First return times on sparse random graphs

TL;DR

The paper analyzes first return times of discrete-time random walks on Erdős-Rényi graphs with fixed mean degree $c$ in the limit $N\to\infty$. It develops a statistical-field-theory framework, employing a supersymmetric auxiliary-field representation and Fyodorov–Mirlin decoupling, to average the first-return generating function over the ER ensemble and connect it to the spectrum of the normalized graph Laplacian. A controlled $1/c$ expansion yields an explicit expression for the averaged generating function $\langle F_{11}(z)\rangle$, with leading behavior governed by Catalan-number combinatorics for returns after $2n$ steps and odd-step returns vanishing in the tree-like limit; second-order corrections closely match numerical simulations on large sparse graphs. The results show that the first-return statistics on ER graphs can be captured by a tractable saddle-point analysis, providing a quantitative bridge between sparse random graphs and their tree-like limits and suggesting avenues for extending these methods to broader random-graph problems.

Abstract

We consider random walks in the form of nearest-neighbor hopping on Erdos-Renyi random graphs of finite fixed mean degree c as the number of vertices N tends to infinity. In this regime, using statistical field theory methods, we develop an analytic theory of the first return time probability distribution. The problem turns out closely related to finding the spectrum of the normalized graph Laplacian that controls the continuum time version of the nearest-neighbor-hopping random walk. In the infinite graph limit, where loops are highly improbable, the returns operate in a manner qualitatively similar to c-regular trees, and the expressions for probabilities resemble those on random c-regular graphs. Because the vertex degrees are not exactly constant, however, the way c enters the formulas differs from the dependence on the graph degree of first return probabilities on random regular graphs.