[Paper Review] Supplementary Material for Random Cayley Graphs Project

Jonathan Hermon, Sam Olesker-Taylor|arXiv (Cornell University)|Oct 11, 2018

Cellular Automata and Applications6 references4 citations

TL;DR

This supplementary paper provides refined analytical results on random walks on integers and cycles, focusing on Shannon entropy, large deviations, exit times, and lattice ball sizes under general Lq norms. It derives precise asymptotic estimates for entropic times and cutoff windows in random Cayley graphs, particularly for symmetric and asymmetric random walks, with key contributions in entropy concentration and stochastic bounds for local times and divisor functions in number-theoretic contexts.

ABSTRACT

This document contains supplementary material for the main articles in our Random Cayley Graphs project. We prove refined results about simple random walks on the integers and on the cycle. We are primarily interested in the entropy of these random walks at certain times and how this entropy changes when the time changes slightly. Additionally, we prove some large deviation and exit time estimates. We prove some results on the size of discrete lattice balls and how this size changes when the radius changes slightly. We do this in a general $L_q$ norm, with $q \in [1,\infty]$. We also prove some other technical results deferred from the main papers. We hope that some of the results, particularly the simple random walk estimates, will be useful in their own right for other researchers.

Motivation & Objective

• To provide rigorous analytical supplements to main works on random Cayley graphs, particularly on entropy dynamics and mixing times.
• To derive precise asymptotic estimates for entropic times and cutoff windows in symmetric and asymmetric random walks on Z and Z_L.
• To establish large deviation bounds and exit time estimates for simple random walks on the integer line and cyclic groups.
• To analyze the size of discrete lattice balls under general Lq norms and their sensitivity to radius perturbations.
• To supply deferred technical proofs on uniform measures in nilpotent groups, local time distributions, and divisor function bounds.

Proposed method

• Applies the local central limit theorem (LCLT) to analyze the distribution of position and entropy in simple random walks on Z.
• Uses Chernoff-type bounds and excursion time analysis to control tail probabilities of local times in random bridges.
• Employs moment generating functions and exponential moment bounds to derive large deviation estimates for random walks on Z.
• Applies union bounds and conditioning arguments to relate bridge local times to unconditioned random walk local times.
• Uses recursive decomposition and group-theoretic techniques to prove uniformity of products of independent uniform variables in nilpotent groups.
• Applies Markov’s inequality and dyadic decomposition to bound the number of divisors of integers in density-1 sets, with explicit dependence on logarithmic and power terms.

Experimental results

Research questions

• RQ1 How does the Shannon entropy of a simple random walk on Z evolve over time, and what are the precise asymptotics of its entropic time and cutoff window?
• RQ2 What are the large deviation probabilities for the position of a simple random walk on Z, particularly in the moderate deviation regime?
• RQ3 How do exit times from intervals behave for simple random walks on Z, and what are the sharp tail estimates for these exit times?
• RQ4 How does the size of discrete lattice balls in Lq norms scale with radius, and how sensitive is this size to small changes in radius?
• RQ5 What are the tail behaviors of local times in random bridges, and how can they be bounded using excursion time and exponential moment techniques?

Key findings

• For symmetric random walks on Z, the entropic time t₀ scales as k · n²ᵏ / (2πe) when k ≪ log n, with a cutoff window of order √(2αt₀/√k).
• When k ≈ λ log n, the entropic time t₀ ≈ k·f(λ) and the cutoff window scales as g(λ)·αt₀/√k, with continuous functions f and g.
• For k ≫ log n, the entropic time t₀ ≈ k/(κ log κ) with κ = k/log n, and the cutoff window scales as √(κ log κ)·αt₀/√k.
• The maximal local time L(r, L) for a bridge of length L on Z satisfies max_r P(L(r, L) > √(2(C+2)L log L)) ≲ L⁻C, implying E[L(r, L)^C] ≲ C^C L^{3C/5}.
• Large deviation estimates for the random walk on Z show that P(|X(s) − s| > x) ≤ exp(−x²/(2s)) for x ≲ s, using exponential moment bounds.
• A density-1 set A_ε ⊆ ℕ exists such that for all n ∈ A_ε and m ≥ 2, ∑_{i≤m} i·1(i∤n) ≤ 40(λε)⁻¹ m (log m)² with high probability, uniformly in m.