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Elephant random walks on infinite Cayley trees

Soumendu Sundar Mukherjee

TL;DR

The paper studies elephant random walks on infinite Cayley trees, showing that memory effects do not alter the asymptotic escape speed, which remains $\frac{d-2}{d}$, the same as simple random walk on the tree. It develops a non-Markovian analysis on non-abelian group geometries by decoupling the urn dynamics governing step counts from the Cayley-tree geometry, yielding sharp rate-of-convergence bounds that exhibit a phase transition at $p_d=\frac{d+1}{2d}$. The authors establish concentration bounds for key quantities via urn-branching embeddings and prove a central limit-type fluctuation result, with detailed return-probability estimates across memory regimes. The work provides a framework for exploring memory in random walks on non-abelian groups, highlighting both universal speed and regime-dependent convergence behavior, and it opens several avenues for studying fluctuations and broader classes of groups and graphs.

Abstract

We introduce a generalisation of Schütz and Trimper's elephant random walk to finitely generated groups. We focus on the simplest non-abelian setting, i.e. groups whose Cayley graphs are homogeneous trees of degree $d \ge 3$. We show that the asymptotic speed of the walk does not depend on the memory parameter $p \in [0, 1)$ and equals $\frac{d - 2}{d}$, the asymptotic speed of simple random walk on these graphs. We also establish upper bounds on the rate of convergence to the limiting speed. These upper bounds depend on $p$ and exhibit a phase transition at the critical value $p_d = \frac{d + 1}{2d}$. Numerical experiments suggest that these upper bounds are tight. Along the way, we also obtain estimates on the return probability.

Elephant random walks on infinite Cayley trees

TL;DR

The paper studies elephant random walks on infinite Cayley trees, showing that memory effects do not alter the asymptotic escape speed, which remains , the same as simple random walk on the tree. It develops a non-Markovian analysis on non-abelian group geometries by decoupling the urn dynamics governing step counts from the Cayley-tree geometry, yielding sharp rate-of-convergence bounds that exhibit a phase transition at . The authors establish concentration bounds for key quantities via urn-branching embeddings and prove a central limit-type fluctuation result, with detailed return-probability estimates across memory regimes. The work provides a framework for exploring memory in random walks on non-abelian groups, highlighting both universal speed and regime-dependent convergence behavior, and it opens several avenues for studying fluctuations and broader classes of groups and graphs.

Abstract

We introduce a generalisation of Schütz and Trimper's elephant random walk to finitely generated groups. We focus on the simplest non-abelian setting, i.e. groups whose Cayley graphs are homogeneous trees of degree . We show that the asymptotic speed of the walk does not depend on the memory parameter and equals , the asymptotic speed of simple random walk on these graphs. We also establish upper bounds on the rate of convergence to the limiting speed. These upper bounds depend on and exhibit a phase transition at the critical value . Numerical experiments suggest that these upper bounds are tight. Along the way, we also obtain estimates on the return probability.

Paper Structure

This paper contains 11 sections, 8 theorems, 81 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The model and our main results
  3. Random walk on finitely generated groups
  4. The ERW on finitely generated groups
  5. ERW on Cayley trees
  6. Asymptotic notation.
  7. Main results
  8. Preliminaries
  9. The urn connection
  10. A concentration result for Xi-n.
  11. Proofs of our main results

Key Result

Theorem 2.1

We have for any $p \in [0, 1)$ that

Figures (4)

  • Figure 1: ERW on $\mathbb{T}_4$: At $w_{n} \ne e$, one either (a) moves away from the root, i.e. $g_{n + 1} \ne \gamma_n$, so that $\Delta_{n + 1} - \Delta_{n} = 1$, or (b) moves towards it, i.e. $g_{n + 1} = \gamma_n$, so that $\Delta_{n + 1} - \Delta_{n} = -1$. The green dotted line depicts the unique geodesic connecting $e$ and $w_n$.
  • Figure 2: Escape rate of the ERW on the Cayley graphs of (a) $\mathbb{Z}_2^{*4}$, (b) $\mathbb{Z} * \mathbb{Z}_2^{*2}$, (c) $\mathbb{Z}^{*2}$, based on $10^4$ Monte Carlo runs with $n = 10^6$ steps each; the red horizontal line corresponds to $\frac{d - 2}{d} = \frac{1}{2}$, the escape rate of SRW. The observed deviation, near $p = 1$, of the mean speed from the theoretical value of $\frac{1}{2}$ reflects the increasingly slower rates of convergence.
  • Figure 3: Histograms of $r_n(\frac{\Delta_n}{n} - \frac{d - 2}{d})$ for various values of $p$ on the Cayley graphs of (a) $\mathbb{Z}_2^{*4}$, (b) $\mathbb{Z} * \mathbb{Z}_2^{*2}$, (c) $\mathbb{Z}^{*2}$, grouped columnwise, based on $10^4$ Monte Carlo runs with $n = 10^6$ steps each. The critical probability here is $p_4 = 0.625$. The solid black lines depict the $N(0, \frac{4(d - 1)}{d^2})$ density.
  • Figure 4: Scatter plot of $r_n \Xi_n$ for (a) $\mathbb{Z}_2^{*4}$, (b) $\mathbb{Z} * \mathbb{Z}_2^{*2}$, (c) $\mathbb{Z}^{*2}$, based on $10^4$ Monte Carlo runs with $n = 10^6$ steps each.

Theorems & Definitions (20)

  • Remark 2.1
  • Theorem 2.1: Escape rate
  • Remark 2.2
  • Theorem 2.2: Rate of convergence to the escape rate
  • Remark 2.3
  • Corollary 2.1
  • Theorem 2.3: Return probability estimates
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • ...and 10 more