[Paper Review] Drift and entropy growth for random walks on groups

Anna Erschler-Dyubina|ArXiv.org|Jan 9, 2001

Mathematical Dynamics and Fractals3 references3 citations

TL;DR

This paper constructs groups with random walks exhibiting extremely slow drift and entropy growth, specifically showing that drift $ L(n) $ can be asymptotically $ n / \underbrace{\ln(\ln(\cdots \ln(n)\cdots))}_{k} $ for any $ k \geq 1 $, and entropy $ H(n) $ can grow as $ n / \underbrace{\ln(\ln(\cdots \ln(n)\cdots))}_{k} $, demonstrating infinitely many distinct sublinear growth rates for both drift and entropy in random walks on groups.

ABSTRACT

In this paper we consider finitary symmetric random walks on groups. We construct new possible asymptotics for the drift. We show that the drift can be very close to linear ant yet sublinear. We also give estimates for entropy growth of these random walks. In particular, we prove that there exist infinitely many asymptotics of entropy.

Motivation & Objective

• To construct groups where the drift $ L(n) $ of symmetric random walks grows slower than any polynomial but faster than $ \sqrt{n} $, specifically achieving $ n / \underbrace{\ln(\ln(\cdots \ln(n)\cdots))}_{k} $ for any $ k \geq 1 $.
• To analyze the corresponding entropy growth $ H(n) $ for such random walks and show it can also achieve the same sublinear asymptotic rates.
• To demonstrate that there are infinitely many distinct asymptotic growth rates for both drift and entropy in random walks on finitely generated groups.
• To extend the known dichotomy between linear and $ \sqrt{n} $-type drift growth by constructing intermediate growth rates using recursive logarithmic iterates.
• To establish sharp bounds on entropy using the second moment of the word length and the volume growth of balls in the group.

Proposed method

• Constructs groups via iterated wreath products of $ \mathbb{Z} $, using a recursive structure to control the growth of the random walk.
• Applies results on two-dimensional simple random walk to estimate local times and range, using the fact that $ \mathbb{E}[R^{(n)}] \asymp n / \ln n $.
• Uses a key auxiliary lemma proving concavity of functions of the form $ x / (\ln^{(k)} x)^\alpha $ for large $ x $, enabling moment estimates.
• Applies concave functionals to local times $ b_z^{(n)} $ of the 2D walk to bound $ \mathbb{E}[\sum_z f(b_z^{(n)})] $, leading to drift estimates.
• Derives entropy bounds via the inequality $ H(n) \geq C \cdot \mathbb{E}[l^2(g)] / n - \ln n $, linking entropy to the second moment of the word length.
• Uses the volume growth $ v(n) $ of balls to bound $ H(n) \leq \ln v(n) $, and combines this with the drift estimate to derive entropy asymptotics.

Experimental results

Research questions

• RQ1 Can the drift $ L(n) $ of a symmetric random walk on a finitely generated group grow at a rate strictly between $ \sqrt{n} $ and linear, and if so, what are the possible intermediate rates?
• RQ2 Can the entropy $ H(n) $ of such a random walk grow sublinearly but arbitrarily close to linear, and what are the possible growth rates?
• RQ3 Are there groups where the drift and entropy grow as $ n / \underbrace{\ln(\ln(\cdots \ln(n)\cdots))}_{k} $ for any $ k \geq 1 $, and do these rates yield distinct asymptotic behaviors?
• RQ4 What is the relationship between the growth of the second moment of the word length and the entropy of the random walk?
• RQ5 Can the concavity properties of iterated logarithmic functions be used to derive sharp bounds on functionals of local times in 2D random walks?

Key findings

• The drift $ L(n) $ of a symmetric random walk on a constructed group can grow asymptotically as $ n / \underbrace{\ln(\ln(\cdots \ln(n)\cdots))}_{k} $ for any fixed $ k \geq 1 $, demonstrating a new class of intermediate growth rates.
• The entropy $ H(n) $ of the same random walk grows as $ n / \underbrace{\ln(\ln(\cdots \ln(n)\cdots))}_{k} $, matching the drift rate asymptotically.
• For each $ k \geq 1 $, there exists a group and a symmetric random walk such that $ H_{G_k}(n) \asymp n / \underbrace{\ln(\ln(\cdots \ln(n)\cdots))}_{k} $, showing infinitely many distinct entropy growth rates.
• The second moment of the word length satisfies $ \mathbb{E}[l^2(g)] \asymp n \cdot H(n) $, linking entropy growth to the spread of the walk.
• The drift satisfies $ L(n) \leq K \sqrt{n(\ln v(n) + \ln n)} $, and for the constructed examples, $ v(n) \asymp n \cdot \underbrace{\ln(\ln(\cdots \ln(n)\cdots))}_{k} $, which matches the entropy growth.
• The construction relies on the concavity of $ x / (\ln^{(k)} x)^\alpha $ for large $ x $, which is proven via analysis of derivatives and recursive logarithmic structures.