η : X η → D
The white clusters in D ∖ K D\setminus K D ∖ K are the path components of
X η ∖ Π η − 1 ( K ) . (10) X_\eta\setminus\Pi_\eta^{-1}(K). \tag*{(10)} X η ∖ Π η − 1 ( K ) . (10) Write η C \eta_C η C C C C C C C D D D X C = X η C X_C=X_{\eta_C} X C = X η C Π C = Π η C \Pi_C=\Pi_{\eta_C} Π C = Π η C d p a t h , C = d p a t h , η C d_{\mathrm{path},C}=d_{\mathrm{path},\eta_C} d path , C = d path , η C C C C D ∖ K D\setminus K D ∖ K X C X_C X C X η C X_{\eta_C} X η C Π C = Π η C ∣ X C \Pi_C=\left.\Pi_{\eta_C}\right|_{X_C} Π C = Π η C ∣ X C X C X_C X C X η X_\eta X η η \eta η η \eta η ∂ D \partial D ∂ D [24 Sections 1.2 and 2.4] .
The space X η X_\eta X η d p a t h , η d_{\mathrm{path},\eta} d path , η [24 Section 2.4] . It is therefore separable and locally path connected, so the components in (10) are open and countable. Enumerate them by the first points they contain in an independent sequence sampled from the normalized gasket measure of full support [24 Section 2.3] . This sequence is almost surely dense. Membership in one component is tested by finite chains of geodesic balls with rational radii centered at these points, whose closures avoid Π η − 1 ( K ) \Pi_\eta^{-1}(K) Π η − 1 ( K ) [13 Section 2.2] , with the inherited orientations; we use the completed CLE probability space.
We use the following contact properties of C L E 6 \mathrm{CLE}_6 CLE 6 [13 Theorem 2(1)] ; the boundary assertion follows from the half-plane three-arm estimate [32 Theorem 3] .
The two-point boundary observable
Fix distinct p , q ∈ ∂ D p,q\in\partial D p , q ∈ ∂ D p p p q q q D D D p , q p,q p , q p p p q q q
In the following, an original white cluster is a white cluster in D ∖ K D\setminus K D ∖ K λ p \lambda_p λ p λ q \lambda_q λ q λ p q \lambda_{pq} λ pq p p p q q q { p , q } \{p,q\} { p , q }
λ p ≥ 2 , λ q ≥ 2 , λ p q ≥ 3. (11) \lambda_p\ge2,\qquad\lambda_q\ge2,\qquad\lambda_{pq}\ge3. \tag*{(11)} λ p ≥ 2 , λ q ≥ 2 , λ pq ≥ 3. (11) Write P 0 , P 1 P_0,P_1 P 0 , P 1
Definition 2.1 (Two-point boundary observable). A separating configuration is of one of the following four types and separates the marked points p , q p,q p , q K K K Figure 1 ):
(a) one original white cluster;
Figure 1. The four separating configurations. Blue and pink indicate distinct original white clusters. In (c)–(d), closing a crosscut in the white cluster along the thick exterior arc separates K K K p , q p,q p , q
(b) two original white clusters;
(c) an original white cluster together with P 0 P_0 P 0
(d) an original white cluster together with P 1 P_1 P 1
Define G ( D , K ; p , q ) G(D,K;p,q) G ( D , K ; p , q )
Away from neighborhoods of the marked points carrying a macroscopic white cluster, Menger’s theorem and planar duality show that failure of λ p q ≥ 3 \lambda_{pq} \ge3 λ pq ≥ 3 λ p ≥ 2 \lambda_p \ge2 λ p ≥ 2 q q q p p p λ q ≥ 2 \lambda_q \ge2 λ q ≥ 2 (11) .
For fixed p , q p,q p , q (11) .
Boundary arcs and integration of the marks
For x , ℓ ∈ ( 0 , 1 ) x,\ell\in(0,1) x , ℓ ∈ ( 0 , 1 ) w ℓ ( x ) = min { x , 1 − x , ℓ , 1 − ℓ } w_\ell(x) = \min\{x,1-x,\ell,1-\ell\} w ℓ ( x ) = min { x , 1 − x , ℓ , 1 − ℓ } G G G Figure 2 ).
Figure 2. Averaging boundary marks gives the piecewise linear weight in Lemma 2.2 . The left panel shows three maximal discarded boundary arcs; omitted arcs fill the gaps up to a null set. Average p p p μ \mu μ q q q μ \mu μ x L xL xL p p p ℓ < 1 / 2 \ell< 1/2 ℓ < 1/2 − w ℓ -w_\ell^{} − w ℓ ℓ , 1 − ℓ \ell,1-\ell ℓ , 1 − ℓ K K K S C S_C S C
Let C C C D ∖ K D \setminus K D ∖ K Π C ( X C ) \Pi_C(X_C) Π C ( X C ) D ∖ Π C ( X C ) D \setminus\Pi_C(X_C) D ∖ Π C ( X C ) K K K ∂ K \partial K ∂ K ∂ D \partial D ∂ D inherited boundary arc J C J_C J C
S C = ∂ D ∖ J C . S_C = \partial D \setminus J_C. S C = ∂ D ∖ J C . The discarded boundary arc S C S_C S C C C C D ∖ K D \setminus K D ∖ K C C C D D D Π C ( X C ) ∩ K = ∅ \Pi_C(X_C) \cap K = \varnothing Π C ( X C ) ∩ K = ∅ η C \eta_C η C S C S_C S C K K K
Write H 3 = H 3 ( D , K ) H_3 = H_3(D,K) H 3 = H 3 ( D , K ) H 3 H_3 H 3 M \mathcal{M} M
Let μ \mu μ L = μ ( ∂ D ) L = \mu(\partial D) L = μ ( ∂ D ) ℓ C = μ ( S C ) / L \ell_C = \mu(S_C)/L ℓ C = μ ( S C ) / L μ \mu μ S 0 S_0 S 0 S 0 S_0 S 0 K K K K K K S 0 S_0 S 0
Each such loop also has only finitely many white clusters in D ∖ K D \setminus K D ∖ K η \eta η X η X_\eta X η F o = Π η − 1 ( ∂ D ) F_o = \Pi_\eta^{-1}(\partial D) F o = Π η − 1 ( ∂ D ) F i = Π η − 1 ( K ) F_i = \Pi_\eta^{-1}(K) F i = Π η − 1 ( K ) Δ > 0 \Delta> 0 Δ > 0 F o F_o F o Δ / 3 \Delta/3 Δ/3 F o F_o F o F i F_i F i X η ∖ F i X_\eta\setminus F_i X η ∖ F i S 0 S_0 S 0
Each nondegenerate maximal arc belongs to a unique white cluster in D ∖ K D \setminus K D ∖ K η \eta η Π η \Pi_\eta Π η Lemma 2.4 , applied before deleting K K K Π η − 1 ( K ) \Pi_\eta^{-1}(K) Π η − 1 ( K ) D ∖ K D \setminus K D ∖ K
Lemma 2.2. On H 3 H_3 H 3 μ \mu μ ∂ D \partial D ∂ D ∂ D \partial D ∂ D R / L Z \mathbb{R}/L\mathbb{Z} R / L Z μ \mu μ 0 < x < 1 0<x<1 0 < x < 1 p + x L p+xL p + xL μ \mu μ x L xL xL p p p
G ‾ ( x ) = 1 L ∫ ∂ D G ( D , K ; p , p + x L ) μ ( d p ) . \overline{G}(x)=\frac{1}{L}\int_{\partial D}G(D,K;p,p+xL)\,\mu(dp). G ( x ) = L 1 ∫ ∂ D G ( D , K ; p , p + xL ) μ ( d p ) . Then for any nonnegative integrable function f f f ( 0 , 1 ) (0,1) ( 0 , 1 )
G ‾ ( x ) = 1 H 3 ∑ C ∈ M w ℓ C ( x ) (12) \overline{G}(x)=\mathbf{1}_{H_3}\sum_{C\in\mathcal{M}}w_{\ell_C}(x) \tag*{(12)} G ( x ) = 1 H 3 C ∈ M ∑ w ℓ C ( x ) (12) ∫ 0 1 f ( x ) G ‾ ( x ) d x = 1 H 3 ∑ C ∈ M Ψ f ( ℓ C ) , Ψ f ( ℓ ) = ∫ 0 1 f ( x ) w ℓ ( x ) d x . (13) \int_0^1f(x)\overline{G}(x)\,dx=\mathbf{1}_{H_3}\sum_{C\in\mathcal{M}}\Psi_f(\ell_C),\qquad\Psi_f(\ell)=\int_0^1f(x)w_\ell(x)\,dx. \tag*{(13)} ∫ 0 1 f ( x ) G ( x ) d x = 1 H 3 C ∈ M ∑ Ψ f ( ℓ C ) , Ψ f ( ℓ ) = ∫ 0 1 f ( x ) w ℓ ( x ) d x . (13) Proof. Fix x ∈ ( 0 , 1 ) x\in(0,1) x ∈ ( 0 , 1 ) H 3 H_3 H 3 μ \mu μ μ \mu μ p ↦ p + x L p\mapsto p+xL p ↦ p + xL p p p q = p + x L q=p+xL q = p + xL p p p q q q
Suppose both marks lie in the interior of S C S_C S C η = η C \eta=\eta_C η = η C I ⋐ S C I\Subset S_C I ⋐ S C C C C I I I C C C A A A X η ∖ Π η − 1 ( K ) X_\eta\setminus\Pi_\eta^{-1}(K) X η ∖ Π η − 1 ( K ) K K K I I I D ∖ Π η ( A ) D\setminus\Pi_\eta(A) D ∖ Π η ( A ) K K K H 3 c H_3^c H 3 c
For a circle of length one and an arc of length ℓ \ell ℓ p p p p + x p+x p + x ( ℓ − x ) + + ( ℓ − 1 + x ) + (\ell-x)_++(\ell-1+x)_+ ( ℓ − x ) + + ( ℓ − 1 + x ) + y + = max { y , 0 } y_+=\max\{y,0\} y + = max { y , 0 } ∑ C ℓ C = 1 \sum_C\ell_C=1 ∑ C ℓ C = 1
∑ C ( ℓ C − ( ℓ C − x ) + − ( ℓ C − 1 + x ) + ) = ∑ C min { x , 1 − x , ℓ C , 1 − ℓ C } . \sum_C\left(\ell_C-(\ell_C-x)_+-(\ell_C-1+x)_+\right)=\sum_C\min\{x,1-x,\ell_C,1-\ell_C\}. C ∑ ( ℓ C − ( ℓ C − x ) + − ( ℓ C − 1 + x ) + ) = C ∑ min { x , 1 − x , ℓ C , 1 − ℓ C } . This gives (12) ; integrating against f f f (13) . □ \square □
For the quantum disk with an independent CLE used below, condition on the Liouville field, which determines the quantum boundary length measure μ \mu μ K K K p p p K K K ε \varepsilon ε p p p H 3 H_3 H 3 D ∖ K D\setminus K D ∖ K p p p [32 Theorem 3] makes this event null for each fixed p p p μ \mu μ
1 H 3 μ ( ∂ D ∖ ⋃ C ∈ M S C ) = 0 almost surely . (14) \mathbf{1}_{H_3}\mu\left(\partial D\setminus\bigcup_{C\in\mathcal{M}}S_C\right)=0\quad\text{almost surely}. \tag*{(14)} 1 H 3 μ ( ∂ D ∖ C ∈ M ⋃ S C ) = 0 almost surely . (14) The countable set of arc endpoints has zero μ \mu μ Lemma 2.2 applies to the quantum disk with its independent CLE.
Cutting a boundary-touching loop
We rewrite the terms in (13) with Π C ( X C ) ∩ K = ∅ \Pi_C(X_C) \cap K = \varnothing Π C ( X C ) ∩ K = ∅
Proposition 2.3 (Averaged cutting identity). Let μ \mu μ (14) . For each white cluster C C C D D D η C \eta_C η C Π C ( X C ) \Pi_C(X_C) Π C ( X C ) K K K U C U_C U C D ∖ η C D \setminus\eta_C D ∖ η C K K K C C C U C U_C U C η C \eta_C η C J C J_C J C p C , q C p_C, q_C p C , q C L L L G ‾ \overline{G} G M \mathcal{M} M Ψ f \Psi_f Ψ f f f f ( 0 , 1 ) (0,1) ( 0 , 1 )
∫ 0 1 f ( x ) G ‾ ( x ) d x = 1 H 3 ∑ C ∈ M : Π C ( X C ) ∩ K ≠ ∅ Ψ f ( ℓ C ) + ∑ C : Π C ( X C ) ∩ K = ∅ Ψ f ( ℓ C ) G ( U C , K ; p C , q C ) . (15) \int_0^1 f(x)\overline{G}(x)\,dx
=
1_{H_3}
\sum_{\substack{C\in\mathcal{M}:\\ \Pi_C(X_C)\cap K\ne\varnothing}}
\Psi_f(\ell_C)
+
\sum_{\substack{C:\\ \Pi_C(X_C)\cap K=\varnothing}}
\Psi_f(\ell_C)G(U_C,K;p_C,q_C). \tag*{(15)} ∫ 0 1 f ( x ) G ( x ) d x = 1 H 3 C ∈ M : Π C ( X C ) ∩ K = ∅ ∑ Ψ f ( ℓ C ) + C : Π C ( X C ) ∩ K = ∅ ∑ Ψ f ( ℓ C ) G ( U C , K ; p C , q C ) . (15) The second sum is over the white clusters C C C η C \eta_C η C
A white cluster C C C D ∖ K D\setminus K D ∖ K K K K D D D K K K η = η C \eta=\eta_C η = η C F = Π η − 1 ( K ) F=\Pi_\eta^{-1}(K) F = Π η − 1 ( K ) W W W X η ∖ F X_\eta\setminus F X η ∖ F F F F X η X_\eta X η F F F
For a cluster C C C (15) , U C U_C U C D ∖ Π C ( X C ) D\setminus\Pi_C(X_C) D ∖ Π C ( X C ) J C J_C J C J C J_C J C newly exposed boundary arc I C I_C I C C C C P I P_I P I P J P_J P J I C I_C I C J C J_C J C
We first describe the white connections in U C U_C U C G ( U C , K ; p C , q C ) G(U_C,K;p_C,q_C) G ( U C , K ; p C , q C )
Lemma 2.4. For a boundary-touching interface loop η C \eta_C η C U C U_C U C Π C ( X C ) ∩ K = ∅ \Pi_C(X_C)\cap K=\varnothing Π C ( X C ) ∩ K = ∅ U C U_C U C K K K C C C η C \eta_C η C U C U_C U C P I P_I P I Π C \Pi_C Π C X C X_C X C
Proof. Write η = η C \eta=\eta_C η = η C § 2.1 imply that η \eta η E E E D D D z ∈ Π E ( X E ) ∩ η z\in\Pi_E(X_E)\cap\eta z ∈ Π E ( X E ) ∩ η D D D E E E η \eta η η \eta η η E \eta_E η E z z z E E E η \eta η
Each such contact z z z Π E \Pi_E Π E d p a t h , E d_{\mathrm{path},E} d path , E η E \eta_E η E z z z [2 Lemma 1.10] . The single visit of η E \eta_E η E z z z D D D X E X_E X E z z z Π E − 1 ( K ) \Pi_E^{-1}(K) Π E − 1 ( K ) D ∖ K D \setminus K D ∖ K E E E η \eta η U C U_C U C D D D C C C
Finally, for each frontier portion to be replaced, choose the preimages of its endpoints according to the prescribed sides, join them by a path in X C X_C X C Π C \Pi_C Π C K K K η \eta η K K K K K K
Figure 3. The two mixed separating configurations used in the proof of Lemma 2.5. In U C U_C U C J C J_C J C I C I_C I C P I P_I P I P J P_J P J E E E H 3 H_3 H 3 E E E I C I_C I C K K K C C C S C ⊊ S E S_C \subsetneq S_E S C ⊊ S E P J P_J P J U C U_C U C P I P_I P I X C X_C X C D D D E , C E,C E , C
Lemma 2.5 (Pointwise cutting identity). For each boundary-touching interface loop η C \eta_C η C U C U_C U C Π C ( X C ) ∩ K = ∅ \Pi_C(X_C) \cap K = \varnothing Π C ( X C ) ∩ K = ∅
1 H 3 ( D , K ) 1 { C ∈ M } = G ( U C , K ; p C , q C ) . (16) \mathbf{1}_{H_3(D,K)}\mathbf{1}_{\{C\in\mathcal{M}\}} = G(U_C,K;p_C,q_C). \tag*{(16)} 1 H 3 ( D , K ) 1 { C ∈ M } = G ( U C , K ; p C , q C ) . (16) The observable on the right uses the original white connections restricted to U C ∖ K U_C \setminus K U C ∖ K
Proof. Apply Lemma 2.4 ; Figure 3 shows the two mixed cases. If H 3 ( D , K ) H_3(D,K) H 3 ( D , K ) U C U_C U C C C C C C C I C I_C I C G ( U C , K ; p C , q C ) = 0 G(U_C,K;p_C,q_C)=0 G ( U C , K ; p C , q C ) = 0 H 3 ( D , K ) H_3(D,K) H 3 ( D , K ) C ∉ M C \notin\mathcal{M} C ∈ / M E E E S C ⊊ S E S_C \subsetneq S_E S C ⊊ S E U C U_C U C I C I_C I C K K K P J P_J P J
@page 12/41 starts_mid=1 ends_mid=1 By compactness and local path connectedness of X E X_E X E z z z Π E − 1 ( K ) \Pi_E^{-1}(K) Π E − 1 ( K ) D ∖ K D \setminus K D ∖ K E E E η \eta η U C U_C U C D D D C C C
Finally, for each frontier portion to be replaced, choose the preimages of its endpoints according to the prescribed sides, join them by a path in X C X_C X C Π C \Pi_C Π C K K K η \eta η K K K K K K
Figure 3 (PDF p. 12). The two mixed separating configurations used in the proof of Lemma 2.5. In U C U_C U C J C J_C J C I C I_C I C P I P_I P I P J P_J P J E E E H 3 H_3 H 3 E E E I C I_C I C K K K C C C S C ⊊ S E S_C \subsetneq S_E S C ⊊ S E P J P_J P J U C U_C U C P I P_I P I X C X_C X C D D D E , C E,C E , C
Lemma 2.5 (Pointwise cutting identity). For each boundary-touching interface loop η C \eta_C η C U C U_C U C Π C ( X C ) ∩ K = ∅ \Pi_C(X_C) \cap K = \varnothing Π C ( X C ) ∩ K = ∅
1 H 3 ( D , K ) 1 { C ∈ M } = G ( U C , K ; p C , q C ) . (17) \mathbf{1}_{H_3(D,K)}\mathbf{1}_{\{C\in\mathcal{M}\}} = G(U_C,K;p_C,q_C). \tag*{(17)} 1 H 3 ( D , K ) 1 { C ∈ M } = G ( U C , K ; p C , q C ) . (17) The observable on the right uses the original white connections restricted to U C ∖ K U_C \setminus K U C ∖ K
Proof. Apply Lemma 2.4 ; Figure 3 shows the two mixed cases. If H 3 ( D , K ) H_3(D,K) H 3 ( D , K ) U C U_C U C C C C C C C I C I_C I C G ( U C , K ; p C , q C ) = 0 G(U_C,K;p_C,q_C)=0 G ( U C , K ; p C , q C ) = 0 H 3 ( D , K ) H_3(D,K) H 3 ( D , K ) C ∉ M C \notin\mathcal{M} C ∈ / M E E E S C ⊊ S E S_C \subsetneq S_E S C ⊊ S E U C U_C U C I C I_C I C K K K P J P_J P J U C U_C U C
1 H 3 ( D , K ) 1 { C ∈ M } = 0 ⟹ G ( U C , K ; p C , q C ) = 0. \mathbf{1}_{H_3(D,K)}\mathbf{1}_{\{C\in\mathcal{M}\}}=0 \quad\Longrightarrow\quad G(U_C,K;p_C,q_C)=0. 1 H 3 ( D , K ) 1 { C ∈ M } = 0 ⟹ G ( U C , K ; p C , q C ) = 0. Conversely, suppose G ( U C , K ; p C , q C ) = 0 G(U_C,K;p_C,q_C)=0 G ( U C , K ; p C , q C ) = 0 U C U_C U C D D D D ∖ U C D\setminus U_C D ∖ U C P I P_I P I X C X_C X C D D D P J P_J P J E E E E E E J C J_C J C I C I_C I C K K K K K K E E E J C J_C J C S C ⊊ S E S_C\subsetneq S_E S C ⊊ S E D D D 1 H 3 ( D , K ) 1 { C ∈ M } = 0 \mathbf{1}_{H_3(D,K)}\mathbf{1}_{\{C\in\mathcal{M}\}}=0 1 H 3 ( D , K ) 1 { C ∈ M } = 0
Proof of Proposition 2.3. Split the sum in Lemma 2.2 according to whether Π C ( X C ) \Pi_C(X_C) Π C ( X C ) K K K K K K C C C D D D
The first term in (15) gives the nonnegative inhomogeneous term in the renewal equation of Proposition 4.3. Each cluster contributing to this term meets both boundary components of the annulus and, on H 3 H_3 H 3 B B B W BBBW B B B W
Conformal welding and the boundary-length kernel
For every 4 < κ < 8 4<\kappa<8 4 < κ < 8 C L E κ \mathrm{CLE}_\kappa CLE κ [30 , 14 , 22 , 21 ] and later developed in e.g. [5 , 6 , 4 LSYZ26] . In particular, we integrate the radial welding identity of [LSYZ26, Theorem 5.19] over all quantum surfaces other than the retained disk, and obtain the explicit joint measure of the boundary lengths and the retained disk. This is given in Proposition 3.2.
