[Paper Review] