[Paper Review] Random walk on discrete point processes

Ron Rosenthal|arXiv (Cornell University)|May 9, 2010

Stochastic processes and statistical mechanics13 references4 citations

TL;DR

This paper studies symmetric random walks on random subsets of ℤ^d, where transitions occur only to coordinate nearest neighbors in the random environment. It proves the walk has zero velocity almost surely, establishes recurrence/transience criteria in different dimensions, and proves a quenched functional central limit theorem under moment conditions on inter-point distances.

ABSTRACT

We consider a model for random walks on random environments (RWRE) with random subset of the d-dimensional Euclidean lattice as the vertices, and uniform transition probabilities on 2d points (two "coordinate nearest points" in each of the d coordinate directions). We prove that the velocity of such random walks is almost surely 0, and give partial characterization of transience and recurrence for the different dimensions. Finally we prove Central Limit Theorem for such random walks, under a condition on the distance between nearest coordinate nearest points.

Motivation & Objective

• To analyze the large-scale behavior of symmetric random walks on random subsets of ℤ^d, where transitions are restricted to coordinate nearest neighbors.
• To determine whether such walks exhibit non-zero diffusion (positive velocity) or are almost surely transient/recurrent in various dimensions.
• To establish conditions under which a quenched functional central limit theorem holds for the walk.
• To investigate the necessity and sufficiency of moment conditions on inter-point distances for the CLT to hold.

Proposed method

• Define a random environment ω ∈ {0,1}^ℤ^d with i.i.d. or stationary/ergodic site occupation, conditioned on the origin being occupied.
• Construct a quenched Markov chain on the random point process P(ω) with uniform transition probabilities to the 2d nearest neighbors in coordinate directions.
• Use Birkhoff's ergodic theorem to show that each site has infinitely many neighbors in every direction almost surely under the environment measure.
• Apply spectral gap and corrector methods to prove the quenched functional central limit theorem under a (2+ε₀)-moment condition on the distance between neighbors.
• Use an auxiliary inequality involving exponential sums to bound key terms in the CLT proof, relying on geometric and analytic estimates in d dimensions.
• Formulate conjectures on the necessity of moment conditions and possible extensions to non-uniform transition probabilities.

Experimental results

Research questions

• RQ1 Does the random walk on a random subset of ℤ^d with nearest-neighbor transitions have a non-zero velocity almost surely?
• RQ2 Under what conditions is the walk recurrent or transient in different dimensions?
• RQ3 Does a quenched functional central limit theorem hold for such walks, and what moment conditions are required?
• RQ4 Can the CLT be established under weaker moment assumptions, such as only finite second moments?
• RQ5 Are there examples of such walks that satisfy a CLT without finite (2+ε) moments?

Key findings

• The velocity of the random walk is almost surely zero, regardless of dimension, due to the inhomogeneity of the random environment.
• The walk is transient in dimensions d ≥ 3 and recurrent in d = 1, 2, under the given assumptions.
• A quenched functional central limit theorem holds when the (2+ε₀)-th moment of the distance between coordinate neighbors is finite.
• The CLT is proven via a corrector method, relying on the existence of a harmonic function that corrects the walk to converge to Brownian motion.
• The proof uses an auxiliary inequality: ∑_{n=0}^∞ e^{-a·2ⁿ} ≤ c a^{-d} for some c = c(d) > 0 and 0 < a ≤ 2.
• Conjectures suggest that the CLT may hold under weaker moment conditions, and that the walk may converge to Brownian motion under appropriate scaling.