Throughout this section,
4 < κ < 8 , γ = 4 κ ∈ ( 2 , 2 ) , α = κ 4 , e = α − 1 , Q L = 2 γ + γ 2 . 4<\kappa<8,\qquad\gamma=\frac{4}{\sqrt{\kappa}}\in(\sqrt{2},2),\qquad\alpha=\frac{\kappa}{4},\qquad e=\alpha-1,\qquad Q_L=\frac{2}{\gamma}+\frac{\gamma}{2}. 4 < κ < 8 , γ = κ 4 ∈ ( 2 , 2 ) , α = 4 κ , e = α − 1 , Q L = γ 2 + 2 γ . Here Q L Q_L Q L h h h ν h \nu_h ν h μ h \mu_h μ h f : D → D ~ f:D\to\widetilde{D} f : D → D h ∘ f − 1 + Q L log ∣ ( f − 1 ) ′ ∣ h\circ f^{-1}+Q_L\log|(f^{-1})'| h ∘ f − 1 + Q L log ∣ ( f − 1 ) ′ ∣ Q D i , j \mathrm{QD}_{i,j} QD i , j i ≥ 0 i\geq0 i ≥ 0 j ≥ 0 j\geq0 j ≥ 0 i , j i,j i , j μ h \mu_h μ h ν h \nu_h ν h Q D 1 , 0 ( L ) \mathrm{QD}_{1,0}(L) QD 1 , 0 ( L ) Q D 1 , 2 ( a , b ) \mathrm{QD}_{1,2}(a,b) QD 1 , 2 ( a , b ) Q D 1 , 0 \mathrm{QD}_{1,0} QD 1 , 0 Q D 1 , 2 \mathrm{QD}_{1,2} QD 1 , 2 L > 0 L>0 L > 0 a , b > 0 a,b>0 a , b > 0 d L dL d L d a d b da\,db d a d b Q D i , j ⊗ C L E κ \mathrm{QD}_{i,j}\otimes\mathrm{CLE}_\kappa QD i , j ⊗ CLE κ M M M ∣ M ∣ |M| ∣ M ∣ [14 ] , we have ∣ Q D 1 , 0 ( L ) ∣ ∝ L − α |\mathrm{QD}_{1,0}(L)|\propto L^{-\alpha} ∣ QD 1 , 0 ( L ) ∣ ∝ L − α
∣ Q D 1 , 2 ( a , b ) ∣ = ( a + b ) ∣ Q D 1 , 0 ( a + b ) ∣ ∝ ( a + b ) 1 − α . |\mathrm{QD}_{1,2}(a,b)|=(a+b)|\mathrm{QD}_{1,0}(a+b)|\propto(a+b)^{1-\alpha}. ∣ QD 1 , 2 ( a , b ) ∣ = ( a + b ) ∣ QD 1 , 0 ( a + b ) ∣ ∝ ( a + b ) 1 − α . We use M 2 d i s k ( W ) \mathcal{M}^{\mathrm{disk}}_2(W) M 2 disk ( W ) W > 0 W>0 W > 0 a a a b b b s s s a + s a+s a + s a + b a+b a + b
Cutting a complete loop
Put W = γ 2 − 2 W=\gamma^2-2 W = γ 2 − 2 k = ( α − 1 ) − 2 k=(\alpha-1)^{-2} k = ( α − 1 ) − 2 0 < W < γ 2 / 2 0<W<\gamma^2/2 0 < W < γ 2 /2 γ 2 − W = 2 \gamma^2-W=2 γ 2 − W = 2 W W W η C \eta_C η C κ = 6 \kappa=6 κ = 6 C C C § 2 .
Lemma 3.1. Under Q D 1 , 0 ⊗ C L E κ \mathrm{QD}_{1,0}\otimes\mathrm{CLE}_{\kappa} QD 1 , 0 ⊗ CLE κ η C \eta_C η C a a a b b b s s s
d s d a d b Ξ s , b ( d E ) ( Q D 1 , 2 ( a , b ) ⊗ C L E κ ) , (18) ds\,da\,db\,\Xi_{s,b}(dE)\left(\mathrm{QD}_{1,2}(a,b)\otimes\mathrm{CLE}_{\kappa}\right), \tag*{(18)} d s d a d b Ξ s , b ( d E ) ( QD 1 , 2 ( a , b ) ⊗ CLE κ ) , (18) where Ξ s , b \Xi_{s,b} Ξ s , b σ \sigma σ a a a E E E
Proof. The loops touching the original boundary are outermost. Conditional on each generation, independent CLEs are sampled in the complementary domains. For such a loop η C \eta_C η C I C I_C I C A C = μ h ( I C ) A_C=\mu_h(I_C) A C = μ h ( I C ) Q D 1 , 0 \mathrm{QD}_{1,0} QD 1 , 0 w w w w w w B C L E κ ( 0 ) \mathrm{BCLE}_{\kappa}(0) BCLE κ ( 0 ) w w w
Q D 1 , 0 f ⊗ μ ↻ = c κ ∫ 0 ∞ W e l d ( Q A ~ f ( W ; ℓ ′ ) , ℓ ′ Q D 1 , 0 f ( ℓ ′ ) ) d ℓ ′ . (19) \mathrm{QD}^{f}_{1,0}\otimes\mu^{\circlearrowright}=c_{\kappa}\int_{0}^{\infty}\mathrm{Weld}\left(\widetilde{\mathrm{QA}}^{f}(W;\ell^{\prime}),\ell^{\prime}\mathrm{QD}^{f}_{1,0}(\ell^{\prime})\right)d\ell^{\prime}. \tag*{(19)} QD 1 , 0 f ⊗ μ ↻ = c κ ∫ 0 ∞ Weld ( QA f ( W ; ℓ ′ ) , ℓ ′ QD 1 , 0 f ( ℓ ′ ) ) d ℓ ′ . (19) Here μ ↻ \mu^{\circlearrowright} μ ↻ ℓ ′ \ell^{\prime} ℓ ′ c κ > 0 c_{\kappa}>0 c κ > 0 f f f Q A ~ f ( W ; ℓ ′ ) \widetilde{\mathrm{QA}}^{f}(W;\ell^{\prime}) QA f ( W ; ℓ ′ ) Q D 1 , 0 f ( ℓ ′ ) \mathrm{QD}^{f}_{1,0}(\ell^{\prime}) QD 1 , 0 f ( ℓ ′ )
The ordinary beads of the annulus are the exterior components with an inherited boundary arc. In the construction of [LSYZ26, Definition 5.3 and equation (5.2)], each bead has one arc on the annulus’s outer boundary and one on its inner boundary. The outer forest becomes the auxiliary forest of the reconstructed disk and is forgotten. The inner boundary is welded to the marked forested disk, and its image is η C \eta_C η C ( a , b ) (a,b) ( a , b ) U C U_C U C J C J_C J C a a a I C I_C I C b b b s s s
We now place a second area mark z z z T T T k d T / T k\,dT/T k d T / T T T T d u M 2 d i s k ( 2 ) du\,\mathcal{M}^{\mathrm{disk}}_2(2) d u M 2 disk ( 2 ) μ U \mu_U μ U U U U M 2 d i s k ( 2 ) ( d U ) μ U ( d z ) = Q D 1 , 2 ( d U , d z ) \mathcal{M}^{\mathrm{disk}}_2(2)(dU)\mu_U(dz)=\mathrm{QD}_{1,2}(dU,dz) M 2 disk ( 2 ) ( d U ) μ U ( d z ) = QD 1 , 2 ( d U , d z ) T T T k d T / T k\,dT/T k d T / T k d T k\,dT k d T d a d b Q D 1 , 2 ( a , b ) da\,db\,\mathrm{QD}_{1,2}(a,b) d a d b QD 1 , 2 ( a , b ) Figure 4 .
Figure 4. Radial welding and the retained component in Lemma 3.1. The colored annulus beads have weight 2; the gray lobes are attached forests. The inner forested disk carries w w w z z z A C − 1 A_C^{-1} A C − 1 w w w η C \eta_C η C A C A_C A C η C \eta_C η C J C J_C J C a a a I C I_C I C b b b p C , q C p_C,q_C p C , q C s s s
Conditional on a a a b b b s s s R ( a + b ) − 1 − α d a d b R(a+b)^{-1-\alpha}\,da\,db R ( a + b ) − 1 − α d a d b M 2 d i s k ( 2 ) \mathcal{M}^{\mathrm{disk}}_2(2) M 2 disk ( 2 ) R > 0 R>0 R > 0 Γ \Gamma Γ T T T S T S_T S T
E [ e − p S T ] = exp { − T R [ − Γ ( 1 − α ) ] α p α − 1 } , p > 0. \mathbb{E}\left[e^{-pS_T}\right]=\exp\left\{-T\frac{R[-\Gamma(1-\alpha)]}{\alpha}p^{\alpha-1}\right\},\qquad p>0. E [ e − p S T ] = exp { − T α R [ − Γ ( 1 − α )] p α − 1 } , p > 0. The variable S T S_T S T ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) S T = s S_T=s S T = s Ξ s , b \Xi_{s,b} Ξ s , b
The union I C \mathcal{I}_C I C A C A_C A C E E E η C \eta_C η C 0 < A C < ∞ 0<A_C<\infty 0 < A C < ∞
∫ I C A C − 1 μ h ( d w ) = 1 \int_{\mathcal{I}_C} A_C^{-1}\,\mu_h(dw)=1 ∫ I C A C − 1 μ h ( d w ) = 1 removes the mark w w w Ξ s , b \Xi_{s,b} Ξ s , b A C − 1 A_C^{-1} A C − 1 w w w Ξ s , b \Xi_{s,b} Ξ s , b
The domain Markov property gives independent CLEs in the exterior components, conditional on the complete loop and its complementary domains. Area marking preserves this conditional independence, since the marks depend only on the quantum surfaces and the selected loop. Sampling the CLE in the selected bead gives (18) . Tonelli’s theorem and monotone convergence extend the identity from finite restrictions to all nonnegative measurable test functions.
Computation of the boundary-length kernel
We next compute ∣ Ξ s , b ∣ |\Xi_{s,b}| ∣ Ξ s , b ∣
C α = − α π sin 2 π α > 0 (20) C_{\alpha}=-\frac{\alpha}{\pi}\sin\frac{2\pi}{\alpha}>0 \tag*{(20)} C α = − π α sin α 2 π > 0 (20) and define the positive boundary-length kernel
B α ( s , b ) = − 2 C α sin ( π α ) π ∫ 0 ∞ u α ( 1 + u ) ( u 2 α − 2 cos ( π α ) u α + 1 ) ( b + s u ) 3 d u . (21) B_{\alpha}(s,b)=-\frac{2C_{\alpha}\sin(\pi\alpha)}{\pi}\int_{0}^{\infty}\frac{u^{\alpha}(1+u)}{(u^{2\alpha}-2\cos(\pi\alpha)u^{\alpha}+1)(b+su)^3}\,du. \tag*{(21)} B α ( s , b ) = − π 2 C α sin ( π α ) ∫ 0 ∞ ( u 2 α − 2 cos ( π α ) u α + 1 ) ( b + s u ) 3 u α ( 1 + u ) d u . (21) Proposition 3.2 (Boundary-length kernel). Under Q D 1 , 0 ⊗ C L E κ \mathrm{QD}_{1,0}\otimes\mathrm{CLE}_{\kappa} QD 1 , 0 ⊗ CLE κ count each complete boundary-touching loop η C \eta_C η C once and retain the component containing the area mark when it is exterior to the loop and has a nondegenerate inherited boundary arc. The joint measure of ( s , a , b ) (s,a,b) ( s , a , b ) and the retained component with its field, marked points and CLE is
B α ( s , b ) d s d a d b ( Q D 1 , 2 ( a , b ) ⊗ C L E κ ) . (22) B_{\alpha}(s,b)\,ds\,da\,db\;(\mathrm{QD}_{1,2}(a,b)\otimes\mathrm{CLE}_{\kappa}). \tag*{(22)} B α ( s , b ) d s d a d b ( QD 1 , 2 ( a , b ) ⊗ CLE κ ) . (22) The boundary marks are the endpoints of the inherited boundary arc. The averaged cutting identity in Proposition 2.3 imposes maximality of the discarded boundary arc.
Proof. Write B ^ α ( s , b ) = ∣ Ξ s , b ∣ \widehat{B}_{\alpha}(s,b)=|\Xi_{s,b}| B α ( s , b ) = ∣ Ξ s , b ∣ [3 Definition 2.27] ), and let G Q D 1 , 0 \mathrm{GQD}_{1,0} GQD 1 , 0 [3 Theorem 2.22 and Proposition 2.34] , after forgetting the distinguished ordinary disk, Q D 1 , 0 f \mathrm{QD}^{f}_{1,0} QD 1 , 0 f G Q D 1 , 0 \mathrm{GQD}_{1,0} GQD 1 , 0 A C − 1 A_C^{-1} A C − 1 w w w λ κ > 0 \lambda_{\kappa}>0 λ κ > 0
Integration over the remaining quantum surfaces. The annulus construction uses the measure k d T / T k\,dT/T k d T / T k = ( α − 1 ) − 2 k=(\alpha-1)^{-2} k = ( α − 1 ) − 2 M 2 d i s k \mathcal{M}^{\mathrm{disk}}_{2} M 2 disk R ( a + b ) − 1 − α d a d b R(a+b)^{-1-\alpha}\,da\,db R ( a + b ) − 1 − α d a d b p , r > 0 p,r>0 p , r > 0
Ψ ( p , r ) = ∬ ( 0 , ∞ ) 2 ( 1 − e − p a − r b ) R ( a + b ) − 1 − α d a d b = R [ − Γ ( 1 − α ) ] α p α − r α p − r . \Psi(p,r)=\iint_{(0,\infty)^2}(1-e^{-pa-rb})R(a+b)^{-1-\alpha}\,da\,db
=\frac{R[-\Gamma(1-\alpha)]}{\alpha}\frac{p^{\alpha}-r^{\alpha}}{p-r}. Ψ ( p , r ) = ∬ ( 0 , ∞ ) 2 ( 1 − e − p a − r b ) R ( a + b ) − 1 − α d a d b = α R [ − Γ ( 1 − α )] p − r p α − r α . Consequently
Ψ ‾ ( p , r ) = E α p − r p α − r α , E α = α ( α − 1 ) R Γ ( 2 − α ) . (23) \overline{\Psi}(p,r)=E_{\alpha}\frac{p-r}{p^{\alpha}-r^{\alpha}},\qquad E_{\alpha}=\frac{\alpha}{(\alpha-1)R\Gamma(2-\alpha)}. \tag*{(23)} Ψ ( p , r ) = E α p α − r α p − r , E α = ( α − 1 ) R Γ ( 2 − α ) α . (23) The singularity at p = r p=r p = r
As in Lemma 3.1 , selecting the exterior bead by its area measure changes k d T / T k\,dT/T k d T / T k d T k\,dT k d T ( S T , V T ) (S_T,V_T) ( S T , V T )
E [ e − p S T − r V T ] = e − T Ψ ( p , r ) . \mathbb{E}\left[e^{-pS_T-rV_T}\right]=e^{-T\Psi(p,r)}. E [ e − p S T − r V T ] = e − T Ψ ( p , r ) . The selected bead contributes b b b Y b + V T Y_b+V_T Y b + V T ( Y v ) v ≥ 0 (Y_v)_{v\geq0} ( Y v ) v ≥ 0 1 / α 1/\alpha 1/ α E [ e − t Y v ] = e − v t 1 / α \mathbb{E}[e^{-tY_v}]=e^{-vt^{1/\alpha}} E [ e − t Y v ] = e − v t 1/ α t > 0 t>0 t > 0 ℓ > 0 \ell>0 ℓ > 0 ∣ GQD ( ℓ ) ∣ = c g ℓ − 2 − 1 / α |\operatorname{GQD}(\ell)|=c_g\ell^{-2-1/\alpha} ∣ GQD ( ℓ ) ∣ = c g ℓ − 2 − 1/ α c g > 0 c_g>0 c g > 0 [3 Lemma 2.28] . The factor ℓ \ell ℓ (19) therefore gives
ℓ ∣ GQD ( ℓ ) ∣ = c g Γ ( 1 + 1 / α ) ∫ 0 ∞ t 1 / α e − t ℓ d t . \ell|\operatorname{GQD}(\ell)|=\frac{c_g}{\Gamma(1+1/\alpha)}\int_0^\infty t^{1/\alpha}e^{-t\ell}\,\mathrm{d}t. ℓ ∣ GQD ( ℓ ) ∣ = Γ ( 1 + 1/ α ) c g ∫ 0 ∞ t 1/ α e − t ℓ d t . Tonelli’s theorem, integration over T T T t = r α t=r^\alpha t = r α
∫ 0 ∞ e − p s B ^ α ( s , b ) d s = λ κ K e x t ∫ 0 ∞ r α p − r p α − r α e − b r d r , K e x t = c g α A g Γ ( 1 + 1 / α ) . \int_0^\infty e^{-ps}\widehat{B}_\alpha(s,b)\,\mathrm{d}s=\lambda_\kappa K_{\mathrm{ext}}\int_0^\infty r^\alpha\frac{p-r}{p^\alpha-r^\alpha}e^{-br}\,\mathrm{d}r,\qquad K_{\mathrm{ext}}=\frac{c_g\alpha A_g}{\Gamma(1+1/\alpha)}. ∫ 0 ∞ e − p s B α ( s , b ) d s = λ κ K ext ∫ 0 ∞ r α p α − r α p − r e − b r d r , K ext = Γ ( 1 + 1/ α ) c g α A g . Thus B ^ α = λ κ K e x t R α \widehat{B}_\alpha=\lambda_\kappa K_{\mathrm{ext}}R_\alpha B α = λ κ K ext R α
R α ( s , b ) = L p → s − 1 [ ∫ 0 ∞ r α p − r p α − r α e − b r d r ] . (24) R_\alpha(s,b)=\mathcal{L}^{-1}_{p\to s}\left[\int_0^\infty r^\alpha\frac{p-r}{p^\alpha-r^\alpha}e^{-br}\,\mathrm{d}r\right]. \tag*{(24)} R α ( s , b ) = L p → s − 1 [ ∫ 0 ∞ r α p α − r α p − r e − b r d r ] . (24) Here L p → s − 1 \mathcal{L}^{-1}_{p\to s} L p → s − 1 p α p^\alpha p α
j α ( x , r ) = − 1 π Im − x − r x α e i π α − r α = − sin ( π α ) π ( x + r ) x α x 2 α − 2 cos ( π α ) x α r α + r 2 α , x , r > 0. j_\alpha(x,r)=-\frac{1}{\pi}\operatorname{Im}\frac{-x-r}{x^\alpha e^{i\pi\alpha}-r^\alpha}
=-\frac{\sin(\pi\alpha)}{\pi}\frac{(x+r)x^\alpha}{x^{2\alpha}-2\cos(\pi\alpha)x^\alpha r^\alpha+r^{2\alpha}},\qquad x,r>0. j α ( x , r ) = − π 1 Im x α e iπ α − r α − x − r = − π sin ( π α ) x 2 α − 2 cos ( π α ) x α r α + r 2 α ( x + r ) x α , x , r > 0. Cauchy’s formula gives the Stieltjes representation ( p − r ) / ( p α − r α ) = ∫ 0 ∞ j α ( x , r ) ( p + x ) − 1 d x (p-r)/(p^\alpha-r^\alpha)=\int_0^\infty j_\alpha(x,r)(p+x)^{-1}\,\mathrm{d}x ( p − r ) / ( p α − r α ) = ∫ 0 ∞ j α ( x , r ) ( p + x ) − 1 d x O ( ∣ p ∣ 1 − α ) O(|p|^{1-\alpha}) O ( ∣ p ∣ 1 − α ) ( p + x ) − 1 = ∫ 0 ∞ e − s ( p + x ) d s (p+x)^{-1}=\int_0^\infty e^{-s(p+x)}\,\mathrm{d}s ( p + x ) − 1 = ∫ 0 ∞ e − s ( p + x ) d s x = u r x=ur x = u r r 2 e − ( b + s u ) r r^2e^{-(b+su)r} r 2 e − ( b + s u ) r
R α ( s , b ) = − 2 sin ( π α ) π ∫ 0 ∞ u α ( 1 + u ) ( u 2 α − 2 cos ( π α ) u α + 1 ) ( b + s u ) 3 d u . R_\alpha(s,b)=-\frac{2\sin(\pi\alpha)}{\pi}\int_0^\infty\frac{u^\alpha(1+u)}{(u^{2\alpha}-2\cos(\pi\alpha)u^\alpha+1)(b+su)^3}\,\mathrm{d}u. R α ( s , b ) = − π 2 sin ( π α ) ∫ 0 ∞ ( u 2 α − 2 cos ( π α ) u α + 1 ) ( b + s u ) 3 u α ( 1 + u ) d u . This function is positive for s , b > 0 s,b>0 s , b > 0 s = 0 s=0 s = 0 p → ∞ p\to\infty p → ∞
Normalization. Let A o L 2 A_oL^2 A o L 2 L L L A g ℓ 2 / α A_g\ell^{2/\alpha} A g ℓ 2/ α ℓ \ell ℓ [3 Theorem 2.23 and Proposition 2.29] ,
A o = 1 4 ( α − 1 ) sin ( π / α ) , A g = α 4 sin ( π / α ) Γ ( 1 − 1 / α ) Γ ( 1 / α ) . A_o=\frac{1}{4(\alpha-1)\sin(\pi/\alpha)},\qquad A_g=\frac{\alpha}{4\sin(\pi/\alpha)}\frac{\Gamma(1-1/\alpha)}{\Gamma(1/\alpha)}. A o = 4 ( α − 1 ) sin ( π / α ) 1 , A g = 4 sin ( π / α ) α Γ ( 1/ α ) Γ ( 1 − 1/ α ) . Removing the area weight in [3 Theorem 2.23] gives the unmarked ordinary disk area an inverse-gamma distribution of shape α \alpha α
ℓ ∣ GQD 1 , 0 ( ℓ ) ∣ = c g A g ℓ − 1 + 1 / α = c g A g Γ ( 1 − 1 / α ) ∫ 0 ∞ t − 1 / α e − t ℓ d t . \ell|\operatorname{GQD}_{1,0}(\ell)|=c_gA_g\ell^{-1+1/\alpha}=\frac{c_gA_g}{\Gamma(1-1/\alpha)}\int_0^\infty t^{-1/\alpha}e^{-t\ell}\,\mathrm{d}t. ℓ ∣ GQD 1 , 0 ( ℓ ) ∣ = c g A g ℓ − 1 + 1/ α = Γ ( 1 − 1/ α ) c g A g ∫ 0 ∞ t − 1/ α e − t ℓ d t . Integrate L e − p L Le^{-pL} L e − p L L L L L L L
∫ 0 ∞ k d T T [ − ∂ p e − T Ψ ( p , r ) ] = k ∂ p log Ψ ( p , r ) . \int_0^\infty\frac{k\,\mathrm{d}T}{T}[-\partial_pe^{-T\Psi(p,r)}]=k\,\partial_p\log\Psi(p,r). ∫ 0 ∞ T k d T [ − ∂ p e − T Ψ ( p , r ) ] = k ∂ p log Ψ ( p , r ) . On the welding side of (19) , the integral of this test function is λ κ K i n p α − 2 \lambda_\kappa K_{\mathrm{in}}p^{\alpha-2} λ κ K in p α − 2
K i n = k c q A q α I α Γ ( 1 − 1 / α ) , I α = ∫ 0 ∞ u α − 2 ( α 1 − u α − 1 1 − u ) d u = − π ( cot π α + cot ( π α ) ) > 0. K_{\mathrm{in}}=\frac{kc_qA_q\alpha I_\alpha}{\Gamma(1-1/\alpha)},\qquad I_\alpha=\int_0^\infty u^{\alpha-2}\left(\frac{\alpha}{1-u^\alpha}-\frac{1}{1-u}\right)\,du=-\pi\left(\cot\frac{\pi}{\alpha}+\cot(\pi\alpha)\right)>0. K in = Γ ( 1 − 1/ α ) k c q A q α I α , I α = ∫ 0 ∞ u α − 2 ( 1 − u α α − 1 − u 1 ) d u = − π ( cot α π + cot ( π α ) ) > 0. The singularities at u = 1 u=1 u = 1 v = u α v=u^\alpha v = u α pv ∫ 0 ∞ v q − 1 ( 1 − v ) − 1 d v = π cot ( π q ) \operatorname{pv}\int_0^\infty v^{q-1}(1-v)^{-1}\,dv=\pi\cot(\pi q) pv ∫ 0 ∞ v q − 1 ( 1 − v ) − 1 d v = π cot ( π q ) 0 < q < 1 0<q<1 0 < q < 1 q = 1 − 1 / α q=1-1/\alpha q = 1 − 1/ α q = α − 1 q=\alpha-1 q = α − 1 I α I_\alpha I α
p t o u c h = 1 + sin ( π ( α + 2 / α ) ) sin ( π α ) . p_{\mathrm{touch}}=1+\frac{\sin(\pi(\alpha+2/\alpha))}{\sin(\pi\alpha)}. p touch = 1 + sin ( π α ) sin ( π ( α + 2/ α )) . by [8 Theorem 1.1] . This probability is independent of the marked point by conformal invariance. Since the ordinary disk perimeter measure is R L − 2 − α d L RL^{-2-\alpha}\,dL R L − 2 − α d L p t o u c h R A o Γ ( 2 − α ) p α − 2 p_{\mathrm{touch}}RA_o\Gamma(2-\alpha)p^{\alpha-2} p touch R A o Γ ( 2 − α ) p α − 2
λ κ K e x t = p t o u c h R A o Γ ( 2 − α ) K i n K e x t = α p t o u c h I α = − α π sin 2 π α = C α . \lambda_\kappa K_{\mathrm{ext}}=\frac{p_{\mathrm{touch}}RA_o\Gamma(2-\alpha)}{K_{\mathrm{in}}}K_{\mathrm{ext}}=\frac{\alpha p_{\mathrm{touch}}}{I_\alpha}=-\frac{\alpha}{\pi}\sin\frac{2\pi}{\alpha}=C_\alpha. λ κ K ext = K in p touch R A o Γ ( 2 − α ) K ext = I α α p touch = − π α sin α 2 π = C α . These integrals are finite and positive for 1 < α < 2 1<\alpha<2 1 < α < 2 B ^ α = B α \widehat{B}_\alpha=B_\alpha B α = B α Lemma 3.1 gives (22) . □
The boundary-length kernel B α B_\alpha B α
B α ( s , b ) = C α ∂ s ∂ b s α − 1 − b α − 1 s α − b α . (25) B_\alpha(s,b)=C_\alpha\partial_s\partial_b\frac{s^{\alpha-1}-b^{\alpha-1}}{s^\alpha-b^\alpha}. \tag*{(25)} B α ( s , b ) = C α ∂ s ∂ b s α − b α s α − 1 − b α − 1 . (25) It is symmetric, homogeneous of degree − 3 -3 − 3 B α ( s , s ) = C α ( α 2 − 1 ) / ( 6 α s 3 ) B_\alpha(s,s)=C_\alpha(\alpha^2-1)/(6\alpha s^3) B α ( s , s ) = C α ( α 2 − 1 ) / ( 6 α s 3 )
∫ 0 ∞ s z B α ( s , b ) d s = b z − 2 sin ( 2 π / α ) sin ( π / α ) z ( 1 − z ) sin ( π z / α ) sin ( π ( 1 − z ) / α ) , 1 − α < Re z < α . (26) \int_0^\infty s^zB_\alpha(s,b)\,ds=b^{z-2}\frac{\sin(2\pi/\alpha)\sin(\pi/\alpha)z(1-z)}{\sin(\pi z/\alpha)\sin(\pi(1-z)/\alpha)},\qquad1-\alpha<\operatorname{Re}z<\alpha. \tag*{(26)} ∫ 0 ∞ s z B α ( s , b ) d s = b z − 2 sin ( π z / α ) sin ( π ( 1 − z ) / α ) sin ( 2 π / α ) sin ( π / α ) z ( 1 − z ) , 1 − α < Re z < α . (26) To obtain this formula, integrate ( b + u s ) − 3 s z (b+us)^{-3}s^z ( b + u s ) − 3 s z s s s 1 2 b z − 2 u − z − 1 Γ ( 1 + z ) Γ ( 2 − z ) \frac{1}{2}b^{z-2}u^{-z-1}\Gamma(1+z)\Gamma(2-z) 2 1 b z − 2 u − z − 1 Γ ( 1 + z ) Γ ( 2 − z ) v = u α v=u^\alpha v = u α b > 0 b>0 b > 0 s α − 2 s^{\alpha-2} s α − 2 s − α − 1 s^{-\alpha-1} s − α − 1 1 − α < Re z < α 1-\alpha<\operatorname{Re}z<\alpha 1 − α < Re z < α
The renewal equation on the Brownian annulus
We derive an integro-differential equation from the Mellin transform of the Brownian annulus measure weighted by the averaged two-point observable G ‾ \overline{G} G § 7 , κ = 6 \kappa=6 κ = 6 γ = 8 / 3 \gamma=\sqrt{8/3} γ = 8/3 B = B 3 / 2 B=B_{3/2} B = B 3/2 B α B_\alpha B α (21) .
Let BA denote the Brownian annulus measure of [7 Definition 1.1 and equation (1.1)] (see also [10 , 19 ] for other equivalent constructions). Write L , m > 0 L,m>0 L , m > 0 x ∈ ( 0 , 1 ) x\in(0,1) x ∈ ( 0 , 1 ) z ( L , x ; m ) z(L,x;m) z ( L , x ; m ) d L d m dL\,dm d L d m G ‾ ( x ) \overline{G}(x) G ( x ) § 2 , as follows. With r = m / L r=m/L r = m / L
d L L d r 2 r ( 1 + r ) . \frac{dL}{L}\frac{dr}{2\sqrt{r(1+r)}}. L d L 2 r ( 1 + r ) d r . Adding a constant to the field rescales both lengths and leaves G ‾ ( x ) \overline{G}(x) G ( x ) Z ‾ ( r , x ) \overline{Z}(r,x) Z ( r , x ) G ‾ ( x ) \overline{G}(x) G ( x ) r r r Lemma 2.2 makes Z ‾ ( r , ⋅ ) \overline{Z}(r,\cdot) Z ( r , ⋅ ) x x x r r r x ∈ ( 0 , 1 ) x \in(0,1) x ∈ ( 0 , 1 ) r r r
z ( L , x ; m ) = L − 2 Z ‾ ( m / L , x ) 2 m / L ( 1 + m / L ) . z(L,x;m)=\frac{L^{-2}\overline{Z}(m/L,x)}{2\sqrt{m/L}(1+m/L)}. z ( L , x ; m ) = 2 m / L ( 1 + m / L ) L − 2 Z ( m / L , x ) . This density satisfies
z ( c L , x ; c m ) = c − 2 z ( L , x ; m ) , c > 0 , z(cL,x;cm)=c^{-2}z(L,x;m),\qquad c>0, z ( c L , x ; c m ) = c − 2 z ( L , x ; m ) , c > 0 , 0 ≤ z ( L , x ; m ) ≤ 1 2 L m ( L + m ) . (27) 0\le z(L,x;m)\le\frac{1}{2\sqrt{Lm}(L+m)}. \tag*{(27)} 0 ≤ z ( L , x ; m ) ≤ 2 L m ( L + m ) 1 . (27) For real σ \sigma σ
F ( σ ) = ∫ 0 ∞ m − σ z ( 1 , 1 / 2 ; m ) d m ∈ [ 0 , ∞ ] . (28) F(\sigma)=\int_{0}^{\infty}m^{-\sigma}z(1,1/2;m)\,dm\in[0,\infty]. \tag*{(28)} F ( σ ) = ∫ 0 ∞ m − σ z ( 1 , 1/2 ; m ) d m ∈ [ 0 , ∞ ] . (28) Recall x ( t ) = e t / ( 1 + e t ) x(t)=e^t/(1+e^t) x ( t ) = e t / ( 1 + e t ) a ( t ) = 1 − x ( t ) a(t)=1-x(t) a ( t ) = 1 − x ( t ) L 0 , T s L_0,T_s L 0 , T s (2) –(3) . For real s ∈ ( − 2 , 2 ) s\in(-2,2) s ∈ ( − 2 , 2 ) f f f (3) also defines T s f T_sf T s f [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ]
Proposition 4.1 (Integro-differential equation). For σ ∈ ( − 2 , 2 ) \sigma\in(-2,2) σ ∈ ( − 2 , 2 ) F ( σ ) < ∞ F(\sigma)<\infty F ( σ ) < ∞
φ − σ ( x ) = ∫ 0 ∞ m − σ z ( 1 , x ; m ) d m , u σ ( t ) = φ − σ ( x ( t ) ) x ( t ) a ( t ) . (29) \varphi_{-\sigma}(x)=\int_{0}^{\infty}m^{-\sigma}z(1,x;m)\,dm,\qquad u_\sigma(t)=\frac{\varphi_{-\sigma}(x(t))}{\sqrt{x(t)a(t)}}. \tag*{(29)} φ − σ ( x ) = ∫ 0 ∞ m − σ z ( 1 , x ; m ) d m , u σ ( t ) = x ( t ) a ( t ) φ − σ ( x ( t )) . (29) Then u σ u_\sigma u σ F ( σ ) = u σ ( 0 ) / 2 F(\sigma)=u_\sigma(0)/2 F ( σ ) = u σ ( 0 ) /2 j σ j_\sigma j σ R \mathbb{R} R
( L 0 − T − σ ) u σ = j σ . (30) (L_0-T_{-\sigma})u_\sigma=j_\sigma. \tag*{(30)} ( L 0 − T − σ ) u σ = j σ . (30) The measure j σ j_\sigma j σ (35) .
Preservation of the filled metric ball
Following [SXZ24, Section 5], we use the Brownian annulus to transfer the measure decomposition of Proposition 3.2 to the annulus setup. The construction of the Brownian annulus removes a filled metric ball and reweights by a function of its perimeter. The same weight applies before and after cutting a loop disjoint from the ball; see Figure 5 .
Figure 5. Cutting a complete boundary-touching loop outside the filled ball. The planar image A = Π C ( X C ) A=\Pi_C(X_C) A = Π C ( X C ) K D K_D K D a a a b b b s s s ∂ A \partial A ∂ A
Sample ( D , h , z ) (D,h,z) ( D , h , z ) Q D 1 , 0 \mathrm{QD}_{1,0} QD 1 , 0 d h d_h d h
E D = { d h ( z , ∂ D ) > 1 } . E_D=\{d_h(z,\partial D)>1\}. E D = { d h ( z , ∂ D ) > 1 } . On this event let K D K_D K D z z z ∂ D \partial D ∂ D m D m_D m D U U U z z z E U E_U E U K U K_U K U m U m_U m U U U U
Lemma 4.2. On E D E_D E D C C C η C \eta_C η C Π C ( X C ) ∩ K D = ∅ \Pi_C(X_C) \cap K_D = \varnothing Π C ( X C ) ∩ K D = ∅ U U U D ∖ Π C ( X C ) D \setminus\Pi_C(X_C) D ∖ Π C ( X C ) z z z E U E_U E U
K U = K D , m U = m D . K_U=K_D,\qquad m_U=m_D. K U = K D , m U = m D . Conversely, in the welding of Proposition 3.2 , E U E_U E U E D E_D E D Π C ( X C ) ∩ K D = ∅ \Pi_C(X_C) \cap K_D = \varnothing Π C ( X C ) ∩ K D = ∅
Proof. Write A = Π C ( X C ) A=\Pi_C(X_C) A = Π C ( X C ) § 2.1 , it is compact, connected and meets ∂ D \partial D ∂ D [16 Section 1.2 and Remark 1.5] , the LQG metric of h ∣ U h|_U h ∣ U d h ( ⋅ , ⋅ ; U ) d_h(\cdot,\cdot;U) d h ( ⋅ , ⋅ ; U )
d h ∣ U ( z , ∂ U ) = d h ( z , A ∪ ∂ D ) = : δ , d_{h|_U}(z,\partial U)=d_h(z,A\cup\partial D)=:\delta, d h ∣ U ( z , ∂ U ) = d h ( z , A ∪ ∂ D ) =: δ , and the unfilled balls in D D D U U U δ \delta δ A ∪ ∂ D A\cup\partial D A ∪ ∂ D U U U U U U D D D A ∪ ∂ D A\cup\partial D A ∪ ∂ D U U U U U U
If E D E_D E D A ∩ K D = ∅ A\cap K_D=\varnothing A ∩ K D = ∅ δ > 1 \delta>1 δ > 1 E U E_U E U E U E_U E U E D E_D E D A A A A A A U U U ∂ D \partial D ∂ D U U U A A A ∂ U \partial U ∂ U K U = K D K_U=K_D K U = K D A ∩ K D = ∅ A\cap K_D=\varnothing A ∩ K D = ∅
Finally choose 1 < ρ < δ 1<\rho<\delta 1 < ρ < δ ρ \rho ρ m U m_U m U m D m_D m D
By [7 Proposition 6.10] , there are a positive constant c 2 c_2 c 2 f : ( 0 , ∞ ) → ( 0 , ∞ ) f:(0,\infty)\to(0,\infty) f : ( 0 , ∞ ) → ( 0 , ∞ ) K D K_D K D
1 E D c 2 m D f ( m D ) Q D 1 , 0 (31) \frac{1_{E_D}}{c_2m_Df(m_D)}\mathbb{Q}_D^{1,0} \tag*{(31)} c 2 m D f ( m D ) 1 E D Q D 1 , 0 (31) gives the Brownian annulus measure B A \mathrm{BA} BA Q D 1 , 0 \mathbb{Q}_D^{1,0} Q D 1 , 0 m D m_D m D Lemma 4.2 allows us to use the same weight on both sides of the measure decomposition in Proposition 3.2 .
Condition on the field and use the inner-boundary approximation of § 2.1 . Removing K D K_D K D K D K_D K D K D K_D K D D ∖ K D D\setminus K_D D ∖ K D (10) . Conformal invariance at each approximation, followed by its limit, gives the usual percolation law on the annulus, conditionally independent of the field given its conformal modulus. Let C τ = [ 0 , τ ] × ( R / Z ) C_\tau=[0,\tau]\times(\mathbb{R}/\mathbb{Z}) C τ = [ 0 , τ ] × ( R / Z ) τ > 0 \tau>0 τ > 0 L F τ \mathrm{LF}_\tau LF τ η \eta η
η ( z ) = e π i z / 12 ∏ n = 1 ∞ ( 1 − e 2 π i n z ) , Im z > 0. \eta(z)=e^{\pi i z/12}\prod_{n=1}^{\infty}(1-e^{2\pi i n z}),\qquad\operatorname{Im}z>0. η ( z ) = e π i z /12 n = 1 ∏ ∞ ( 1 − e 2 π in z ) , Im z > 0. The Brownian annulus identity [7 Theorem 1.3] is
B A = 1 2 η ( 2 i τ ) L F τ ( d h ) 1 { τ > 0 } d τ . (32) \mathrm{BA}=\frac{1}{\sqrt{2}}\eta(2i\tau)\mathrm{LF}_\tau(dh)\,1_{\{\tau>0\}}\,d\tau. \tag*{(32)} BA = 2 1 η ( 2 i τ ) LF τ ( d h ) 1 { τ > 0 } d τ . (32) The renewal equation
We apply Proposition 2.3 to the Brownian annulus. Averaging the original marks gives the piecewise linear weights w ℓ w_\ell w ℓ ℓ ∈ ( 0 , 1 ) \ell\in(0,1) ℓ ∈ ( 0 , 1 ) Lemma 2.2 ; integrating the first boundary mark of the retained component contributes its perimeter.
