[Paper Review] Excited random walk in three dimensions has positive speed

Gady Kozma|ArXiv.org|Oct 20, 2003

Stochastic processes and statistical mechanics4 references20 citations

TL;DR

This paper proves that three-dimensional excited random walk with a constant bias (epsilon > 0) has positive speed almost surely, meaning the walk's position in the biased direction grows linearly with time. The proof uses a multi-layered block decomposition with a restart mechanism to overcome failures in individual blocks, leveraging coupling with simple random walk and a novel two-dimensional covering estimate to control recurrence and ensure linear drift.

ABSTRACT

Excited random walk is a random walk that has a positive drift to the right when it reaches a vertex it hasn't been to before. We show that in three dimensions the walk drifts to the right in non-zero speed.

Motivation & Objective

• To establish that the one-dimensional projection of excited random walk in three dimensions grows linearly with time, i.e., has positive speed.
• To overcome the challenge that standard block decomposition fails due to cascading failures in bad blocks.
• To develop a restart mechanism that allows the process to recover after rare bad blocks by reinitializing from two-dimensional considerations.
• To prove a sharp two-dimensional covering estimate for independent simple random walks, which is central to controlling recurrence and drift.
• To show that the lower limit of the walk's position divided by time is almost surely positive, resolving a key open problem in the model.

Proposed method

• Uses a multi-layered block decomposition strategy, where each layer handles failures in the previous layer via a restart mechanism.
• Employs a coupling argument to compare the excited walk with a simple symmetric random walk, ensuring the excited walk stays strictly to the right.
• Applies a key two-dimensional result: two independent simple random walks of lengths n and m = exp(log^μ n) cover a common region with high probability, and the first walk has at least m^{3/4} 'holes' in a √m neighborhood.
• Implements a probabilistic bound on the number of 'bad' blocks (those failing to make sufficient progress) using exponential tail estimates involving √log n.
• Uses Borel-Cantelli lemma on a dyadic sequence of times (n = 2^i) to show that the walk eventually satisfies the linear speed condition almost surely.
• Introduces a parameterized speed α_n that depends on block size and failure probability, ensuring the speed remains bounded away from zero.

Experimental results

Research questions

• RQ1 Does three-dimensional excited random walk exhibit positive speed, i.e., does R(n)_1 / n have a positive liminf almost surely?
• RQ2 Can the standard block decomposition method be salvaged despite the risk of cascading failures in bad blocks?
• RQ3 What is the sharpness of the two-dimensional covering estimate for independent simple random walks of different lengths?
• RQ4 Is the lower limit of R(n)_1 / n almost surely a limit, or could it oscillate?
• RQ5 Can the speed be bounded away from zero uniformly, even in the presence of rare, large deviations?

Key findings

• The one-dimensional projection of an ε-excited random walk in three dimensions satisfies liminf_{n→∞} R(n)_1 / n > 0 almost surely, proving positive speed.
• The proof establishes a speed lower bound of order c(ε) > 0, independent of n, with the constant depending only on ε.
• The two-dimensional covering estimate (Theorem 2) shows that with high probability, a long simple random walk of length m = exp(log^μ n) covers at least m^{3/4} vertices not visited by a shorter walk of length n.
• The failure probability for the speed estimate decays as exp(−c√log n), which is summable, enabling the use of the Borel-Cantelli lemma.
• The restart mechanism successfully handles rare bad blocks by reinitializing the process from a two-dimensional perspective, ensuring long-term progress.
• The result is sharp in the sense that the decay rate exp(−c√log n) cannot be improved without stronger assumptions, as shown by the matching lower bound on the probability of full coverage.