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Maximal Entropy Random Walks in Z: Random and non-random environments

Duboux Thibaut, Lucas Gerin, Yoann Offret

TL;DR

This work analyzes Maximal Entropy Random Walks (MERWs) on the infinite lattice $\mathbb{Z}$ with loops, addressing both deterministic and random loop environments. It develops explicit Perron–Frobenius eigenvector representations in deterministic $M$-nice settings (where $\lambda=2+M$) and shows MERWs in i.i.d. loop environments are random walks in ergodic environments, yielding positive linear speed; non-extremal MERWs decompose as mixtures of extremal cases with positive speed when diverging. The Bernoulli loop case reveals a sharp speed transition: as the loop probability $p$ vanishes, the speed tends to a positive limit $\sqrt{1-4/(2+M)^2}$, while as $p\to1$, the speed scales like $\sim 3(1-p)/(2+M)$, contrasting with the $p=0$ recurrent baseline. Overall, the paper connects MERWs to spectral theory and the Parabolic Anderson Model, showing randomness in one dimension can prevent localization and induce sustained transport, and it provides explicit, computable representations for speeds and eigenvectors to enable further analysis.

Abstract

The Maximal Entropy Random Walk (MERW) is a natural process on a finite graph, introduced a few years ago with motivations from theoretical physics. The construction of this process relies on Perron-Frobenius theory for adjacency matrices. Generalizing to infinite graphs is rather delicate, and in this article, we treat in a fairly exhaustive manner the case of the MERW on Z with loops, for both random and nonrandom loops. Thanks to an explicit combinatorial representation of the corresponding Perron-Frobenius eigenvectors, we are able to precisely determine the asymptotic behavior of these walks. We show, in particular, that essentially all MERWs on Z with loops have positive speed.

Maximal Entropy Random Walks in Z: Random and non-random environments

TL;DR

This work analyzes Maximal Entropy Random Walks (MERWs) on the infinite lattice with loops, addressing both deterministic and random loop environments. It develops explicit Perron–Frobenius eigenvector representations in deterministic -nice settings (where ) and shows MERWs in i.i.d. loop environments are random walks in ergodic environments, yielding positive linear speed; non-extremal MERWs decompose as mixtures of extremal cases with positive speed when diverging. The Bernoulli loop case reveals a sharp speed transition: as the loop probability vanishes, the speed tends to a positive limit , while as , the speed scales like , contrasting with the recurrent baseline. Overall, the paper connects MERWs to spectral theory and the Parabolic Anderson Model, showing randomness in one dimension can prevent localization and induce sustained transport, and it provides explicit, computable representations for speeds and eigenvectors to enable further analysis.

Abstract

The Maximal Entropy Random Walk (MERW) is a natural process on a finite graph, introduced a few years ago with motivations from theoretical physics. The construction of this process relies on Perron-Frobenius theory for adjacency matrices. Generalizing to infinite graphs is rather delicate, and in this article, we treat in a fairly exhaustive manner the case of the MERW on Z with loops, for both random and nonrandom loops. Thanks to an explicit combinatorial representation of the corresponding Perron-Frobenius eigenvectors, we are able to precisely determine the asymptotic behavior of these walks. We show, in particular, that essentially all MERWs on Z with loops have positive speed.

Paper Structure

This paper contains 15 sections, 14 theorems, 112 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Maximal Entropy Random Walks on finite and infinite graphs
  3. Our results: MERWs on $\mathbb{Z}$
  4. MERW on $\mathbb{Z}$: deterministic loop environments
  5. Preliminaries
  6. The two extremal eigenvectors $\psi^+$ and $\psi^-$
  7. MERW on $\mathbb{Z}$: first properties
  8. Combinatorial description of $\psi^+$ and $\psi^-$
  9. MERW on $\mathbb{Z}$: i.i.d. loop environments
  10. MERW associated with the random extremal eigenvectors
  11. Analysis of MERW associated with a non-extremal eigenvector
  12. The case of a Bernoulli i.i.d. loop environment
  13. Discussion and remaining questions
  14. Acknowledgements.
  15. Non-random periodic loop environments

Key Result

Proposition 1.3

The supremum of $\mathcal{E}(q)$, among all adapted positive recurrent Markov kernels $q$ on $G$, is equal to $\log(\lambda)$, and one can even restrict the supremum to all adapted positive recurrent Markov chains on finite strongly connected subgraphs $H \subset G$. Furthermore:

Figures (5)

  • Figure 1: Left: $200$ independent simulations of the MERW $(X_n^+)_{n\geq 0}$ up to $n=600$ in the same realization of the random i.i.d. loop environment. There is a loop of weight $M=20$ at each vertex with probability $p=0.02$. Right: Same parameters except $p=0.05$. These simulations support \ref{['asympbernoulli']}: the asymptotic speed $v_{p,M}$ of the MERW is decreasing in $p$.
  • Figure 2: Sketchs of plots of $g_s$.
  • Figure 3: Sketch of plots of $p\longmapsto v_{p,M}$ obtained by Monte Carlo simulations.
  • Figure 4: Sketch of plots of $M\longmapsto v_{p,M}$ obtained by Monte Carlo simulations.
  • Figure 5: The reduced graph

Theorems & Definitions (39)

  • Remark 1.1
  • Definition 1.2
  • Proposition 1.3: MERWs maximize entropy (see Duboux)
  • Definition 1.4
  • Remark 1.5
  • Definition 2.1
  • Lemma 2.2
  • proof : Proof of \ref{['prop:nice']}
  • Proposition 2.3
  • proof : Proof of \ref{['solution']}
  • ...and 29 more