For a nonnegative measurable symmetric function f f f ( 0 , 1 ) (0,1) ( 0 , 1 )
z f ( L ; m ) = ∫ 0 1 f ( x ) z ( L , x ; m ) d x , Ψ f ( ℓ ) = ∫ 0 1 f ( x ) min { x , 1 − x , ℓ , 1 − ℓ } d x . z_f(L;m)=\int_0^1 f(x)z(L,x;m)\,dx,\qquad\Psi_f(\ell)=\int_0^1 f(x)\min\{x,1-x,\ell,1-\ell\}\,dx. z f ( L ; m ) = ∫ 0 1 f ( x ) z ( L , x ; m ) d x , Ψ f ( ℓ ) = ∫ 0 1 f ( x ) min { x , 1 − x , ℓ , 1 − ℓ } d x . Let t f ( L ; m ) t_f(L;m) t f ( L ; m ) d L d m dL\,dm d L d m (15) against B A \mathrm{BA} BA z f z_f z f
Proposition 4.3 (Renewal equation). As nonnegative measures in ( L , m ) (L,m) ( L , m )
z f ( L ; m ) = t f ( L ; m ) + ∫ 0 L d s ∫ 0 ∞ d b ( L − s + b ) B ( s , b ) Ψ f ( s / L ) z ( L − s + b , L − s L − s + b ; m ) . (33) z_f(L;m)=t_f(L;m)+\int_0^L ds\int_0^\infty db\,(L-s+b)B(s,b)\Psi_f(s/L)z\left(L-s+b,\frac{L-s}{L-s+b};m\right). \tag*{(33)} z f ( L ; m ) = t f ( L ; m ) + ∫ 0 L d s ∫ 0 ∞ d b ( L − s + b ) B ( s , b ) Ψ f ( s / L ) z ( L − s + b , L − s + b L − s ; m ) . (33) For the densities defined above, the identity holds almost everywhere in ( L , m ) (L,m) ( L , m ) [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) m m m L = 1 L=1 L = 1
Proof. Apply the averaged cutting identity of Proposition 2.3 with K = K D K=K_D K = K D μ = ν h \mu=\nu_h μ = ν h t f t_f t f Ψ f ( s C / L ) \Psi_f(s_C/L) Ψ f ( s C / L ) s C = ν h ( S C ) s_C=\nu_h(S_C) s C = ν h ( S C ) G G G η C \eta_C η C
The retained component is marked at the endpoints p C , q C p_C,q_C p C , q C Ψ f ( s C / L ) \Psi_f(s_C/L) Ψ f ( s C / L )
Integrate the loop sum using the measure decomposition in Proposition 3.2 . By Lemma 4.2 , the original and retained components have the same inner perimeter m m m (31) is unchanged. Write a a a b b b L ′ = a + b L'=a+b L ′ = a + b L ′ L' L ′ a a a ( a , b ) (a,b) ( a , b ) L ′ z ( L ′ , a / L ′ ; m ) L'z(L',a/L';m) L ′ z ( L ′ , a / L ′ ; m ) a = L − s a=L-s a = L − s L ′ = L − s + b L'=L-s+b L ′ = L − s + b (33) for integrable f f f f f f f ∧ n f\wedge n f ∧ n n → ∞ n\to\infty n → ∞
The Mellin transform
We prove Proposition 4.1 from (33) by taking the Mellin transform and differentiating twice with respect to x x x
Proof of Proposition 4.1. Symmetry and concavity give Z ( r , x ) ≤ Z ( r , 1 / 2 ) Z(r,x)\le Z(r,1/2) Z ( r , x ) ≤ Z ( r , 1/2 ) F ( σ ) F(\sigma) F ( σ ) φ − σ \varphi_{-\sigma} φ − σ 1 / 2 1/2 1/2 u σ u_\sigma u σ 2 F ( σ ) 2F(\sigma) 2 F ( σ )
∫ 0 ∞ m − σ z ( L , x ; m ) d m = L − 1 − σ φ − σ ( x ) . \int_0^\infty m^{-\sigma}z(L,x;m)\,dm=L^{-1-\sigma}\varphi_{-\sigma}(x). ∫ 0 ∞ m − σ z ( L , x ; m ) d m = L − 1 − σ φ − σ ( x ) . Let ϑ σ \vartheta_\sigma ϑ σ
∫ 0 1 f ( x ) ϑ σ ( x ) d x = ∫ 0 ∞ m − σ t f ( 1 ; m ) d m . \int_0^1 f(x)\vartheta_\sigma(x)\,dx=\int_0^\infty m^{-\sigma}t_f(1;m)\,dm. ∫ 0 1 f ( x ) ϑ σ ( x ) d x = ∫ 0 ∞ m − σ t f ( 1 ; m ) d m . As a nonnegative integral of the symmetric concave weights w ℓ w_\ell w ℓ ϑ σ \vartheta_\sigma ϑ σ 0 ≤ ϑ σ ≤ φ − σ 0\le\vartheta_\sigma\le\varphi_{-\sigma} 0 ≤ ϑ σ ≤ φ − σ − ϑ σ ′ ′ -\vartheta_\sigma'' − ϑ σ ′′ ( 0 , 1 ) (0,1) ( 0 , 1 )
Apply the Mellin transform to (33) , set L = 1 L=1 L = 1 w ℓ w_\ell w ℓ
− φ − σ ′ ′ ( x ) = − ϑ σ ′ ′ ( x ) + ∫ 0 ∞ ( 1 − x + b ) − σ B ( x , b ) φ − σ ( 1 − x 1 − x + b ) d b + ∫ 0 ∞ ( x + b ) − σ B ( 1 − x , b ) φ − σ ( x x + b ) d b . -\varphi_{-\sigma}''(x)=-\vartheta_\sigma''(x)
+\int_0^\infty(1-x+b)^{-\sigma}B(x,b)\varphi_{-\sigma}\left(\frac{1-x}{1-x+b}\right)\,db
+\int_0^\infty(x+b)^{-\sigma}B(1-x,b)\varphi_{-\sigma}\left(\frac{x}{x+b}\right)\,db. − φ − σ ′′ ( x ) = − ϑ σ ′′ ( x ) + ∫ 0 ∞ ( 1 − x + b ) − σ B ( x , b ) φ − σ ( 1 − x + b 1 − x ) d b + ∫ 0 ∞ ( x + b ) − σ B ( 1 − x , b ) φ − σ ( x + b x ) d b . Truncate the arc sums, differentiate the weights w ℓ w_\ell w ℓ φ − σ \varphi_{-\sigma} φ − σ ( 1 − x + b ) − σ (1-x+b)^{-\sigma} ( 1 − x + b ) − σ L ′ L' L ′ (33) by the factor L ′ − 1 − σ L'^{-1-\sigma} L ′ − 1 − σ
Use B ( s , b ) = ( s b ) − 3 / 2 H ( log ( s / b ) ) B(s,b)=(sb)^{-3/2}H(\log(s/b)) B ( s , b ) = ( s b ) − 3/2 H ( log ( s / b )) H H H (2) . The substitutions b = ( 1 − x ) ( 1 − y ) / y b=(1-x)(1-y)/y b = ( 1 − x ) ( 1 − y ) / y b = x ( 1 − y ) / y b=x(1-y)/y b = x ( 1 − y ) / y φ − σ \varphi_{-\sigma} φ − σ T − σ u σ T_{-\sigma}u_\sigma T − σ u σ φ \varphi φ
L 0 [ φ ( x ( t ) ) x ( t ) a ( t ) ] = − [ x ( t ) a ( t ) ] 3 / 2 φ ′ ′ ( x ( t ) ) . L_0\left[\frac{\varphi(x(t))}{\sqrt{x(t)a(t)}}\right]=-[x(t)a(t)]^{3/2}\varphi''(x(t)). L 0 [ x ( t ) a ( t ) φ ( x ( t )) ] = − [ x ( t ) a ( t ) ] 3/2 φ ′′ ( x ( t )) . Define j σ j_\sigma j σ χ \chi χ
∫ R χ ( t ) j σ ( d t ) = ∫ 0 1 x ( 1 − x ) χ ( log x 1 − x ) ( − ν σ ′ ′ ) ( d x ) . \int_{\mathbb{R}} \chi(t)\,j_\sigma(dt)=\int_0^1 \sqrt{x(1-x)}\,\chi\left(\log\frac{x}{1-x}\right)(-\nu_\sigma'')(dx). ∫ R χ ( t ) j σ ( d t ) = ∫ 0 1 x ( 1 − x ) χ ( log 1 − x x ) ( − ν σ ′′ ) ( d x ) . It is nonnegative, even and locally finite. This gives (30) . In particular, an arc of normalized quantum boundary length ℓ \ell ℓ ℓ ( 1 − ℓ ) \sqrt{\ell(1-\ell)} ℓ ( 1 − ℓ ) t = ± log ( ℓ / ( 1 − ℓ ) ) t=\pm\log(\ell/(1-\ell)) t = ± log ( ℓ / ( 1 − ℓ )) ℓ = 1 / 2 \ell=1/2 ℓ = 1/2
Estimates for the Mellin transform
We prove boundedness of u σ u_\sigma u σ j σ j_\sigma j σ (34) below with the unmarked three-arm probability. The proofs use GMC moment estimates and the boundary one-arm and polychromatic four-arm exponents. Throughout, κ = 6 \kappa=6 κ = 6 γ 2 = 8 / 3 \gamma^2=8/3 γ 2 = 8/3
Consider the cylinder C τ = [ 0 , τ ] × ( R / Z ) \mathcal{C}_\tau=[0,\tau]\times(\mathbb{R}/\mathbb{Z}) C τ = [ 0 , τ ] × ( R / Z ) ∂ o C τ = { 0 } × ( R / Z ) \partial_o\mathcal{C}_\tau=\{0\}\times(\mathbb{R}/\mathbb{Z}) ∂ o C τ = { 0 } × ( R / Z ) ∂ i C τ = { τ } × ( R / Z ) \partial_i\mathcal{C}_\tau=\{\tau\}\times(\mathbb{R}/\mathbb{Z}) ∂ i C τ = { τ } × ( R / Z ) § 2 conformally to C τ \mathcal{C}_\tau C τ p 3 ( τ ) p_3(\tau) p 3 ( τ ) h l a t h^{\mathrm{lat}} h lat { t } × ( R / Z ) \{t\}\times(\mathbb{R}/\mathbb{Z}) { t } × ( R / Z ) M o , M i M_o,M_i M o , M i Z o = M o ( ∂ o C τ ) Z_o=M_o(\partial_o\mathcal{C}_\tau) Z o = M o ( ∂ o C τ ) Z i = M i ( ∂ i C τ ) Z_i=M_i(\partial_i\mathcal{C}_\tau) Z i = M i ( ∂ i C τ ) 0 < x < 1 0<x<1 0 < x < 1 M o / Z o M_o/Z_o M o / Z o x Z o xZ_o x Z o G G G G ‾ τ ( x ) \overline{G}_\tau(x) G τ ( x ) σ \sigma σ
W σ ( τ , x ) = E [ ( Z o / Z i ) σ G ‾ τ ( x ) ] . (34) W_\sigma(\tau,x)=\mathbb{E}\left[(Z_o/Z_i)^\sigma\overline{G}_\tau(x)\right]. \tag*{(34)} W σ ( τ , x ) = E [ ( Z o / Z i ) σ G τ ( x ) ] . (34) The expectation is over the field and independent percolation, with τ \tau τ
Let M i \mathcal{M}_i M i C ∈ M C\in\mathcal{M} C ∈ M Π C ( X C ) \Pi_C(X_C) Π C ( X C ) ∂ i C τ \partial_i\mathcal{C}_\tau ∂ i C τ M i = ∅ \mathcal{M}_i=\varnothing M i = ∅ H 3 H_3 H 3 C ∈ M i C\in\mathcal{M}_i C ∈ M i
ℓ C = M o ( S C ) Z o ∈ ( 0 , 1 ) , t C = log ℓ C 1 − ℓ C . \ell_C=\frac{M_o(S_C)}{Z_o}\in(0,1),\qquad t_C=\log\frac{\ell_C}{1-\ell_C}. ℓ C = Z o M o ( S C ) ∈ ( 0 , 1 ) , t C = log 1 − ℓ C ℓ C . For every real σ \sigma σ χ \chi χ R \mathbb{R} R
∫ R χ ( t ) j σ ( d t ) = 2 γ ∫ 0 ∞ η ( 2 i τ ) e 2 π σ 2 τ / 3 × E [ ( Z o / Z i ) σ 1 H 3 ∑ C ∈ M i ℓ C ( 1 − ℓ C ) ( χ ( t C ) + χ ( − t C ) ) ] d τ . (35) \int_{\mathbb{R}}\chi(t)\,j_\sigma(dt)
=\frac{\sqrt{2}}{\gamma}\int_0^\infty\eta(2i\tau)e^{2\pi\sigma^2\tau/3}
\times\mathbb{E}\left[(Z_o/Z_i)^\sigma\mathbf{1}_{H_3}\sum_{C\in\mathcal{M}_i}\sqrt{\ell_C(1-\ell_C)}\bigl(\chi(t_C)+\chi(-t_C)\bigr)\right]d\tau. \tag*{(35)} ∫ R χ ( t ) j σ ( d t ) = γ 2 ∫ 0 ∞ η ( 2 i τ ) e 2 π σ 2 τ /3 × E [ ( Z o / Z i ) σ 1 H 3 C ∈ M i ∑ ℓ C ( 1 − ℓ C ) ( χ ( t C ) + χ ( − t C ) ) ] d τ . (35) This defines a nonnegative even measure with values in [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ] (40) , and then using (32) , shows that it agrees with the measure in Proposition 4.1 for σ ∈ ( − 2 , 2 ) \sigma\in(-2,2) σ ∈ ( − 2 , 2 ) F ( σ ) < ∞ F(\sigma)<\infty F ( σ ) < ∞ g ( t ) = e − ∣ t ∣ / 2 g(t)=e^{-|t|/2} g ( t ) = e − ∣ t ∣/2 L 0 L_0 L 0
X ( σ ) = σ 2 − 1 / 4 3 . (36) X(\sigma)=\frac{\sigma^2-1/4}{3}. \tag*{(36)} X ( σ ) = 3 σ 2 − 1/4 . (36) Proposition 5.1. For every compact interval J ⊂ ( 3 / 2 , 2 ) J\subset(3/2,2) J ⊂ ( 3/2 , 2 ) C , c , K > 0 C,c,K>0 C , c , K > 0 τ 0 > c \tau_0>c τ 0 > c σ ∈ J \sigma\in J σ ∈ J 0 < x < 1 0<x<1 0 < x < 1
W σ ( τ , x ) ≤ C x ( 1 − x ) p 3 ( τ − c ) , τ ≥ τ 0 , (37) W_\sigma(\tau,x)\le C\sqrt{x(1-x)}p_3(\tau-c),\qquad\tau\ge\tau_0, \tag*{(37)} W σ ( τ , x ) ≤ C x ( 1 − x ) p 3 ( τ − c ) , τ ≥ τ 0 , (37) W σ ( τ , x ) ≤ C τ − K x ( 1 − x ) , 0 < τ ≤ τ 0 . W_{\sigma}(\tau,x) \le C\tau^{-K}\sqrt{x(1-x)}, \qquad0<\tau\le\tau_0. W σ ( τ , x ) ≤ C τ − K x ( 1 − x ) , 0 < τ ≤ τ 0 . Consequently, if σ ∈ J \sigma\in J σ ∈ J X ( σ ) < ξ 3 X(\sigma)<\xi_3 X ( σ ) < ξ 3 φ − σ ( x ) / x ( 1 − x ) \varphi_{-\sigma}(x)/\sqrt{x(1-x)} φ − σ ( x ) / x ( 1 − x ) ( 0 , 1 ) (0,1) ( 0 , 1 ) u σ u_\sigma u σ R \mathbb{R} R
Proposition 5.2. For every compact interval J ⊂ ( 3 / 2 , 2 ) J\subset(3/2,2) J ⊂ ( 3/2 , 2 ) a > 0 a>0 a > 0 σ ↦ j σ \sigma\mapsto j_\sigma σ ↦ j σ J J J j j j R \mathbb{R} R ∫ e a ∣ t ∣ ∣ j ∣ ( d t ) < ∞ \int e^{a|t|}|j|(dt)<\infty ∫ e a ∣ t ∣ ∣ j ∣ ( d t ) < ∞ ∣ j ∣ |j| ∣ j ∣ σ ∈ J \sigma\in J σ ∈ J g ∗ j σ g*j_\sigma g ∗ j σ
Proposition 5.3. For every compact interval J ⊂ ( 3 / 2 , 2 ) J\subset(3/2,2) J ⊂ ( 3/2 , 2 ) c J , C J > 0 c_J,C_J>0 c J , C J > 0 c , c ′ > 0 c,c'>0 c , c ′ > 0 τ \tau τ σ ∈ J \sigma\in J σ ∈ J
c J p 3 ( τ − c ) ≤ W σ ( τ , 1 / 2 ) ≤ C J p 3 ( τ − c ′ ) . (38) c_Jp_3(\tau-c)\le W_\sigma(\tau,1/2)\le C_Jp_3(\tau-c'). \tag*{(38)} c J p 3 ( τ − c ) ≤ W σ ( τ , 1/2 ) ≤ C J p 3 ( τ − c ′ ) . (38) The right endpoint of the interval of convergence of the Mellin transform F F F
σ c = 1 / 4 + 3 ξ 3 ∈ ( 3 / 2 , 2 ) . (39) \sigma_c=\sqrt{1/4+3\xi_3}\in(3/2,2). \tag*{(39)} σ c = 1/4 + 3 ξ 3 ∈ ( 3/2 , 2 ) . (39) We prove these propositions in § 6.2 . Below, we first provide the needed technical input from GMC and percolation estimates.
GMC estimates
Integration of the additive constant. In the representation of the Brownian annulus (32) , decompose the field as h = B + h l a t + c h=B+h^{\mathrm{lat}}+c h = B + h lat + c B t B_t B t { t } × ( R / Z ) \{t\}\times(\mathbb{R}/\mathbb{Z}) { t } × ( R / Z ) t ∈ [ 0 , τ ] t\in[0,\tau] t ∈ [ 0 , τ ] B 0 = 0 B_0=0 B 0 = 0 Var ( B t ) = 2 π t \operatorname{Var}(B_t)=2\pi t Var ( B t ) = 2 π t h l a t h^{\mathrm{lat}} h lat c c c
L = e γ c / 2 Z o , m = e γ c / 2 + γ B τ / 2 Z i . L=e^{\gamma c/2}Z_o,\qquad m=e^{\gamma c/2+\gamma B_\tau/2}Z_i. L = e γ c /2 Z o , m = e γ c /2 + γ B τ /2 Z i . For a nonnegative measurable function A A A L F τ [ A ; L , m ] L_{F_\tau}[A;L,m] L F τ [ A ; L , m ] A A A L F τ L_{F_\tau} L F τ e γ c / 2 = L / Z o e^{\gamma c/2}=L/Z_o e γ c /2 = L / Z o
∫ 0 ∞ m − σ L F τ [ A ; L , m ] d m = 2 γ L − σ − 1 e 2 π σ 2 τ / 3 E [ ( Z o / Z i ) σ A ] . (40) \int_0^\infty m^{-\sigma}L_{F_\tau}[A;L,m]\,dm=\frac{2}{\gamma}L^{-\sigma-1}e^{2\pi\sigma^2\tau/3}\mathbb{E}\left[(Z_o/Z_i)^\sigma A\right]. \tag*{(40)} ∫ 0 ∞ m − σ L F τ [ A ; L , m ] d m = γ 2 L − σ − 1 e 2 π σ 2 τ /3 E [ ( Z o / Z i ) σ A ] . (40) The Gaussian expectation is E [ e − γ σ B τ / 2 ] = e 2 π σ 2 τ / 3 \mathbb{E}[e^{-\gamma\sigma B_\tau/2}]=e^{2\pi\sigma^2\tau/3} E [ e − γ σ B τ /2 ] = e 2 π σ 2 τ /3 η ( 2 i τ ) ≍ e − π τ / 6 \eta(2i\tau)\asymp e^{-\pi\tau/6} η ( 2 i τ ) ≍ e − π τ /6 τ → ∞ \tau\to\infty τ → ∞ X ( σ ) X(\sigma) X ( σ ) (36) .
Mixed GMC boundary-length moments. The next lemma estimates the piecewise linear weight w ℓ ( x ) = min { x , 1 − x , ℓ , 1 − ℓ } w_\ell(x)=\min\{x,1-x,\ell,1-\ell\} w ℓ ( x ) = min { x , 1 − x , ℓ , 1 − ℓ } e 2 n / 3 e^{2n/3} e 2 n /3 γ / 2 \gamma/2 γ /2 4 / 3 4/3 4/3
ζ ( r ) = 5 r − 2 r 2 3 . \zeta(r)=\frac{5r-2r^2}{3}. ζ ( r ) = 3 5 r − 2 r 2 . The constants below are uniform for a uniformly bounded regular part of the covariance.
Lemma 5.4. Let M M M I I I I c I^{c} I c 0 < p , q < 3 / 2 0<p,q<3/2 0 < p , q < 3/2 p + q < 5 / 2 p+q<5/2 p + q < 5/2
sup I E [ M ( I ) p M ( I c ) q ] < ∞ , (41) \sup_{I}\mathbb{E}[M(I)^{p}M(I^{c})^{q}]<\infty, \tag*{(41)} I sup E [ M ( I ) p M ( I c ) q ] < ∞ , (41) where the supremum is over all arcs. If 3 / 2 < σ < 2 3/2<\sigma<2 3/2 < σ < 2 0 < x ≤ 1 / 2 0<x\leq1/2 0 < x ≤ 1/2 I n I_n I n e − n e^{-n} e − n
∑ n ≥ 0 e 2 n / 3 E [ ( M ( I n ) + M ( I n c ) ) σ min { x , M ( I n ) M ( I n ) + M ( I n c ) , M ( I n c ) M ( I n ) + M ( I n c ) } ] ≤ C σ x . (42) \sum_{n\geq0}e^{2n/3}\mathbb{E}\left[(M(I_n)+M(I_n^{c}))^{\sigma}\min\left\{x,\frac{M(I_n)}{M(I_n)+M(I_n^{c})},\frac{M(I_n^{c})}{M(I_n)+M(I_n^{c})}\right\}\right]\leq C_{\sigma}\sqrt{x}. \tag*{(42)} n ≥ 0 ∑ e 2 n /3 E [ ( M ( I n ) + M ( I n c ) ) σ min { x , M ( I n ) + M ( I n c ) M ( I n ) , M ( I n ) + M ( I n c ) M ( I n c ) } ] ≤ C σ x . (42) The constant is locally uniform in σ ∈ ( 3 / 2 , 2 ) \sigma\in(3/2,2) σ ∈ ( 3/2 , 2 ) ∣ I n ∣ |I_n| ∣ I n ∣
Proof. Decompose the two complementary arcs into Whitney intervals near their common endpoints. Let I n , J m I_n,J_m I n , J m e − n e^{-n} e − n e − m e^{-m} e − m n ≥ m n\geq m n ≥ m e − m e^{-m} e − m e − m e^{-m} e − m
E [ M ( I n ) p M ( J m ) q ] ≤ C e − m ζ ( p + q ) e − ( n − m ) ζ ( p ) . (43) \mathbb{E}[M(I_n)^{p}M(J_m)^{q}]\leq Ce^{-m\zeta(p+q)}e^{-(n-m)\zeta(p)}. \tag*{(43)} E [ M ( I n ) p M ( J m ) q ] ≤ C e − m ζ ( p + q ) e − ( n − m ) ζ ( p ) . (43) Moments on a fixed interval are finite for powers below 3 / 2 3/2 3/2 [26 equation (1.6) and Theorem 1.1] and bounded covariance comparison. For more distant intervals, increasing the cross-covariance to that of the comparison model increases the mixed moment, since the mixed derivative in the two masses is positive. Bounded changes of the covariances within either interval change the estimate by at most a constant factor. Bound powers of sums of masses of the Whitney intervals by sums of their powers with weights C ϵ e ϵ n C_{\epsilon}e^{\epsilon n} C ϵ e ϵ n ζ ( p ) \zeta(p) ζ ( p ) ζ ( q ) \zeta(q) ζ ( q ) ζ ( p + q ) \zeta(p+q) ζ ( p + q ) ϵ > 0 \epsilon>0 ϵ > 0 (41) . For coordinate intervals, take a comparison circle whose circumference is at least twice their total span.
For the second assertion, fix σ ∈ ( 3 / 2 , 2 ) \sigma\in(3/2,2) σ ∈ ( 3/2 , 2 ) 0 < x ≤ 1 / 2 0<x\leq1/2 0 < x ≤ 1/2 A , B > 0 A,B>0 A , B > 0
F x ( A , B ) = ( A + B ) σ min { x , A / ( A + B ) , B / ( A + B ) } . F_x(A,B)=(A+B)^{\sigma}\min\{x,A/(A+B),B/(A+B)\}. F x ( A , B ) = ( A + B ) σ min { x , A / ( A + B ) , B / ( A + B )} . This function is increasing in each mass and homogeneous of degree σ \sigma σ
F ~ x ( A , B ) = ( A + B ) σ x − 1 + 2 + A / B + B / A \widetilde{F}_x(A,B)=\frac{(A+B)^{\sigma}}{x^{-1}+2+A/B+B/A} F x ( A , B ) = x − 1 + 2 + A / B + B / A ( A + B ) σ satisfies F x / 3 ≤ F ~ x ≤ F x F_x/3\leq\widetilde{F}_x\leq F_x F x /3 ≤ F x ≤ F x
∣ A i B j ∂ A i ∂ B j F ~ x ∣ ≤ C σ F ~ x , i , j ∈ Z ≥ 0 , i + j ≤ 2. \left|A^{i}B^{j}\partial_A^{i}\partial_B^{j}\widetilde{F}_x\right|\leq C_{\sigma}\widetilde{F}_x,\qquad i,j\in\mathbb{Z}_{\geq0},\quad i+j\leq2. A i B j ∂ A i ∂ B j F x ≤ C σ F x , i , j ∈ Z ≥ 0 , i + j ≤ 2. The first logarithmic derivatives of its denominator are averages of slopes in { − 1 , 0 , 1 } \{-1,0,1\} { − 1 , 0 , 1 } E F x \mathbb{E}F_x E F x x x x
Let β \beta β Var ( β t ) = t \operatorname{Var}(\beta_t)=t Var ( β t ) = t
Y t = − 5 3 t + 4 3 β t , t ≥ 0. Y_t=-\frac{5}{3}t+\sqrt{\frac{4}{3}}\beta_t,\qquad t\geq0. Y t = − 3 5 t + 3 4 β t , t ≥ 0. The factor e 2 n / 3 e^{2n/3} e 2 n /3
∫ 0 ∞ e 2 t / 3 P ( Y t ∈ d y ) d t = 1 { y < 0 } e − y / 2 d y + 1 { y > 0 } e − 2 y d y . (44) \int_{0}^{\infty} e^{2t/3}\mathbb{P}(Y_t\in dy)\,dt=\mathbf{1}_{\{y<0\}}e^{-y/2}\,dy+\mathbf{1}_{\{y>0\}}e^{-2y}\,dy. \tag*{(44)} ∫ 0 ∞ e 2 t /3 P ( Y t ∈ d y ) d t = 1 { y < 0 } e − y /2 d y + 1 { y > 0 } e − 2 y d y . (44) This follows by integrating the Gaussian density; the discriminant is ( 5 / 3 ) 2 − 2 ( 2 / 3 ) ( 4 / 3 ) = 1 (5/3)^2-2(2/3)(4/3)=1 ( 5/3 ) 2 − 2 ( 2/3 ) ( 4/3 ) = 1 r > 0 r>0 r > 0 y = log ( x / r ) y=\log(x/r) y = log ( x / r )
∫ 0 ∞ e 2 t / 3 E [ min { x , r e Y t } ] d t ≤ 4 x r . (45) \int_{0}^{\infty}e^{2t/3}\mathbb{E}[\min\{x,re^{Y_t}\}]\,dt\le4\sqrt{xr}. \tag*{(45)} ∫ 0 ∞ e 2 t /3 E [ min { x , r e Y t }] d t ≤ 4 x r . (45) Let h x ( r ) = ( 1 + r ) σ min { x , r / ( 1 + r ) , 1 / ( 1 + r ) } h_x(r)=(1+r)^\sigma\min\{x,r/(1+r),1/(1+r)\} h x ( r ) = ( 1 + r ) σ min { x , r / ( 1 + r ) , 1/ ( 1 + r )} r ≤ 1 r\le1 r ≤ 1 min { x , r } \min\{x,r\} min { x , r } r > 1 r>1 r > 1 C σ x r σ − 1 / 2 C_\sigma\sqrt{x}\,r^{\sigma-1/2} C σ x r σ − 1/2 ζ ( σ − 1 / 2 ) > 2 / 3 \zeta(\sigma-1/2)>2/3 ζ ( σ − 1/2 ) > 2/3
∫ 0 ∞ e 2 t / 3 E [ h x ( r e Y t ) ] d t ≤ C σ x ( r 1 / 2 + r σ − 1 / 2 ) . (46) \int_{0}^{\infty}e^{2t/3}\mathbb{E}[h_x(re^{Y_t})]\,dt\le C_\sigma\sqrt{x}\left(r^{1/2}+r^{\sigma-1/2}\right). \tag*{(46)} ∫ 0 ∞ e 2 t /3 E [ h x ( r e Y t )] d t ≤ C σ x ( r 1/2 + r σ − 1/2 ) . (46) The same bound holds for the discrete-time sum: for 0 ≤ s ≤ 1 0\le s\le1 0 ≤ s ≤ 1 Y n + s − Y n Y_{n+s}-Y_n Y n + s − Y n [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] h x ( e v r ) ≍ h x ( r ) h_x(e^v r)\asymp h_x(r) h x ( e v r ) ≍ h x ( r ) ∣ v ∣ ≤ 1 |v|\le1 ∣ v ∣ ≤ 1 s s s
Partition I n c I_n^c I n c e − m e^{-m} e − m I n I_n I n 0 ≤ m < n 0\le m<n 0 ≤ m < n O ( e − n ) O(e^{-n}) O ( e − n ) m = n m=n m = n B m B_m B m ϵ > 0 \epsilon>0 ϵ > 0 S ϵ = ∑ m ≥ 0 e − ϵ m S_\epsilon=\sum_{m\ge0}e^{-\epsilon m} S ϵ = ∑ m ≥ 0 e − ϵ m ∑ m B m ≤ S ϵ sup m e ϵ m B m \sum_m B_m\le S_\epsilon\sup_m e^{\epsilon m}B_m ∑ m B m ≤ S ϵ sup m e ϵ m B m F x F_x F x
F x ( A , ∑ m B m ) ≤ S ϵ σ ∑ m e σ ϵ m F x ( A , B m ) . F_x\left(A,\sum_m B_m\right)\le S_\epsilon^\sigma\sum_m e^{\sigma\epsilon m}F_x(A,B_m). F x ( A , m ∑ B m ) ≤ S ϵ σ m ∑ e σ ϵ m F x ( A , B m ) . For separated scales, compare the rescaled fields with a common Gaussian contribution down to scale e − m e^{-m} e − m Y n − m Y_{n-m} Y n − m U , V U,V U , V e − m ζ ( σ ) e^{-m\zeta(\sigma)} e − m ζ ( σ ) c c c (46) gives
∑ n ≥ m + c e 2 n / 3 E F x ( M ( I n ) , B m ) ≤ C σ x e − m [ ζ ( σ ) − 2 / 3 ] E [ U 1 / 2 V σ − 1 / 2 + U σ − 1 / 2 V 1 / 2 ] . \sum_{n\ge m+c}e^{2n/3}\mathbb{E}F_x(M(I_n),B_m)
\le C_\sigma\sqrt{x}\,e^{-m[\zeta(\sigma)-2/3]}\mathbb{E}\left[U^{1/2}V^{\sigma-1/2}+U^{\sigma-1/2}V^{1/2}\right]. n ≥ m + c ∑ e 2 n /3 E F x ( M ( I n ) , B m ) ≤ C σ x e − m [ ζ ( σ ) − 2/3 ] E [ U 1/2 V σ − 1/2 + U σ − 1/2 V 1/2 ] . The powers on the right are below 3 / 2 3/2 3/2 (41) bounds these moments uniformly. The bounded number of neighboring scales is controlled by the same mixed-moment bound and the inequality
F x ( A , B ) ≤ C σ x ( A 1 / 2 B σ − 1 / 2 + A σ − 1 / 2 B 1 / 2 ) . (47) F_x(A,B)\le C_\sigma\sqrt{x}\left(A^{1/2}B^{\sigma-1/2}+A^{\sigma-1/2}B^{1/2}\right). \tag*{(47)} F x ( A , B ) ≤ C σ x ( A 1/2 B σ − 1/2 + A σ − 1/2 B 1/2 ) . (47) Including the factor e σ ϵ m e^{\sigma\epsilon m} e σ ϵ m
∑ m ≥ 0 e − [ ζ ( σ ) − 2 / 3 − σ ϵ ] m . \sum_{m\ge0}e^{-[\zeta(\sigma)-2/3-\sigma\epsilon]m}. m ≥ 0 ∑ e − [ ζ ( σ ) − 2/3 − σ ϵ ] m . For σ \sigma σ J ⊂ ( 3 / 2 , 2 ) J\subset(3/2,2) J ⊂ ( 3/2 , 2 ) ϵ > 0 \epsilon>0 ϵ > 0 ϵ max J < inf σ ∈ J [ ζ ( σ ) − 2 / 3 ] \epsilon\max J<\inf_{\sigma\in J}[\zeta(\sigma)-2/3] ϵ max J < inf σ ∈ J [ ζ ( σ ) − 2/3 ] (42) with the stated local uniformity. Apply the interpolation first to finitely many arcs and regularized chaos; the moment bounds justify the limits.
The two boundary fields at small modulus. For small modulus, decompose the boundary fields into a common Gaussian field and independent remainders. Rescale the circumference to 2 π 2\pi 2 π T = 2 π τ T=2\pi\tau T = 2 π τ h o , h i h_o,h_i h o , h i h l a t h^{\mathrm{lat}} h lat K a b ( u ) K_{ab}(u) K ab ( u ) a , b ∈ { o , i } a,b\in\{o,i\} a , b ∈ { o , i } u u u
K o o ( u ) = K i i ( u ) = 2 ∑ n ≥ 1 coth ( n T ) n cos ( n u ) , K_{oo}(u)=K_{ii}(u)=2\sum_{n\ge1}\frac{\coth(nT)}{n}\cos(nu), K oo ( u ) = K ii ( u ) = 2 n ≥ 1 ∑ n coth ( n T ) cos ( n u ) , K o i ( u ) = 2 ∑ n ≥ 1 csch ( n T ) n cos ( n u ) . K_{oi}(u)=2\sum_{n\ge1}\frac{\operatorname{csch}(nT)}{n}\cos(nu). K o i ( u ) = 2 n ≥ 1 ∑ n csch ( n T ) cos ( n u ) . Since coth v − csch v = tanh ( v / 2 ) \coth v-\operatorname{csch}v=\tanh(v/2) coth v − csch v = tanh ( v /2 )
h o = C T + E o , h i = C T + E i , (48) h_o=C_T+E_o,\qquad h_i=C_T+E_i, \tag*{(48)} h o = C T + E o , h i = C T + E i , (48) where C T , E o , E i C_T,E_o,E_i C T , E o , E i C T C_T C T 2 csch ( n T ) / n 2\operatorname{csch}(nT)/n 2 csch ( n T ) / n E o E_o E o E i E_i E i 2 tanh ( n T / 2 ) / n 2\tanh(nT/2)/n 2 tanh ( n T /2 ) / n J J J osc J C T = sup J C T − inf J C T \operatorname{osc}_J C_T=\sup_J C_T-\inf_J C_T osc J C T = sup J C T − inf J C T a , c > 0 a,c>0 a , c > 0 C T C_T C T O ( T ) O(T) O ( T )
sup ∣ J ∣ ≤ c T E exp { a osc J C T } ≤ C a , c . (49) \sup_{|J|\le cT}\mathbb{E}\exp\{a\operatorname{osc}_J C_T\}\le C_{a,c}. \tag*{(49)} ∣ J ∣ ≤ c T sup E exp { a osc J C T } ≤ C a , c . (49) The increment variance is at most C ∣ u − v ∣ 2 / T 2 C|u-v|^2/T^2 C ∣ u − v ∣ 2 / T 2 (49) . Splitting the Fourier series at n = T − 1 n=T^{-1} n = T − 1 O ( T ) O(T) O ( T ) E o , E i E_o,E_i E o , E i T T T
Partition the boundary circle into arcs J j J_j J j T T T u j ∈ J j u_j\in J_j u j ∈ J j
a j = exp { γ C T ( u j ) / 2 } . a_j=\exp\{\gamma C_T(u_j)/2\}. a j = exp { γ C T ( u j ) /2 } . On J j J_j J j a j a_j a j E o E_o E o E i E_i E i e − γ D j / 2 e^{-\gamma D_j/2} e − γ D j /2 e γ D j / 2 e^{\gamma D_j/2} e γ D j /2 D j = osc J j C T D_j=\operatorname{osc}_{J_j}C_T D j = osc J j C T ε γ 2 / 4 e γ h ε / 2 d u \varepsilon^{\gamma^2/4}e^{\gamma h_\varepsilon/2}\,du ε γ 2 /4 e γ h ε /2 d u h ε h_\varepsilon h ε ε \varepsilon ε c T = T 1 + γ 2 / 4 = T 5 / 3 c_T=T^{1+\gamma^2/4}=T^{5/3} c T = T 1 + γ 2 /4 = T 5/3 u = T v u=Tv u = T v T T T C j = M i ( J j ) C_j=M_i(J_j) C j = M i ( J j ) p , q > 0 p,q>0 p , q > 0
Z i p + q ≥ C j p C k q . (50) Z_i^{p+q}\ge C_j^p C_k^q. \tag*{(50)} Z i p + q ≥ C j p C k q . (50) For factors of degree p , q p,q p , q J j , J k J_j,J_k J j , J k a j p a k q a_j^pa_k^q a j p a k q c T p + q c_T^{p+q} c T p + q E o E_o E o E i E_i E i [27 ] and bounded covariance comparison. The remaining factors depend only on D j , D k D_j,D_k D j , D k (49) . Summing over the arcs gives a bound polynomial in T − 1 T^{-1} T − 1 (34) and (35) .
A boundary one-arm estimate for the discarded arcs
Recall from § 2.3 that S C S_C S C C C C H 3 H_3 H 3 S C S_C S C ℓ C \ell_C ℓ C
G ‾ T ( x ) = 1 H 3 ∑ C ∈ M min { x , 1 − x , ℓ C , 1 − ℓ C } . \overline{G}_T(x)=\mathbf{1}_{H_3}\sum_{C\in\mathcal{M}}\min\{x,1-x,\ell_C,1-\ell_C\}. G T ( x ) = 1 H 3 C ∈ M ∑ min { x , 1 − x , ℓ C , 1 − ℓ C } . The following lemma bounds the expected number of maximal discarded boundary arcs at each scale, restricted to the event H 3 H_3 H 3
Lemma 5.5. For an integer n ≥ 0 n \ge0 n ≥ 0 N n N_n N n S C S_C S C S C S_C S C e − n e^{-n} e − n C , c > 0 C,c>0 C , c > 0 n n n τ \tau τ τ \tau τ
E [ 1 H 3 N n ] ≤ C e 2 n / 3 p 3 ( τ − c ) . (51) \mathbb{E}[1_{H_3}N_n] \le Ce^{2n/3}p_3(\tau-c). \tag*{(51)} E [ 1 H 3 N n ] ≤ C e 2 n /3 p 3 ( τ − c ) . (51) For τ ≤ 1 \tau\le1 τ ≤ 1 τ e − n \tau e^{-n} τ e − n C e 2 n / 3 Ce^{2n/3} C e 2 n /3 τ \tau τ
Proof. Put δ = e − n \delta=e^{-n} δ = e − n O ( δ − 1 ) O(\delta^{-1}) O ( δ − 1 ) Q Q Q δ / 100 \delta/100 δ /100 Q Q Q δ \delta δ R R R Q Q Q D ∖ K D \setminus K D ∖ K Q Q Q Q Q Q δ \delta δ H 3 H_3 H 3
Each such endpoint therefore requires a black crossing of the half-annulus, with probability at most C R δ 1 / 3 CR\delta^{1/3} C R δ 1/3 [31 ] . Disjointness and the lower bound on arc lengths give a bounded number of initial endpoints, and likewise terminal endpoints, in each Q Q Q H 3 H_3 H 3 p 3 ( τ − c ) p_3(\tau-c) p 3 ( τ − c ) Q Q Q (51) . For τ ≤ 1 \tau\le1 τ ≤ 1 R R R τ \tau τ
Proofs of the three estimates
Proof of Proposition 5.1. For small modulus, use T = 2 π τ T=2\pi\tau T = 2 π τ (48) . Consider an arc of length T e − n Te^{-n} T e − n J j J_j J j J k J_k J k J j J_j J j (46) conditionally on C T , E i C_T,E_i C T , E i n n n e 2 n / 3 e^{2n/3} e 2 n /3
C σ σ T σ x e C σ ( D j + D k ) ( a j 1 / 2 a k σ − 1 / 2 + a j σ − 1 / 2 a k 1 / 2 ) . C_{\sigma} \sigma_T^{\sigma}\sqrt{x}e^{C_{\sigma}(D_j+D_k)}\left(a_j^{1/2}a_k^{\sigma-1/2}+a_j^{\sigma-1/2}a_k^{1/2}\right). C σ σ T σ x e C σ ( D j + D k ) ( a j 1/2 a k σ − 1/2 + a j σ − 1/2 a k 1/2 ) . Apply (50) separately to the two terms. The factors a j , a k a_j,a_k a j , a k c T σ c_T^\sigma c T σ 3 / 2 3/2 3/2 c T cT c T (47) ; there are at most O ( T − 1 ) O(T^{-1}) O ( T − 1 ) C e 2 n / 3 Ce^{2n/3} C e 2 n /3 3 / 2 < σ < 2 3/2<\sigma<2 3/2 < σ < 2
W σ ( τ , x ) ≤ C σ τ − K σ x ( 1 − x ) , 0 < τ ≤ τ 0 , (52) W_{\sigma}(\tau,x) \le C_{\sigma}\tau^{-K_{\sigma}}\sqrt{x(1-x)}, \qquad0<\tau\le\tau_0, \tag*{(52)} W σ ( τ , x ) ≤ C σ τ − K σ x ( 1 − x ) , 0 < τ ≤ τ 0 , (52) for some finite K σ K_{\sigma} K σ τ 0 > 0 \tau_0>0 τ 0 > 0 τ 0 \tau_0 τ 0 x \sqrt{x} x
For long cylinders, the joint covariance of the boundary traces differs from that of two independent Gaussian fields on the circle by a uniformly bounded smooth function. Condition on percolation to fix the arcs S C S_C S C (51) with the scale sum (42) gives the bound for x ≤ 1 / 2 x\le1/2 x ≤ 1/2 x x x 1 − x 1-x 1 − x (37) .
By (40) and (32) , φ − σ ( x ) \varphi_{-\sigma}(x) φ − σ ( x ) W σ ( τ , x ) W_{\sigma}(\tau,x) W σ ( τ , x ) ( 2 / γ ) η ( 2 i τ ) e 2 π σ 2 τ / 3 d τ (\sqrt{2}/\gamma)\eta(2i\tau)e^{2\pi\sigma^{2}\tau/3}\,d\tau ( 2 / γ ) η ( 2 i τ ) e 2 π σ 2 τ /3 d τ τ ↓ 0 \tau\downarrow0 τ ↓ 0
η ( 2 i τ ) = ( 2 τ ) − 1 / 2 η ( i / ( 2 τ ) ) ≍ τ − 1 / 2 e − π / ( 24 τ ) . (53) \eta(2i\tau)=(2\tau)^{-1/2}\eta(i/(2\tau))\asymp\tau^{-1/2}e^{-\pi/(24\tau)}. \tag*{(53)} η ( 2 i τ ) = ( 2 τ ) − 1/2 η ( i / ( 2 τ )) ≍ τ − 1/2 e − π / ( 24 τ ) . (53) This exponential decay makes the small-modulus integral finite. As τ → ∞ \tau\to\infty τ → ∞ X ( σ ) < ξ 3 X(\sigma)<\xi_3 X ( σ ) < ξ 3 x ( 1 − x ) \sqrt{x(1-x)} x ( 1 − x ) u σ u_\sigma u σ
Proof of Proposition 5.2. Choose max J − 3 / 2 < θ < 1 / 2 \max J-3/2<\theta<1/2 max J − 3/2 < θ < 1/2 A , B > 0 A,B>0 A , B > 0 ℓ = A / ( A + B ) \ell=A/(A+B) ℓ = A / ( A + B )
( A + B ) σ [ ℓ ( 1 − ℓ ) ] θ ≤ C J ( A σ − θ B θ + A θ B σ − θ ) . (54) (A+B)^\sigma[\ell(1-\ell)]^\theta\le C_J(A^{\sigma-\theta}B^\theta+A^\theta B^{\sigma-\theta}). \tag*{(54)} ( A + B ) σ [ ℓ ( 1 − ℓ ) ] θ ≤ C J ( A σ − θ B θ + A θ B σ − θ ) . (54) The powers in (54) satisfy Lemma 5.4 . Including the inner negative moment bounds the field expectation uniformly over boundary arcs on long cylinders. Let N ∂ N_\partial N ∂ N ∂ N_\partial N ∂ H 3 H_3 H 3 B B B W BBBW B B B W C J E [ N ∂ 1 A 4 ( τ − c ) ] C_J\mathbb{E}[N_\partial\mathbf{1}_{A_4}(\tau^{-c})] C J E [ N ∂ 1 A 4 ( τ − c )] A 4 A_4 A 4 B B B W BBBW B B B W q > 1 q>1 q > 1
E [ N ∂ 1 A 4 ] ≤ E [ N ∂ q ] 1 / q P ( A 4 ) 1 − 1 / q . \mathbb{E}[N_\partial\mathbf{1}_{A_4}]\le\mathbb{E}[N_\partial^q]^{1/q}\mathbb{P}(A_4)^{1-1/q}. E [ N ∂ 1 A 4 ] ≤ E [ N ∂ q ] 1/ q P ( A 4 ) 1 − 1/ q . The polychromatic four-arm exponent is 5 / 4 5/4 5/4 [9 Theorem 2] . Given ϵ > 0 \epsilon>0 ϵ > 0 q q q
E [ ( Z o Z i ) σ 1 H 3 ∑ C ∈ M i [ ℓ C ( 1 − ℓ C ) ] θ ] ≤ C J , ϵ e − 2 π ( 5 / 4 − ϵ ) τ . (55) \mathbb{E}\left[\left(\frac{Z_o}{Z_i}\right)^\sigma\mathbf{1}_{H_3}\sum_{C\in\mathcal{M}_i}[\ell_C(1-\ell_C)]^\theta\right]\le C_{J,\epsilon}e^{-2\pi(5/4-\epsilon)\tau}. \tag*{(55)} E [ ( Z i Z o ) σ 1 H 3 C ∈ M i ∑ [ ℓ C ( 1 − ℓ C ) ] θ ] ≤ C J , ϵ e − 2 π ( 5/4 − ϵ ) τ . (55) At small modulus, the argument leading to (50) , now with powers σ − θ , θ \sigma-\theta,\theta σ − θ , θ τ − 1 \tau^{-1} τ − 1
Multiplying (55) by η ( 2 i τ ) e 2 π σ 2 τ / 3 \eta(2i\tau)e^{2\pi\sigma^2\tau/3} η ( 2 i τ ) e 2 π σ 2 τ /3 (35) , leaves, at large modulus, the exponential
exp { [ 2 π 3 ( σ 2 − 4 ) + 2 π ϵ ] τ } . \exp\left\{\left[\frac{2\pi}{3}(\sigma^2-4)+2\pi\epsilon\right]\tau\right\}. exp { [ 3 2 π ( σ 2 − 4 ) + 2 π ϵ ] τ } . For sufficiently small ϵ \epsilon ϵ J J J (53) controls small modulus. Since θ < 1 / 2 \theta<1/2 θ < 1/2 0 < a < 1 / 2 − θ 0<a<1/2-\theta 0 < a < 1/2 − θ
e a ∣ log ( ℓ / ( 1 − ℓ ) ) ∣ ℓ ( 1 − ℓ ) ≤ C a [ ℓ ( 1 − ℓ ) ] θ e^{a|\log(\ell/(1-\ell))|}\sqrt{\ell(1-\ell)}\le C_a[\ell(1-\ell)]^\theta e a ∣ l o g ( ℓ / ( 1 − ℓ )) ∣ ℓ ( 1 − ℓ ) ≤ C a [ ℓ ( 1 − ℓ ) ] θ then proves the asserted weighted norm bound. To prove holomorphy, choose a compact interval J ′ ⊂ ( 3 / 2 , 2 ) J'\subset(3/2,2) J ′ ⊂ ( 3/2 , 2 ) J J J θ , a \theta,a θ , a J ′ J' J ′ σ − < min J ≤ max J < σ + \sigma_-<\min J\le\max J<\sigma_+ σ − < min J ≤ max J < σ + J ′ J' J ′ j σ j_\sigma j σ
η ( 2 i τ ) e 2 π σ + 2 τ / 3 ( ( Z o / Z i ) σ − + ( Z o / Z i ) σ + ) . \eta(2i\tau)e^{2\pi\sigma_+^2\tau/3}\left((Z_o/Z_i)^{\sigma_-}+(Z_o/Z_i)^{\sigma_+}\right). η ( 2 i τ ) e 2 π σ + 2 τ /3 ( ( Z o / Z i ) σ − + ( Z o / Z i ) σ + ) . The resulting positive measure has finite e a ∣ t ∣ e^{a|t|} e a ∣ t ∣ (55) and the small-modulus bound. For a complex parameter σ ′ \sigma' σ ′ J J J
∣ e 2 π ( σ ′ ) 2 τ / 3 ∣ ≤ e 2 π ( Re σ ′ ) 2 τ / 3 . \left|e^{2\pi(\sigma')^2\tau/3}\right|\le e^{2\pi(\operatorname{Re}\sigma')^2\tau/3}. e 2 π ( σ ′ ) 2 τ /3 ≤ e 2 π ( Re σ ′ ) 2 τ /3 . Each fixed-order derivative introduces powers of τ \tau τ log ( Z o / Z i ) \log(Z_o/Z_i) log ( Z o / Z i ) ( Re σ ′ ) 2 < σ + 2 (\operatorname{Re}\sigma')^2<\sigma_+^2 ( Re σ ′ ) 2 < σ + 2 σ − < Re σ ′ < σ + \sigma_-<\operatorname{Re}\sigma'<\sigma_+ σ − < Re σ ′ < σ + ∣ log u ∣ k ≤ C k , δ ( u δ + u − δ ) |\log u|^k\le C_{k,\delta}(u^\delta+u^{-\delta}) ∣ log u ∣ k ≤ C k , δ ( u δ + u − δ ) (53) dominates the polynomial factors. For each fixed derivative order, a constant multiple of this measure dominates the derivatives and difference quotients. Dominated differentiation in weighted total variation proves the asserted holomorphy.
