Langevin Dynamics of Non-reciprocally Coupled Spherical Spin Glasses

Introduction

Non-reciprocal interactions arise naturally in biological systems. For instance, in predator-prey systems, the interaction between species benefits one population and harms the other [1], and in neural networks the influence of one neuron on another need not be symmetric [19]. Such interactions also play a role in collective motion and synchronization, where they can generate sustained oscillations, traveling waves, and other time-dependent phases including chaos [14, 21].

A particular natural class of non-reciprocal systems consists of multiple interacting species, with symmetric interactions among members of each species and antisymmetric interactions between members of different species. This structure has been previously known in the context of excitatory-inhibitory neural networks, for example in [20], where such interactions were shown to give rise to dynamics ranging from stable fixed points to limit cycles. Models with such non-reciprocal interactions have recently been studied in the statistical physics literature as well, for example, [5] where two non-reciprocally coupled Ising models were studied. In another direction, Lorenzana et al [15] analyzed the Langevin dynamics of two identical copies of the spherical Sherrington–Kirkpatrick spin glass model with a non-reciprocal coupling. Their analysis showed evidence that depending on the network structure spin glass systems could retain glassy-dynamics phenomena such as aging, now accompanied by oscillations. Contrary to their findings, previous studies had predicted that any amount of non-reciprocity tends to destroy glassiness. In this paper, we rigorously analyze the dynamics introduced in [15] and extend its analysis to the mixed $p$-spin model.

For $x=(x_1,\ldots,x_N)\in\mathbb{R}^N$, let $H_N$ denote the Hamiltonian of a spin glass with mixed $p$-spin interactions, i.e.,

where $m \ge2$, $a_m \ne0$, and the disorder parameters $J_{i_1\ldots i_p}=J_{\{i_1\ldots i_p\}}$ are independent (modulo the permutation of indices) centered normal random variables with variance $N^{-p+1}\prod_k l_k!$, with $(l_1,l_2,\ldots)$ denoting the multiplicities of each element in the multiset $\{i_1,\ldots,i_p\}$. Alternatively, if we set $\xi(r)=\sum_{p=1}^{m}a_p^2r^p/p!$, $H_N$ is the centered Gaussian process defined on $\mathbb{R}^N$ with covariance given by

where $\cdot$ denotes the standard inner product in $\mathbb{R}^N$.

In this paper we study the Langevin dynamics of two identical copies of $H_N$ coupled with a non-reciprocal parameter. Precisely, let $f:\mathbb{R}\to\mathbb{R}_+$ be a differentiable function such that $f'$ is locally Lipschitz and there exist constants $\kappa<\infty$ and $\rho,\delta>0$ such that

Let $x(t)=(x^1(t),x^2(t))\in\mathbb{R}^{2N}$, where $x^a(t)=(x^a_1(t),\ldots,x^a_N(t))\in\mathbb{R}^N$ is the spin configuration of the species $a\in\{1,2\}$ at time $t$. At an inverse temperature $\beta>0$ and coupling strength $\alpha>0$, we consider the Langevin dynamics given by

where

with $\otimes$ denoting the Kronecker product of matrices, $I_N$ the identity matrix of size $N$,

and $B(t)$ a standard Brownian motion in $\mathbb{R}^{2N}$. The term containing $f$ in (2) is a Lagrange multiplier that implements a ‘soft’ spherical constraint on the spin configuration $x(t)$. We assume that the initial condition $x(0)$, drawn according to a distribution $\mu_{2N}$ on $\mathbb{R}^{2N}$, the disorder $J:=\{J_{i_1,\ldots,i_p}:1\le p\le m,\ 1\le i_1,\ldots,i_p\le N\}$ in $H_N$ and the Brownian motion $B(t)$ are all independent of each other.

Note that when $\alpha=0$, the drift in (2) is of gradient form, with the two copies sharing a common confining potential and following the same stochastic gradient descent dynamics. For $\alpha>0$, the additional forces $+\alpha x^2$ and $-\alpha x^1$ favor alignment of the first species with the second and anti-alignment of the second with the first, creating now an asymmetric imbalance.

Define the following quantities for $a,b\in\{1,2\}$:

Denoting by $M^{ab}$ the $(a,b)$th entry of the matrix $M$, we can assemble the above quantities in four $2\times2$ matrices $C_N(s,t)$, $\chi_N(s,t)$, $A_N(s,t)$ and $F_N(s,t)$, where

and the others are defined similarly. Finally, we denote $K_N(t)=C_N(t,t)$.