Finally, RSW gives positive probability to a separated B B B W BBBW B B B W j σ j_\sigma j σ g g g
Proof of Proposition 5.3. It remains to prove the lower bound. As τ → ∞ \tau\to\infty τ → ∞ M o , M i M_o,M_i M o , M i h l a t h^{\mathrm{lat}} h lat ϵ > 0 \epsilon>0 ϵ > 0 M > 1 M>1 M > 1 c 0 > 0 c_0>0 c 0 > 0
R τ = { M − 1 < Z o , Z i < M , sup ∣ I ∣ ≤ 2 ϵ M o ( I ) < Z o / 3 } \mathcal{R}_\tau=\left\{M^{-1}<Z_o,Z_i<M,\quad\sup_{|I|\le2\epsilon}M_o(I)<Z_o/3\right\} R τ = { M − 1 < Z o , Z i < M , ∣ I ∣ ≤ 2 ϵ sup M o ( I ) < Z o /3 } has probability at least c 0 c_0 c 0 τ \tau τ M , ϵ M,\epsilon M , ϵ τ \tau τ
On R τ \mathcal{R}_\tau R τ 2 ϵ 2\epsilon 2 ϵ [9 Section 2.1] in a fixed-width outer strip give three black arms whose outer endpoints u 1 , u 2 , u 3 u_1,u_2,u_3 u 1 , u 2 , u 3 p , u 1 , u 2 , q , u 3 p,u_1,u_2,q,u_3 p , u 1 , u 2 , q , u 3 c ϵ p 3 ( τ − c ) c_\epsilon p_3(\tau-c) c ϵ p 3 ( τ − c ) ϵ \epsilon ϵ p p p u 1 , u 3 u_1,u_3 u 1 , u 3 p p p q q q u 2 , u 3 u_2,u_3 u 2 , u 3 q q q p p p u 1 u_1 u 1 q q q u 2 , u 3 u_2,u_3 u 2 , u 3 G = 1 G=1 G = 1 (11) . Averaging over the first mark preserves the bound. Since ( Z o / Z i ) σ ≥ M − 2 max J (Z_o/Z_i)^\sigma\ge M^{-2}\max J ( Z o / Z i ) σ ≥ M − 2 max J R τ \mathcal{R}_\tau R τ
By [9 Theorems 4 and 5] , p 3 ( τ ) = exp { − 2 π ξ 3 τ + o ( τ ) } p_3(\tau)=\exp\{-2\pi\xi_3\tau+o(\tau)\} p 3 ( τ ) = exp { − 2 π ξ 3 τ + o ( τ )} 2 / 3 < ξ 3 < 5 / 4 2/3<\xi_3<5/4 2/3 < ξ 3 < 5/4 3 / 2 < σ < 2 3/2<\sigma<2 3/2 < σ < 2 (40) and (38) imply convergence of F F F X ( σ ) < ξ 3 X(\sigma)<\xi_3 X ( σ ) < ξ 3 X ( σ ) > ξ 3 X(\sigma)>\xi_3 X ( σ ) > ξ 3 (53) . This proves (39) .
The spectral threshold and the main theorem
We identify the convergence endpoint of the Mellin transform F F F s ∗ s_* s ∗ Theorem 6.1 .
The Mellin transform and the spectral threshold
Recall the Mellin transform F F F (28) , and the function u σ u_\sigma u σ j σ j_\sigma j σ Proposition 4.1 . They satisfy
F ( σ ) = 1 2 u σ ( 0 ) , ( L 0 − T − σ ) u σ = j σ . (56) F(\sigma)=\tfrac{1}{2}u_\sigma(0),\qquad(L_0-T_{-\sigma})u_\sigma=j_\sigma. \tag*{(56)} F ( σ ) = 2 1 u σ ( 0 ) , ( L 0 − T − σ ) u σ = j σ . (56) Propositions 5.1 and 5.3 show that u σ u_\sigma u σ 3 / 2 < σ < σ c 3/2 < \sigma< \sigma_c 3/2 < σ < σ c F F F
σ c = 1 / 4 + 3 ξ 3 ∈ ( 3 / 2 , 2 ) . (57) \sigma_c = \sqrt{1/4 + 3\xi_3} \in(3/2, 2). \tag*{(57)} σ c = 1/4 + 3 ξ 3 ∈ ( 3/2 , 2 ) . (57) By Proposition 5.2, j σ j_\sigma j σ ( 3 / 2 , 2 ) (3/2, 2) ( 3/2 , 2 ) ∫ e η 0 ∣ t ∣ ∣ j ∣ ( d t ) \int e^{\eta_0 |t|}|j|(dt) ∫ e η 0 ∣ t ∣ ∣ j ∣ ( d t ) η 0 > 0 \eta_0 > 0 η 0 > 0 j σ j_\sigma j σ σ c \sigma_c σ c
Recall L 0 = − d 2 / d t 2 + 1 / 4 L_0 = -d^2/dt^2 + 1/4 L 0 = − d 2 / d t 2 + 1/4 T s T_s T s (2) –(3) . For ∣ Re s ∣ < 2 |\operatorname{Re} s| < 2 ∣ Re s ∣ < 2
g ( t ) = e − ∣ t ∣ / 2 , P s f = g ∗ ( T s f ) . g(t) = e^{-|t|/2}, \qquad P_s f = g * (T_s f). g ( t ) = e − ∣ t ∣/2 , P s f = g ∗ ( T s f ) . Here g g g L 0 L_0 L 0 R \mathbb{R} R ∗ * ∗ R > 0 R > 0 R > 0 g R g_R g R L 0 L_0 L 0 I R = ( − R , R ) I_R = (-R,R) I R = ( − R , R ) K R , s = g R T s K_{R,s} = g_R T_s K R , s = g R T s C ( [ − R , R ] ) C([-R,R]) C ([ − R , R ]) I R I_R I R s ∈ C s \in\mathbb{C} s ∈ C r R ( s ) r_R(s) r R ( s ) s s s T s ( t , u ) T_s(t,u) T s ( t , u ) P s ( t , u ) P_s(t,u) P s ( t , u ) T s T_s T s P s P_s P s
We use the spaces
Y η = { f ∈ C e v e n ( R ) : ∥ f ∥ η : = sup t e η ∣ t ∣ ∣ f ( t ) ∣ < ∞ } , X = { u ∈ C e v e n ( R ) : lim ∣ t ∣ → ∞ u ( t ) ∈ C } , \mathcal{Y}_\eta= \{f \in C_{\mathrm{even}}(\mathbb{R}) : \|f\|_\eta:= \sup_t e^{\eta|t|}|f(t)| < \infty\}, \qquad\mathcal{X} = \{u \in C_{\mathrm{even}}(\mathbb{R}) : \lim_{|t|\to\infty} u(t) \in\mathbb{C}\}, Y η = { f ∈ C even ( R ) : ∥ f ∥ η := t sup e η ∣ t ∣ ∣ f ( t ) ∣ < ∞ } , X = { u ∈ C even ( R ) : ∣ t ∣ → ∞ lim u ( t ) ∈ C } , where η > 0 \eta> 0 η > 0 C e v e n ( R ) C_{\mathrm{even}}(\mathbb{R}) C even ( R ) X \mathcal{X} X
Theorem 6.1 (Spectral threshold). There is s ∗ ∈ ( 1 , 2 ) s_* \in(1,2) s ∗ ∈ ( 1 , 2 ) ϵ ∈ { − 1 , 1 } \epsilon\in\{-1,1\} ϵ ∈ { − 1 , 1 } P ϵ s ∗ u = u P_{\epsilon s_*}u = u P ϵ s ∗ u = u ψ ϵ \psi_\epsilon ψ ϵ ψ ϵ \psi_\epsilon ψ ϵ P ϵ s ∗ u = u P_{\epsilon s_*}u = u P ϵ s ∗ u = u ϵ \epsilon ϵ s ∈ ( 0 , 2 ) s \in(0,2) s ∈ ( 0 , 2 ) P ϵ s u = u P_{\epsilon s}u = u P ϵs u = u
*For 0 ≤ s < s ∗ 0 \le s < s_* 0 ≤ s < s ∗ G ϵ s f = ∑ n ≥ 0 P ϵ s n f \mathcal{G}_{\epsilon s}f = \sum_{n\geq0} P_{\epsilon s}^n f G ϵs f = ∑ n ≥ 0 P ϵs n f I − P ϵ s I-P_{\epsilon s} I − P ϵs Y η \mathcal{Y}_\eta Y η X \mathcal{X} X [ 0 , s ∗ ) [0,s_*) [ 0 , s ∗ ) η > 0 \eta> 0 η > 0 ( I − P ϵ s ) u = f (I-P_{\epsilon s})u = f ( I − P ϵs ) u = f r R ( s ) r_R(s) r R ( s ) r R ( s ) = r R ( − s ) r_R(s) = r_R(-s) r R ( s ) = r R ( − s )
s ∗ = inf { s > 0 : sup R > 0 r R ( s ) > 1 } , sup R > 0 r R ( s ) > 1 ( s ∗ < s < 2 ) . (58) s_* = \inf\{s > 0 : \sup_{R>0} r_R(s) > 1\}, \qquad\sup_{R>0} r_R(s) > 1 \quad(s_* < s < 2). \tag*{(58)} s ∗ = inf { s > 0 : R > 0 sup r R ( s ) > 1 } , R > 0 sup r R ( s ) > 1 ( s ∗ < s < 2 ) . (58) We postpone the proof to § 7.4 . Below, we first finish the proof of Theorem 1.1 .
Proof of Theorem 1.1
We need the following basic fact on the Laplace transform.
Lemma 6.2. Let μ \mu μ [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) ∫ e s v μ ( d v ) \int e^{sv}\mu(dv) ∫ e s v μ ( d v ) β \beta β β \beta β
Proof. If it were analytic on a disk of radius r r r β \beta β 0 < δ < r / 3 0 < \delta< r/3 0 < δ < r /3 a = β − δ a = \beta-\delta a = β − δ a + 3 δ / 2 > β a+3\delta/2 > \beta a + 3 δ /2 > β ∫ v n e a v μ ( d v ) / n ! ≥ 0 \int v^n e^{av}\mu(dv)/n! \geq0 ∫ v n e a v μ ( d v ) / n ! ≥ 0 a + 3 δ / 2 a+3\delta/2 a + 3 δ /2
Proof of Theorem 1.1. For 3 / 2 < σ < σ c 3/2 < \sigma< \sigma_c 3/2 < σ < σ c u σ u_\sigma u σ (56) with g g g
u σ = P − σ u σ + b σ , b σ = g ∗ j σ > 0. (59) u_\sigma= P_{-\sigma}u_\sigma+ b_\sigma,\qquad b_\sigma= g * j_\sigma> 0. \tag*{(59)} u σ = P − σ u σ + b σ , b σ = g ∗ j σ > 0. (59) The difference w w w L 0 w = 0 L_0w = 0 L 0 w = 0 e t / 2 e^{t/2} e t /2 e − t / 2 e^{-t/2} e − t /2
First, σ c ≤ s ∗ \sigma_c \le s_* σ c ≤ s ∗ * Let ℓ R \ell_R ℓ R K R , − σ K_{R,-\sigma} K R , − σ r R ( − σ ) r_R(-\sigma) r R ( − σ ) g R ≤ g g_R \le g g R ≤ g (59) gives
( 1 − r R ( − σ ) ) ℓ R ( u σ ) ≥ ℓ R ( b σ ) > 0. (1-r_R(-\sigma))\ell_R(u_\sigma) \ge\ell_R(b_\sigma) > 0. ( 1 − r R ( − σ )) ℓ R ( u σ ) ≥ ℓ R ( b σ ) > 0. All these pairings are on I R I_R I R r R ( − σ ) < 1 r_R(-\sigma) < 1 r R ( − σ ) < 1 R R R r R ( σ ) r_R(\sigma) r R ( σ ) σ c > s ∗ \sigma_c > s_* σ c > s ∗ max { 3 / 2 , s ∗ } < σ < σ c \max\{3/2,s_*\} < \sigma< \sigma_c max { 3/2 , s ∗ } < σ < σ c R R R r R ( σ ) > 1 r_R(\sigma) > 1 r R ( σ ) > 1 σ c ≤ s ∗ \sigma_c \le s_* σ c ≤ s ∗ s ∗ > 3 / 2 s_* > 3/2 s ∗ > 3/2
Next, σ c ≥ s ∗ \sigma_c \ge s_* σ c ≥ s ∗ * Suppose otherwise, and take a compact interval J ⋐ ( 3 / 2 , s ∗ ) J \Subset(3/2,s_*) J ⋐ ( 3/2 , s ∗ ) σ c \sigma_c σ c j σ j_\sigma j σ
sup t e η ∣ t ∣ ∣ ( g ∗ j ) ( t ) ∣ ≤ ∫ e η ∣ u ∣ ∣ j ∣ ( d u ) , 0 < η < 1 / 2 , \sup_t e^{\eta|t|}|(g*j)(t)| \le\int e^{\eta|u|}|j|(du),\qquad0 < \eta< 1/2, t sup e η ∣ t ∣ ∣ ( g ∗ j ) ( t ) ∣ ≤ ∫ e η ∣ u ∣ ∣ j ∣ ( d u ) , 0 < η < 1/2 , make the continuation b ζ = g ∗ j ζ b_\zeta= g*j_\zeta b ζ = g ∗ j ζ ζ \zeta ζ J J J Y η Y_\eta Y η η \eta η j σ j_\sigma j σ (59) give u ζ = G − ζ b ζ u_\zeta= G_{-\zeta}b_\zeta u ζ = G − ζ b ζ ζ < σ c \zeta< \sigma_c ζ < σ c σ c \sigma_c σ c 1 2 ( G − ζ b ζ ) ( 0 ) \frac{1}{2}(G_{-\zeta}b_\zeta)(0) 2 1 ( G − ζ b ζ ) ( 0 ) F F F σ c \sigma_c σ c
To exclude this continuation, split the defining integral for F F F m = 1 m=1 m = 1 (27) , the part over m ≥ 1 m \ge1 m ≥ 1 Re ζ > − 1 / 2 \operatorname{Re}\zeta> -1/2 Re ζ > − 1/2 v = − log m v=-\log m v = − log m
F < ( ζ ) : = ∫ 0 1 m − ζ z ( 1 , 1 / 2 ; m ) d m = ∫ 0 ∞ e ζ v μ ( d v ) , μ ( d v ) = e − v z ( 1 , 1 / 2 ; e − v ) d v . F_{<}(\zeta) := \int_0^1 m^{-\zeta}z(1,1/2;m)\,dm = \int_0^\infty e^{\zeta v}\,\mu(dv),\qquad\mu(dv)=e^{-v}z(1,1/2;e^{-v})\,dv. F < ( ζ ) := ∫ 0 1 m − ζ z ( 1 , 1/2 ; m ) d m = ∫ 0 ∞ e ζ v μ ( d v ) , μ ( d v ) = e − v z ( 1 , 1/2 ; e − v ) d v . The Laplace transform of this positive measure has σ c \sigma_c σ c
Thus σ c = s ∗ \sigma_c=s_* σ c = s ∗ (57) yields ξ 3 = ( s ∗ 2 − 1 / 4 ) / 3 \xi_3=(s_*^2-1/4)/3 ξ 3 = ( s ∗ 2 − 1/4 ) /3 D \mathcal{D} D s ∗ s_* s ∗
The limiting convolution kernel and bounded solutions
Recall x ( t ) = e t / ( 1 + e t ) x(t)=e^t/(1+e^t) x ( t ) = e t / ( 1 + e t ) a ( t ) = 1 − x ( t ) a(t)=1-x(t) a ( t ) = 1 − x ( t ) (2) . For fixed t , u t,u t , u P s ( t + v , u + v ) → q 0 ( t − u ) P_s(t+v,u+v)\to q_0(t-u) P s ( t + v , u + v ) → q 0 ( t − u ) v → ± ∞ v\to\pm\infty v → ± ∞
q 0 ( t ) = 3 3 8 π ( cosh ( t / 2 ) + 1 / 2 ) , L 0 q 0 = H , g ∗ H = q 0 . (60) q_0(t)=\frac{3\sqrt{3}}{8\pi(\cosh(t/2)+1/2)},\qquad L_0q_0=H,\qquad g*H=q_0. \tag*{(60)} q 0 ( t ) = 8 π ( cosh ( t /2 ) + 1/2 ) 3 3 , L 0 q 0 = H , g ∗ H = q 0 . (60) The weight in T s T_s T s
W s ( t , u ) = x ( t ) 1 + s x ( u ) 1 − s + a ( t ) 1 + s a ( u ) 1 − s . (61) W_s(t,u)=x(t)^{1+s}x(u)^{1-s}+a(t)^{1+s}a(u)^{1-s}. \tag*{(61)} W s ( t , u ) = x ( t ) 1 + s x ( u ) 1 − s + a ( t ) 1 + s a ( u ) 1 − s . (61) Thus T s ( t , u ) = H ( t − u ) W s ( t , u ) T_s(t,u)=H(t-u)W_s(t,u) T s ( t , u ) = H ( t − u ) W s ( t , u ) T s T_s T s P s P_s P s
To prove Theorem 6.1, we first study the convolution operator with kernel q 0 ( t − u ) q_0(t-u) q 0 ( t − u ) Q f = q 0 ∗ f Qf=q_0*f Q f = q 0 ∗ f f ^ ( k ) = ∫ e i k t f ( t ) d t \widehat{f}(k)=\int e^{ikt}f(t)\,dt f ( k ) = ∫ e ik t f ( t ) d t
q ^ 0 ( k ) = 3 1 + 2 cosh ( 4 π k / 3 ) , ∫ q 0 = 1 , σ 0 2 : = ∫ t 2 q 0 ( t ) d t = 32 π 2 27 . (62) \widehat{q}_0(k)=\frac{3}{1+2\cosh(4\pi k/3)},\qquad\int q_0=1,\qquad\sigma_0^2:=\int t^2q_0(t)\,dt=\frac{32\pi^2}{27}. \tag*{(62)} q 0 ( k ) = 1 + 2 cosh ( 4 π k /3 ) 3 , ∫ q 0 = 1 , σ 0 2 := ∫ t 2 q 0 ( t ) d t = 27 32 π 2 . (62) The walk with increment density q 0 q_0 q 0 V ( t ) = log ( 1 + t 2 ) V(t)=\log(1+t^2) V ( t ) = log ( 1 + t 2 ) Q V ( t ) − V ( t ) = − σ 0 2 / t 2 + o ( t − 2 ) QV(t)-V(t)=-\sigma_0^2/t^2+o(t^{-2}) Q V ( t ) − V ( t ) = − σ 0 2 / t 2 + o ( t − 2 ) [ − R , R ] [-R,R] [ − R , R ] V ( t ) / log ( 1 + R 2 ) V(t)/\log(1+R^2) V ( t ) / log ( 1 + R 2 ) R → ∞ R\to\infty R → ∞ Q Q Q Q f ≤ f Qf\leq f Q f ≤ f Q Q Q
The kernel of the perturbation R s = P s − Q R_s=P_s-Q R s = P s − Q ∣ t ∣ + ∣ u ∣ |t|+|u| ∣ t ∣ + ∣ u ∣ ∣ Re s ∣ < 2 |\operatorname{Re}s|<2 ∣ Re s ∣ < 2 g g g H ( v − u ) ( W s ( v , u ) − 1 ) H(v-u)(W_s(v,u)-1) H ( v − u ) ( W s ( v , u ) − 1 ) ∣ Re s ∣ ≤ r |\operatorname{Re}s|\leq r ∣ Re s ∣ ≤ r 1 ≤ r < 2 1\leq r<2 1 ≤ r < 2 v ≥ u ≥ 0 v\geq u\geq0 v ≥ u ≥ 0 e − ( v − u ) − 2 u e^{-(v-u)-2u} e − ( v − u ) − 2 u e − ( 2 − r ) ( v − u ) − 2 u e^{-(2-r)(v-u)-2u} e − ( 2 − r ) ( v − u ) − 2 u v ≥ 0 ≥ u v\geq0\geq u v ≥ 0 ≥ u e − v − ( 2 − r ) ∣ u ∣ e^{-v-(2-r)|u|} e − v − ( 2 − r ) ∣ u ∣ e − ( 2 − r ) v − ∣ u ∣ e^{-(2-r)v-|u|} e − ( 2 − r ) v − ∣ u ∣
The potential kernel of Q Q Q
A ( t ) = 27 32 π 2 t coth 3 t 4 , ( I − Q ) A = − q 0 , A ( t ) = ∣ t ∣ σ 0 2 + O ( ( 1 + ∣ t ∣ ) e − 3 ∣ t ∣ / 2 ) . (63) A(t)=\frac{27}{32\pi^2}t\coth\frac{3t}{4},\qquad(I-Q)A=-q_0,\qquad A(t)=\frac{|t|}{\sigma_0^2}+O((1+|t|)e^{-3|t|/2}). \tag*{(63)} A ( t ) = 32 π 2 27 t coth 4 3 t , ( I − Q ) A = − q 0 , A ( t ) = σ 0 2 ∣ t ∣ + O (( 1 + ∣ t ∣ ) e − 3∣ t ∣/2 ) . (63) Define A ( 0 ) A(0) A ( 0 ) (62) . Define an operator on even exponentially decaying functions by
( S f ) ( t ) = f ( t ) − ∫ R [ A ( t − u ) + A ( t + u ) 2 − A ( t ) ] f ( u ) d u . (64) (Sf)(t)=f(t)-\int_{\mathbb{R}}\left[\frac{A(t-u)+A(t+u)}{2}-A(t)\right]f(u)\,du. \tag*{(64)} ( S f ) ( t ) = f ( t ) − ∫ R [ 2 A ( t − u ) + A ( t + u ) − A ( t ) ] f ( u ) d u . (64) The second difference cancels the linear tails, so S : Y η → Y η S:\mathcal{Y}_\eta\to\mathcal{Y}_\eta S : Y η → Y η η > 0 \eta>0 η > 0
( I − Q ) S f = f − q 0 ∫ R f . (65) (I-Q)Sf=f-q_0\int_{\mathbb{R}}f. \tag*{(65)} ( I − Q ) S f = f − q 0 ∫ R f . (65) Every bounded even solution of ( I − P s ) u = f (I-P_s)u=f ( I − P s ) u = f f ∈ Y η f\in\mathcal{Y}_\eta f ∈ Y η ∣ Re s ∣ < 2 |\operatorname{Re}s|<2 ∣ Re s ∣ < 2 w = f + R s u w=f+R_su w = f + R s u ( I − Q ) u = w (I-Q)u=w ( I − Q ) u = w w w w χ \chi χ ∥ ⋅ ∥ 1 \|\cdot\|_1 ∥ ⋅ ∥ 1 L 1 ( R ) L^1(\mathbb{R}) L 1 ( R ) χ ( t / R ) \chi(t/R) χ ( t / R )
∣ ∫ w ( t ) χ ( t / R ) d t ∣ ≤ ∥ u ∥ ∞ ∥ ( I − Q ) χ ( ⋅ / R ) ∥ 1 ≤ σ 0 2 2 ∥ u ∥ ∞ ∥ ∂ t 2 χ ( ⋅ / R ) ∥ 1 = O ( R − 1 ) . \left|\int w(t)\chi(t/R)\,dt\right|\leq\|u\|_\infty\|(I-Q)\chi(\cdot/R)\|_1\leq\frac{\sigma_0^2}{2}\|u\|_\infty\|\partial_t^2\chi(\cdot/R)\|_1=O(R^{-1}). ∫ w ( t ) χ ( t / R ) d t ≤ ∥ u ∥ ∞ ∥ ( I − Q ) χ ( ⋅ / R ) ∥ 1 ≤ 2 σ 0 2 ∥ u ∥ ∞ ∥ ∂ t 2 χ ( ⋅ / R ) ∥ 1 = O ( R − 1 ) . Thus ∫ w = 0 \int w=0 ∫ w = 0 (65) makes u − S w u-Sw u − S w Q Q Q
u = c + S ( f + R s u ) , (66) u=c+S(f+R_su), \tag*{(66)} u = c + S ( f + R s u ) , (66) where c ∈ C c\in\mathbb{C} c ∈ C X \mathcal{X} X φ ( x ( t ) ) = x ( t ) a ( t ) u ( t ) \varphi(x(t))=\sqrt{x(t)a(t)}u(t) φ ( x ( t )) = x ( t ) a ( t ) u ( t ) u u u
Proof of Theorem 6.1
Proof of Theorem 6.1. Comparison on finite intervals. The Dirichlet Green kernel of L 0 L_0 L 0 I R = ( − R , R ) I_R=(-R,R) I R = ( − R , R )
g R ( t , v ) = 2 sinh ( ( min ( t , v ) + R ) / 2 ) sinh ( ( R − max ( t , v ) ) / 2 ) sinh R . g_R(t,v)=\frac{2\sinh((\min(t,v)+R)/2)\sinh((R-\max(t,v))/2)}{\sinh R}. g R ( t , v ) = sinh R 2 sinh (( min ( t , v ) + R ) /2 ) sinh (( R − max ( t , v )) /2 ) . The operator K R , s K_{R,s} K R , s s s s r R ( s ) r_R(s) r R ( s ) R R R r ∞ ( s ) = sup R r R ( s ) r_\infty(s) = \sup_R r_R(s) r ∞ ( s ) = sup R r R ( s ) T T T T s T = T − s T_s^T = T_{-s} T s T = T − s A B AB A B B A BA B A r R ( s ) = r R ( − s ) r_R(s) = r_R(-s) r R ( s ) = r R ( − s ) s s s r ∞ r_\infty r ∞
After translating I R I_R I R K R , s K_{R,s} K R , s T s ( t , u ) T_s(t,u) T s ( t , u ) H ( t − u ) H(t-u) H ( t − u ) R → ∞ R \to\infty R → ∞ L 0 L_0 L 0 χ R ( t ) = χ ( t / R ) \chi_R(t) = \chi(t/R) χ R ( t ) = χ ( t / R )
∬ H ( t − u ) χ R ( t ) χ R ( u ) d t d u ∫ ( ∣ χ R ′ ∣ 2 + 1 4 χ R 2 ) d t ⟶ 1 , \frac{\iint H(t-u)\chi_R(t)\chi_R(u)\,\mathrm{d}t\,\mathrm{d}u}{\int\left(|\chi_R'|^2+\frac{1}{4}\chi_R^2\right)\,\mathrm{d}t} \longrightarrow1, ∫ ( ∣ χ R ′ ∣ 2 + 4 1 χ R 2 ) d t ∬ H ( t − u ) χ R ( t ) χ R ( u ) d t d u ⟶ 1 , since ∫ H = 1 / 4 \int H = 1/4 ∫ H = 1/4 Q Q Q r ∞ ( s ) ≥ 1 r_\infty(s) \ge1 r ∞ ( s ) ≥ 1 s s s ∣ s ∣ ≤ 1 |s| \le1 ∣ s ∣ ≤ 1 W s ≤ 1 W_s \le1 W s ≤ 1 P s ≤ Q P_s \le Q P s ≤ Q r ∞ ( s ) = 1 r_\infty(s) = 1 r ∞ ( s ) = 1 r ∞ ( s ) = 1 r_\infty(s) = 1 r ∞ ( s ) = 1 g R g_R g R e ∣ t − u ∣ / 2 e^{|t-u|/2} e ∣ t − u ∣/2 v s v_s v s
v s ( 0 ) = 1 , P s v s ≤ v s , v s ( t ) ≥ e − ∣ t ∣ / 2 . (67) v_s(0)=1,\qquad P_s v_s\le v_s,\qquad v_s(t)\ge e^{-|t|/2}. \tag*{(67)} v s ( 0 ) = 1 , P s v s ≤ v s , v s ( t ) ≥ e − ∣ t ∣/2 . (67) Fatou’s lemma gives the inequality.
A return estimate. We need a lower bound at infinity for positive supersolutions, uniform in the sign and in 1 ≤ s < 2 1 \le s < 2 1 ≤ s < 2 M f M_f M f f f f g g g 1 ≤ s < 2 1 \le s < 2 1 ≤ s < 2 a a a W s W_s W s
P s ≥ K + : = g M a 3 ( H ∗ ) , P − s ≥ K − : = Q M a 3 , K ± ≤ Q . P_s \ge K_+ := gM_{a^3}(H*),\qquad P_{-s}\ge K_- := QM_{a^3},\qquad K_\pm\le Q. P s ≥ K + := g M a 3 ( H ∗ ) , P − s ≥ K − := Q M a 3 , K ± ≤ Q . Both kernels are obtained from the q 0 q_0 q 0 h r e t ( t ) = 1 + exp ( min ( t , 0 ) / 4 ) h_{\mathrm{ret}}(t)=1+\exp(\min(t,0)/4) h ret ( t ) = 1 + exp ( min ( t , 0 ) /4 ) ∫ e v / 4 q 0 ( v ) d v = 3 / 2 \int e^{v/4}q_0(v)\,\mathrm{d}v=3/2 ∫ e v /4 q 0 ( v ) d v = 3/2
Q h r e t ( t ) − h r e t ( t ) ≥ 1 2 e t / 4 − C e t / 2 , t ≤ 0. Qh_{\mathrm{ret}}(t)-h_{\mathrm{ret}}(t)\ge\frac{1}{2}e^{t/4}-Ce^{t/2},\qquad t\le0. Q h ret ( t ) − h ret ( t ) ≥ 2 1 e t /4 − C e t /2 , t ≤ 0. Since 1 − a ( u ) 3 ≤ 3 min ( e u , 1 ) 1-a(u)^3\le3\min(e^u,1) 1 − a ( u ) 3 ≤ 3 min ( e u , 1 ) ∫ e − ∣ t − u ∣ / 2 min ( e u , 1 ) d u ≤ 4 e t / 2 \int e^{-|t-u|/2}\min(e^u,1)\,\mathrm{d}u\le4e^{t/2} ∫ e − ∣ t − u ∣/2 min ( e u , 1 ) d u ≤ 4 e t /2 ( Q − K ϵ ) h r e t (Q-K_\epsilon)h_{\mathrm{ret}} ( Q − K ϵ ) h ret C e t / 2 Ce^{t/2} C e t /2 R 0 > 0 R_0>0 R 0 > 0
K ϵ h r e t ( t ) − h r e t ( t ) ≥ 1 2 e t / 4 − C e t / 2 > 0 , t ≤ − R 0 . (68) K_\epsilon h_{\mathrm{ret}}(t)-h_{\mathrm{ret}}(t)\ge\frac{1}{2}e^{t/4}-Ce^{t/2}>0,\qquad t\le-R_0. \tag*{(68)} K ϵ h ret ( t ) − h ret ( t ) ≥ 2 1 e t /4 − C e t /2 > 0 , t ≤ − R 0 . (68) For the chain ( X n ) n ≥ 0 (X_n)_{n\ge0} ( X n ) n ≥ 0 K ϵ K_\epsilon K ϵ P t K ϵ P_t^{K_\epsilon} P t K ϵ t t t τ \tau τ [ − R 0 , ∞ ) [-R_0,\infty) [ − R 0 , ∞ ) ζ \zeta ζ q 0 q_0 q 0 τ ∧ ζ \tau\wedge\zeta τ ∧ ζ P t K ϵ ( τ < ζ ) ≥ 1 / 2 P_t^{K_\epsilon}(\tau<\zeta)\ge1/2 P t K ϵ ( τ < ζ ) ≥ 1/2 t < − R 0 t<-R_0 t < − R 0 J > 0 J>0 J > 0
sup b ≥ 0 ∫ b + J ∞ q 0 ( v ) d v ∫ b ∞ q 0 ( v ) d v ≤ C e − J / 2 \sup_{b\ge0}\frac{\int_{b+J}^{\infty}q_0(v)\,\mathrm{d}v}{\int_b^{\infty}q_0(v)\,\mathrm{d}v}\le Ce^{-J/2} b ≥ 0 sup ∫ b ∞ q 0 ( v ) d v ∫ b + J ∞ q 0 ( v ) d v ≤ C e − J /2 bounds the overshoot of the walk with increment density q 0 q_0 q 0 J J J
P t K ϵ ( X τ ∈ [ − R 0 , − R 0 + J ] , τ < ζ ) ≥ 1 4 , t < − R 0 . (69) P_t^{K_\epsilon}(X_\tau\in[-R_0,-R_0+J],\ \tau<\zeta)\ge\frac{1}{4},\qquad t<-R_0. \tag*{(69)} P t K ϵ ( X τ ∈ [ − R 0 , − R 0 + J ] , τ < ζ ) ≥ 4 1 , t < − R 0 . (69) Optional stopping now shows that every positive continuous even P ε s P_{\varepsilon s} P ε s [ − R 0 , − R 0 + J ] [-R_0,-R_0+J] [ − R 0 , − R 0 + J ]
We use this estimate to show that r R ( s ) > 1 r_R(s)>1 r R ( s ) > 1 R > 0 R>0 R > 0 s < 2 s<2 s < 2 r ∞ ( s ) = 1 r_\infty(s)=1 r ∞ ( s ) = 1 1 ≤ s < 2 1\le s<2 1 ≤ s < 2 (69) to the supersolution in (67) ; then v s ( t ) ≥ c 0 > 0 v_s(t)\ge c_0>0 v s ( t ) ≥ c 0 > 0 t < − R 0 t<-R_0 t < − R 0 s s s v ∈ [ 0 , 1 ] v\in[0,1] v ∈ [ 0 , 1 ] x x x W s W_s W s P s ( 0 , u ) ≥ c 1 e ( 2 − s ) u P_s(0,u)\ge c_1e^{(2-s)u} P s ( 0 , u ) ≥ c 1 e ( 2 − s ) u u < − R 0 u<-R_0 u < − R 0 c 1 > 0 c_1>0 c 1 > 0 s s s 1 ≥ c 0 c 1 e − R 0 ( 2 − s ) / ( 2 − s ) 1\ge c_0c_1e^{-R_0(2-s)}/(2-s) 1 ≥ c 0 c 1 e − R 0 ( 2 − s ) / ( 2 − s ) s ↑ 2 s\uparrow2 s ↑ 2 s ˉ ∈ ( 1 , 2 ) \bar{s}\in(1,2) s ˉ ∈ ( 1 , 2 ) r ∞ ( s ˉ ) > 1 r_\infty(\bar{s})>1 r ∞ ( s ˉ ) > 1 r R ( s ) = r R ( − s ) r_R(s)=r_R(-s) r R ( s ) = r R ( − s ) − s ˉ -\bar{s} − s ˉ
Bounded Green potentials below the threshold. Choose η > 0 \eta>0 η > 0 R s R_s R s s ∈ [ − s ˉ , s ˉ ] s\in[-\bar{s},\bar{s}] s ∈ [ − s ˉ , s ˉ ] w η ( t ) = e − η ∣ t ∣ w_\eta(t)=e^{-\eta|t|} w η ( t ) = e − η ∣ t ∣ ∑ n ≥ 0 P s n w η \sum_{n\ge0}P_s^nw_\eta ∑ n ≥ 0 P s n w η s = 0 , ± 1 s=0,\pm1 s = 0 , ± 1 Q Q Q v v v e − η ∣ t ∣ e^{-\eta|t|} e − η ∣ t ∣ 0 < v ≤ 1 0<v\le1 0 < v ≤ 1 Q v − v ≥ c w η Qv-v\ge c w_\eta Q v − v ≥ c w η q 0 q_0 q 0 η \eta η P P P C C C
( I − P ) ( C − v ) ≥ ( Q v − v ) + ( C − 1 ) ( 1 − P 1 ) ≥ c ′ w η (I-P)(C-v)\ge(Qv-v)+(C-1)(1-P1)\ge c'w_\eta ( I − P ) ( C − v ) ≥ ( Q v − v ) + ( C − 1 ) ( 1 − P 1 ) ≥ c ′ w η everywhere. This supersolution bounds the minimal potential. Recurrence of the walk with increment density q 0 q_0 q 0 (66) , the resulting Green operators G 0 , G ± 1 \mathcal{G}_0,\mathcal{G}_{\pm1} G 0 , G ± 1 Y η \mathcal{Y}_\eta Y η X \mathcal{X} X
For ε = ± 1 \varepsilon=\pm1 ε = ± 1
h ε , s = ∑ n ≥ 0 P ε s n w η , b ε = sup { s ∈ [ 0 , s ˉ ] : ∥ h ε , s ∥ ∞ < ∞ } . h_{\varepsilon,s}=\sum_{n\ge0}P_{\varepsilon s}^nw_\eta,\qquad b_\varepsilon=\sup\{s\in[0,\bar{s}]:\|h_{\varepsilon,s}\|_\infty<\infty\}. h ε , s = n ≥ 0 ∑ P ε s n w η , b ε = sup { s ∈ [ 0 , s ˉ ] : ∥ h ε , s ∥ ∞ < ∞ } . Hölder’s inequality applied to the iterated integrals for the positive kernels shows that the set of parameters for which h ε , s h_{\varepsilon,s} h ε , s h ε , s h_{\varepsilon,s} h ε , s ∣ f ∣ ≤ ∥ f ∥ η w η |f|\le\|f\|_\eta w_\eta ∣ f ∣ ≤ ∥ f ∥ η w η G ε s : Y η → X \mathcal{G}_{\varepsilon s}:\mathcal{Y}_\eta\to\mathcal{X} G ε s : Y η → X h ε , s h_{\varepsilon,s} h ε , s s ′ s' s ′ s s s
[ I − G ε s ∣ P ε s ′ − P ε s ∣ ] − 1 h ε , s [I-\mathcal{G}_{\varepsilon s}|P_{\varepsilon s'}-P_{\varepsilon s}|]^{-1}h_{\varepsilon,s} [ I − G ε s ∣ P ε s ′ − P ε s ∣ ] − 1 h ε , s is a bounded supersolution for the potential at s ′ s' s ′ O ( ∣ s ′ − s ∣ ) O(|s'-s|) O ( ∣ s ′ − s ∣ ) Y η \mathcal{Y}_\eta Y η b ε > 1 b_\varepsilon>1 b ε > 1 r R ( ε s ) < 1 r_R(\varepsilon s)<1 r R ( ε s ) < 1 I R I_R I R K R , ε s K_{R,\varepsilon s} K R , ε s r R ( ε s ˉ ) > 1 r_R(\varepsilon\bar{s})>1 r R ( ε s ˉ ) > 1 R R R r R r_R r R b ε < s ˉ b_\varepsilon<\bar{s} b ε < s ˉ
For s < b ε s<b_\varepsilon s < b ε P ε s u = u P_{\varepsilon s}u=u P ε s u = u s ≤ 1 s\le1 s ≤ 1 Q Q Q s ≥ 1 s\ge1 s ≥ 1 (69) bounds h ε , s h_{\varepsilon,s} h ε , s P ε s P_{\varepsilon s} P ε s P ε s n h ε , s = ∑ k ≥ n P ε s k w η → 0 P_{\varepsilon s}^nh_{\varepsilon,s}=\sum_{k\ge n}P_{\varepsilon s}^kw_\eta\to0 P ε s n h ε , s = ∑ k ≥ n P ε s k w η → 0
For z z z [ − s ˉ , s ˉ ] [-\bar{s},\bar{s}] [ − s ˉ , s ˉ ]
A z = G 0 ( P z − P 0 ) : X → X , G z = ( I − A z ) − 1 G 0 . (70) \mathcal{A}_z=\mathcal{G}_0(P_z-P_0):\mathcal{X}\to\mathcal{X},\qquad\mathcal{G}_z=(I-\mathcal{A}_z)^{-1}\mathcal{G}_0. \tag*{(70)} A z = G 0 ( P z − P 0 ) : X → X , G z = ( I − A z ) − 1 G 0 . (70) The second identity holds wherever I − A z I-\mathcal{A}_z I − A z η \eta η s = 0 s=0 s = 0 ( I − P z ) u = f (I-P_z)u=f ( I − P z ) u = f ( I − A z ) u = G 0 f (I-\mathcal{A}_z)u=\mathcal{G}_0f ( I − A z ) u = G 0 f b ε b_\varepsilon b ε
A positive bounded solution at the threshold. As s ↑ b ε s \uparrow b_\varepsilon s ↑ b ε ∥ h ε , s ∥ ∞ \|h_{\varepsilon,s}\|_\infty ∥ h ε , s ∥ ∞ b ε b_\varepsilon b ε h ε , s = G 0 w η + A ε , s h ε , s h_{\varepsilon,s}=G_0w_\eta+A_{\varepsilon,s}h_{\varepsilon,s} h ε , s = G 0 w η + A ε , s h ε , s P ε b ε u = u P_{\varepsilon b_\varepsilon}u=u P ε b ε u = u X X X (69) and (66) give a positive common limit at both ends and an exponentially decaying remainder. Normalize that limit to one and denote the resulting function by ψ ε \psi_\varepsilon ψ ε
Recurrence and equality of the thresholds. The Doob transform is the Markov kernel
P ~ ε ( t , u ) = P ε b ε ( t , u ) ψ ε ( u ) ψ ε ( t ) . (71) \widetilde P_\varepsilon(t,u)=P_{\varepsilon b_\varepsilon}(t,u)\frac{\psi_\varepsilon(u)}{\psi_\varepsilon(t)}. \tag*{(71)} P ε ( t , u ) = P ε b ε ( t , u ) ψ ε ( t ) ψ ε ( u ) . (71) Because ψ ε − 1 \psi_\varepsilon-1 ψ ε − 1 R ε b ε R_{\varepsilon b_\varepsilon} R ε b ε P ~ ε \widetilde P_\varepsilon P ε σ 0 2 \sigma_0^2 σ 0 2 Q Q Q P ~ ε \widetilde P_\varepsilon P ε P ε b ε u = u P_{\varepsilon b_\varepsilon}u=u P ε b ε u = u ψ ε \psi_\varepsilon ψ ε
The positive bounded solution gives r ∞ ( ε b ε ) = 1 r_\infty(\varepsilon b_\varepsilon)=1 r ∞ ( ε b ε ) = 1 b ε < s < 2 b_\varepsilon<s<2 b ε < s < 2 r ∞ ( ε s ) = 1 r_\infty(\varepsilon s)=1 r ∞ ( ε s ) = 1 v ε s v_{\varepsilon s} v ε s (67) and put θ = b ε / s \theta=b_\varepsilon/s θ = b ε / s W s W_s W s
P ε b ε ( v ε s θ ) ≤ ( P ε s v ε s ) θ ( P 0 1 ) 1 − θ < v ε s θ . P_{\varepsilon b_\varepsilon}(v_{\varepsilon s}^{\theta})\leq(P_{\varepsilon s}v_{\varepsilon s})^\theta(P_0 1)^{1-\theta}<v_{\varepsilon s}^{\theta}. P ε b ε ( v ε s θ ) ≤ ( P ε s v ε s ) θ ( P 0 1 ) 1 − θ < v ε s θ . Here P 0 1 ( t ) < 1 P_0 1(t)<1 P 0 1 ( t ) < 1 t t t ψ ε \psi_\varepsilon ψ ε r ∞ ( ε s ) > 1 r_\infty(\varepsilon s)>1 r ∞ ( ε s ) > 1 s > b ε s>b_\varepsilon s > b ε s ≥ 2 s\geq2 s ≥ 2 b ε = inf { s > 0 : sup R r R ( ε s ) > 1 } b_\varepsilon=\inf\{s>0:\sup_R r_R(\varepsilon s)>1\} b ε = inf { s > 0 : sup R r R ( ε s ) > 1 } r R ( s ) = r R ( − s ) r_R(s)=r_R(-s) r R ( s ) = r R ( − s ) b + = b − = : s ∗ b_+=b_-=:s_* b + = b − =: s ∗ (58) . A positive bounded solution at any larger parameter would contradict r R > 1 r_R>1 r R > 1 ε s ∗ \varepsilon s_* ε s ∗ G s G_s G s s = ε s ∗ s=\varepsilon s_* s = ε s ∗
Proof of Proposition 1.2
To prove Proposition 1.2, we separate the limit at infinity from the decaying part of a bounded solution of P s ψ = ψ P_s\psi=\psi P s ψ = ψ D ( s ) \mathcal{D}(s) D ( s )
The bordered determinant
For ∣ Re s ∣ < 2 |\operatorname{Re}s|<2 ∣ Re s ∣ < 2 P s ψ = ψ P_s\psi=\psi P s ψ = ψ ψ = c + u \psi=c+u ψ = c + u c c c u u u R s = P s − Q R_s=P_s-Q R s = P s − Q S S S (66) , the equation P s ψ = ψ P_s\psi=\psi P s ψ = ψ
u = S R s ( c + u ) , ∫ R R s ( c + u ) = 0. u=SR_s(c+u),\qquad\int_{\mathbb{R}}R_s(c+u)=0. u = S R s ( c + u ) , ∫ R R s ( c + u ) = 0. The second identity is the solvability condition for ( I − Q ) ψ = R s ψ (I-Q)\psi=R_s\psi ( I − Q ) ψ = R s ψ
K s = S R s , b s = K s 1 , c s ( u ) = ∫ R R s ( t , u ) d t , \mathcal{K}_s=SR_s,\qquad b_s=\mathcal{K}_s1,\qquad c_s(u)=\int_{\mathbb{R}}R_s(t,u)\,dt, K s = S R s , b s = K s 1 , c s ( u ) = ∫ R R s ( t , u ) d t , d s = ∫ R c s ( u ) d u , ℓ s ( f ) = ∫ R c s ( u ) f ( u ) d u . (72) d_s=\int_{\mathbb{R}}c_s(u)\,du,\qquad\ell_s(f)=\int_{\mathbb{R}}c_s(u)f(u)\,du. \tag*{(72)} d s = ∫ R c s ( u ) d u , ℓ s ( f ) = ∫ R c s ( u ) f ( u ) d u . (72) Here ℓ s \ell_s ℓ s 1 1 1 C ⊕ L e v e n 2 ( R ) \mathbb{C}\oplus L^2_{\mathrm{even}}(\mathbb{R}) C ⊕ L even 2 ( R )
D ( s ) = det ( d s ℓ s − b s I − K s ) , ∣ Re s ∣ < 2. (73) \mathcal{D}(s)=\det\begin{pmatrix} d_s & \ell_s \\ -b_s & I-\mathcal{K}_s \end{pmatrix},\qquad|\operatorname{Re}s|<2. \tag*{(73)} D ( s ) = det ( d s − b s ℓ s I − K s ) , ∣ Re s ∣ < 2. (73) This block operator is the identity plus a trace-class operator. Indeed, the kernel of S R s S R_s S R s K s \mathcal{K}_s K s ( 1 − ∂ t 2 + t 2 ) − 1 (1-\partial_t^2+t^2)^{-1} ( 1 − ∂ t 2 + t 2 ) − 1
All entries are given by explicit kernels and their integrals. If ( c , u ) (c,u) ( c , u ) K s \mathcal{K}_s K s u u u ( I − P s ) ( c + u ) = 0 (I-P_s)(c+u)=0 ( I − P s ) ( c + u ) = 0 P s ψ = ψ P_s\psi=\psi P s ψ = ψ
D ( s ) = 0 ⟺ P s u = u has a nonzero bounded even solution. (74) \mathcal{D}(s)=0\quad\Longleftrightarrow\quad P_su=u\text{ has a nonzero bounded even solution.} \tag*{(74)} D ( s ) = 0 ⟺ P s u = u has a nonzero bounded even solution. (74) The scalar row and column account for the constant solutions of ( I − Q ) u = 0 (I-Q)u=0 ( I − Q ) u = 0 Theorem 6.1 shows that s ∗ s_* s ∗ D \mathcal{D} D
Simplicity of the first positive zero
Theorem 6.1 gives a one-dimensional space of bounded solutions of P s ∗ u = u P_{s_*}u=u P s ∗ u = u s ∗ s_* s ∗ D \mathcal{D} D P s ∗ ′ ψ P'_{s_*}\psi P s ∗ ′ ψ B ∗ B^* B ∗ B s ′ B'_s B s ′ s s s
Proposition A.1. The determinant D \mathcal{D} D s ∗ s_* s ∗ ψ , ϕ \psi,\phi ψ , ϕ P s ∗ ψ = ψ P_{s_*}\psi=\psi P s ∗ ψ = ψ P s ∗ ∗ ϕ = ϕ P^*_{s_*}\phi=\phi P s ∗ ∗ ϕ = ϕ f ∈ Y η f\in\mathcal{Y}_\eta f ∈ Y η η > 0 \eta>0 η > 0
G s f = ψ ∫ R ϕ f M ( s ∗ − s ) + O X ( 1 ) , M = ∫ R ϕ P s ∗ ′ ψ > 0. (75) G_sf=\frac{\psi\int_{\mathbb{R}}\phi f}{M(s_*-s)}+O_{\mathcal{X}}(1),\qquad M=\int_{\mathbb{R}}\phi P'_{s_*}\psi>0. \tag*{(75)} G s f = M ( s ∗ − s ) ψ ∫ R ϕ f + O X ( 1 ) , M = ∫ R ϕ P s ∗ ′ ψ > 0. (75) Here O X ( 1 ) O_{\mathcal{X}}(1) O X ( 1 ) X \mathcal{X} X s ∗ s_* s ∗
Proof. The adjoint solution and the block operator. Normalize the solutions ψ ± \psi_\pm ψ ± Theorem 6.1 to have limit one at infinity. Set ψ = ψ + \psi=\psi_+ ψ = ψ +
ϕ = T − s ∗ ψ − . \phi=T_{-s_*}\psi_-. ϕ = T − s ∗ ψ − . Since g ∗ ϕ = ψ − g*\phi=\psi_- g ∗ ϕ = ψ − P s ∗ ∗ = T − s ∗ g ∗ P^*_{s_*}=T_{-s_*}g^* P s ∗ ∗ = T − s ∗ g ∗ P s ∗ ∗ ϕ = ϕ P^*_{s_*}\phi=\phi P s ∗ ∗ ϕ = ϕ T − s ∗ T_{-s_*} T − s ∗ ϕ > 0 \phi>0 ϕ > 0 T − s ∗ − H ∗ T_{-s_*}-H^* T − s ∗ − H ∗ ψ − − 1 \psi_--1 ψ − − 1
ϕ ( t ) = 1 4 + O ( e − η ∣ t ∣ ) \phi(t)=\frac{1}{4}+O(e^{-\eta|t|}) ϕ ( t ) = 4 1 + O ( e − η ∣ t ∣ ) for sufficiently small η > 0 \eta>0 η > 0 ψ \psi ψ ϕ \phi ϕ
To compute the coefficient of ( s ∗ − s ) − 1 (s_*-s)^{-1} ( s ∗ − s ) − 1 B s \mathcal{B}_s B s v = − R s ∗ ∗ ϕ v=-R^*_{s_*}\phi v = − R s ∗ ∗ ϕ ( I − Q ) ϕ = R s ∗ ∗ ϕ (I-Q)\phi=R^*_{s_*}\phi ( I − Q ) ϕ = R s ∗ ∗ ϕ ∫ v = 0 \int v=0 ∫ v = 0 ( I − Q ) ( ϕ − 1 / 4 ) (I-Q)(\phi-1/4) ( I − Q ) ( ϕ − 1/4 ) S ∗ S^* S ∗
S ∗ v = v − A ∗ v + ∫ R A ( t ) v ( t ) d t , ( I − Q ) S ∗ v = v . S^*v=v-A*v+\int_{\mathbb{R}}A(t)v(t)\,dt,\qquad(I-Q)S^*v=v. S ∗ v = v − A ∗ v + ∫ R A ( t ) v ( t ) d t , ( I − Q ) S ∗ v = v . Evenness and ∫ v = 0 \int v=0 ∫ v = 0 S ∗ v S^{*}v S ∗ v ϕ + S ∗ v \phi+S^{*}v ϕ + S ∗ v Q Q Q β \beta β
v = K s ∗ ∗ v − β c s ∗ , β d s ∗ − ∫ v b s ∗ = 0. v=\mathcal{K}_{s_*}^{*}v-\beta c_{s_*},\qquad\beta d_{s_*}-\int vb_{s_*}=0. v = K s ∗ ∗ v − β c s ∗ , β d s ∗ − ∫ v b s ∗ = 0. The second identity follows by integrating the first. Hence ( β , v ) (\beta,v) ( β , v ) B s ∗ B_{s_*} B s ∗ ( I − P s ) u = f (I-P_s)u=f ( I − P s ) u = f f ∈ Y η f\in\mathcal{Y}_{\eta} f ∈ Y η B s B_s B s ( − ∫ f , S f ) (-\int f,Sf) ( − ∫ f , S f ) ( β , v ) (\beta,v) ( β , v ) − ∫ ϕ f -\int\phi f − ∫ ϕ f
The derivative pairing. For one application of P s P_s P s y y y p ∈ { x , a } p\in\{x,a\} p ∈ { x , a } s = 0 s=0 s = 0 R 2 × { x , a } \mathbb{R}^{2}\times\{x,a\} R 2 × { x , a }
μ t ( d y d u , p ) = g ( t − y ) H ( y − u ) p ( y ) p ( u ) d y d u , m 0 ( t ) = μ t ( 1 ) = P 01 ( t ) < 1. \mu_t(dy\,du,p)=g(t-y)H(y-u)p(y)p(u)\,dy\,du,\qquad m_0(t)=\mu_t(1)=P_{01}(t)<1. μ t ( d y d u , p ) = g ( t − y ) H ( y − u ) p ( y ) p ( u ) d y d u , m 0 ( t ) = μ t ( 1 ) = P 01 ( t ) < 1. Here μ t ( 1 ) \mu_t(1) μ t ( 1 ) μ t \mu_t μ t e s ∗ [ log p ( y ) − log p ( u ) ] e^{s_*[\log p(y)-\log p(u)]} e s ∗ [ l o g p ( y ) − l o g p ( u )] ψ \psi ψ
μ ~ t ( d y d u , p ) = e s ∗ [ log p ( y ) − log p ( u ) ] ψ ( u ) ψ ( t ) μ t ( d y d u , p ) . \widetilde{\mu}_t(dy\,du,p)=e^{s_*[\log p(y)-\log p(u)]}\frac{\psi(u)}{\psi(t)}\mu_t(dy\,du,p). μ t ( d y d u , p ) = e s ∗ [ l o g p ( y ) − l o g p ( u )] ψ ( t ) ψ ( u ) μ t ( d y d u , p ) . The Markov kernel t ↦ μ ~ t ( u ∈ ⋅ ) t\mapsto\widetilde{\mu}_t(u\in\cdot) t ↦ μ t ( u ∈ ⋅ ) π ( t ) d t = ψ ( t ) ϕ ( t ) d t \pi(t)dt=\psi(t)\phi(t)dt π ( t ) d t = ψ ( t ) ϕ ( t ) d t ν \nu ν μ \mu μ ν ≪ μ \nu\ll\mu ν ≪ μ
D ( ν ∥ μ ) = ∫ log d ν d μ d ν = D ( ν ∥ μ / μ ( 1 ) ) − log μ ( 1 ) . D(\nu\Vert\mu)=\int\log\frac{d\nu}{d\mu}\,d\nu=D(\nu\Vert\mu/\mu(1))-\log\mu(1). D ( ν ∥ μ ) = ∫ log d μ d ν d ν = D ( ν ∥ μ / μ ( 1 )) − log μ ( 1 ) . Relative entropy and invariance give
s ∗ ∫ R ϕ P s ∗ ′ ψ = ∫ R π ( t ) D ( μ ~ t ∥ μ t ) d t ≥ ∫ R π ( t ) [ − log m 0 ( t ) ] d t > 0. (76) s_*\int_{\mathbb{R}}\phi P_{s_*}'\psi=\int_{\mathbb{R}}\pi(t)D(\widetilde{\mu}_t\Vert\mu_t)\,dt\geq\int_{\mathbb{R}}\pi(t)[-\log m_0(t)]\,dt>0. \tag*{(76)} s ∗ ∫ R ϕ P s ∗ ′ ψ = ∫ R π ( t ) D ( μ t ∥ μ t ) d t ≥ ∫ R π ( t ) [ − log m 0 ( t )] d t > 0. (76) The terms involving log ψ \log\psi log ψ p p p
∑ p = x , a ∬ H ( y − u ) p ( y ) 1 + s p ( u ) 1 − s ∣ log p ( y ) − log p ( u ) ∣ d y d u < ∞ , 0 < s < 2. \sum_{p=x,a}\iint H(y-u)p(y)^{1+s}p(u)^{1-s}\lvert\log p(y)-\log p(u)\rvert\,dy\,du<\infty,\qquad0<s<2. p = x , a ∑ ∬ H ( y − u ) p ( y ) 1 + s p ( u ) 1 − s ∣ log p ( y ) − log p ( u )∣ d y d u < ∞ , 0 < s < 2. In u ≥ y ≥ 0 u\geq y\geq0 u ≥ y ≥ 0 a a a C e − 2 y − ( 2 − s ) ( u − y ) ( u − y ) Ce^{-2y-(2-s)(u-y)}(u-y) C e − 2 y − ( 2 − s ) ( u − y ) ( u − y ) x x x e − y min ( 1 , u − y ) e^{-y}\min(1,u-y) e − y min ( 1 , u − y ) log ψ \log\psi log ψ log ψ ( u ) − log ψ ( t ) \log\psi(u)-\log\psi(t) log ψ ( u ) − log ψ ( t )
The simple pole. The right null vector of B s ∗ B_{s_*} B s ∗ ( 1 , ψ − 1 ) (1,\psi-1) ( 1 , ψ − 1 ) B s ∗ ′ B_{s_*}' B s ∗ ′
⟨ ( β , v ) , B s ∗ ′ ( 1 , ψ − 1 ) ⟩ = β ∫ R s ∗ ′ ψ − ∫ v S R s ∗ ′ ψ = ∫ ϕ P s ∗ ′ ψ = M > 0. \langle(\beta,v),B_{s_*}'(1,\psi-1)\rangle=\beta\int R_{s_*}'\psi-\int vSR_{s_*}'\psi=\int\phi P_{s_*}'\psi=M>0. ⟨( β , v ) , B s ∗ ′ ( 1 , ψ − 1 )⟩ = β ∫ R s ∗ ′ ψ − ∫ v S R s ∗ ′ ψ = ∫ ϕ P s ∗ ′ ψ = M > 0. The nonzero derivative pairing and the one-dimensional kernel give a simple zero of the determinant. The pairing − ∫ ϕ f -\int\phi f − ∫ ϕ f
Numerical evaluation
We approximate the positive bounded even solution ψ \psi ψ L 0 ψ = T s ∗ ψ L_0\psi=T_{s_*}\psi L 0 ψ = T s ∗ ψ R > 0 R>0 R > 0 ψ ( t ) = 1 \psi(t)=1 ψ ( t ) = 1 t ≥ R t\ge R t ≥ R
ψ ( t ) = 1 + ∫ t R ( u − t ) h ( u ) d u , h = ψ ′ ′ , 0 ≤ t ≤ R . \psi(t)=1+\int_t^R (u-t)h(u)\,\mathrm{d}u,\qquad h=\psi'',\qquad0\le t\le R. ψ ( t ) = 1 + ∫ t R ( u − t ) h ( u ) d u , h = ψ ′′ , 0 ≤ t ≤ R . We extend ψ \psi ψ R \mathbb{R} R h = ψ / 4 − T s ψ h=\psi/4-T_s\psi h = ψ /4 − T s ψ T s 1 T_s1 T s 1 R + 70 R+70 R + 70 s s s h h h ψ ′ ( 0 ) = 0 \psi'(0)=0 ψ ′ ( 0 ) = 0 ∫ 0 R h ( u ) d u = 0 \int_0^R h(u)\,\mathrm{d}u=0 ∫ 0 R h ( u ) d u = 0 s s s
Table 1.
Refining the quadrature and extending the integration range leave the displayed digits for R = 36 R=36 R = 36 s ∗ ≈ 1.5993283761 s_*\approx1.5993283761 s ∗ ≈ 1.5993283761 ξ 3 ≈ 0.76928375 \xi_3\approx0.76928375 ξ 3 ≈ 0.76928375 (5) .
The candidate exponent for general CLEκ _\kappa κ
The welding calculation in § 3 gives an explicit integral operator for every CLEκ _\kappa κ 4 < κ < 8 4<\kappa<8 4 < κ < 8 κ _\kappa κ
Set
α = κ 4 ∈ ( 1 , 2 ) , γ = 4 κ , e = α − 1 , d = 2 − α , q = − 2 cos π α . (77) \alpha=\frac{\kappa}{4}\in(1,2),\qquad\gamma=\frac{4}{\sqrt{\kappa}},\qquad e=\alpha-1,\qquad d=2-\alpha,\qquad\sqrt{q}=-2\cos\frac{\pi}{\alpha}. \tag*{(77)} α = 4 κ ∈ ( 1 , 2 ) , γ = κ 4 , e = α − 1 , d = 2 − α , q = − 2 cos α π . (77) Here γ \gamma γ q ∈ ( 0 , 4 ) q\in(0,4) q ∈ ( 0 , 4 ) x ( t ) = e t / ( 1 + e t ) x(t)=e^t/(1+e^t) x ( t ) = e t / ( 1 + e t ) a ( t ) = 1 − x ( t ) a(t)=1-x(t) a ( t ) = 1 − x ( t ) L 0 = − d 2 / d t 2 + 1 / 4 L_0=-\mathrm{d}^2/\mathrm{d}t^2+1/4 L 0 = − d 2 / d t 2 + 1/4
C α = − α sin ( 2 π / α ) π , C_\alpha=-\frac{\alpha\sin(2\pi/\alpha)}{\pi}, C α = − π α sin ( 2 π / α ) , H α ( t ) = C α α 2 α cosh ( α t / 2 ) sinh ( t / 2 ) − sinh ( α t / 2 ) cosh ( t / 2 ) sinh 3 ( α t / 2 ) , H α ( 0 ) = C α ( α 2 − 1 ) 6 α . (78) H_\alpha(t)=\frac{C_\alpha\alpha}{2}\frac{\alpha\cosh(\alpha t/2)\sinh(t/2)-\sinh(\alpha t/2)\cosh(t/2)}{\sinh^3(\alpha t/2)},\qquad H_\alpha(0)=\frac{C_\alpha(\alpha^2-1)}{6\alpha}. \tag*{(78)} H α ( t ) = 2 C α α sinh 3 ( α t /2 ) α cosh ( α t /2 ) sinh ( t /2 ) − sinh ( α t /2 ) cosh ( t /2 ) , H α ( 0 ) = 6 α C α ( α 2 − 1 ) . (78) with the continuous value at zero, and, whenever the integral converges absolutely,
( T α , s f ) ( t ) = ∫ R H α ( t − u ) [ x ( t ) 1 + s x ( u ) 1 − s + a ( t ) 1 + s a ( u ) 1 − s ] f ( u ) d u . (79) (T_{\alpha,s}f)(t)=\int_{\mathbb{R}}H_\alpha(t-u)\left[x(t)^{1+s}x(u)^{1-s}+a(t)^{1+s}a(u)^{1-s}\right]f(u)\,\mathrm{d}u. \tag*{(79)} ( T α , s f ) ( t ) = ∫ R H α ( t − u ) [ x ( t ) 1 + s x ( u ) 1 − s + a ( t ) 1 + s a ( u ) 1 − s ] f ( u ) d u . (79) The powers use the real logarithms of x x x a a a α = 3 / 2 \alpha=3/2 α = 3/2 H α H_\alpha H α T α , s T_{\alpha,s} T α , s Theorem 1.1 .
Let s A ( α ) ∈ ( α − 1 , 2 α − 1 ) s_A(\alpha)\in(\alpha-1,2\alpha-1) s A ( α ) ∈ ( α − 1 , 2 α − 1 )
L 0 ψ = T α , s ψ (80) L_0\psi=T_{\alpha,s}\psi \tag*{(80)} L 0 ψ = T α , s ψ (80) has a strictly positive even solution ψ ∈ C 2 ( R ) \psi\in C^2(\mathbb{R}) ψ ∈ C 2 ( R ) c > 0 c > 0 c > 0
ψ ( t ) = { c e − ( α − 3 / 2 ) ∣ t ∣ + o ( e − ∣ α − 3 / 2 ∣ ∣ t ∣ ) , α ≠ 3 / 2 , c + o ( 1 ) , α = 3 / 2 , ∣ t ∣ → ∞ . (81) \psi(t) =
\begin{cases}
ce^{-(\alpha-3/2)|t|}+o\left(e^{-|\alpha-3/2||t|}\right), & \alpha\ne3/2,\\
c+o(1), & \alpha=3/2,
\end{cases}
\qquad|t| \to\infty. \tag*{(81)} ψ ( t ) = { c e − ( α − 3/2 ) ∣ t ∣ + o ( e − ∣ α − 3/2∣∣ t ∣ ) , c + o ( 1 ) , α = 3/2 , α = 3/2 , ∣ t ∣ → ∞. (81) One can prove existence and uniqueness of s A ( α ) s_A(\alpha) s A ( α )
X A ( κ ) : = X κ / 4 ( s A ( κ / 4 ) ) , X α ( s ) = s 2 − ( α − 1 ) 2 2 α . (82) X_A(\kappa):=X_{\kappa/4}\left(s_A(\kappa/4)\right), \qquad X_\alpha(s)=\frac{s^2-(\alpha-1)^2}{2\alpha}. \tag*{(82)} X A ( κ ) := X κ /4 ( s A ( κ /4 ) ) , X α ( s ) = 2 α s 2 − ( α − 1 ) 2 . (82) At κ = 6 \kappa=6 κ = 6 Theorem 1.1 gives s A ( 3 / 2 ) = s ∗ s_A(3/2)=s_* s A ( 3/2 ) = s ∗ X A ( 6 ) = ξ 3 X_A(6)=\xi_3 X A ( 6 ) = ξ 3 X A ( κ ) → 2 X_A(\kappa)\to2 X A ( κ ) → 2 κ → 8 \kappa\to8 κ → 8 X A ( κ ) → 1 / 8 X_A(\kappa)\to1/8 X A ( κ ) → 1/8 κ → 4 \kappa\to4 κ → 4
Table 3.
The last column reports estimates from the author’s Monte Carlo simulations.
Beijing International Center for Mathematical Research, Peking University, No.5 Yiheyuan Road, Haidian District, Beijing 100871, P.R.China
Email address: caigeifei1917@pku.edu.cn
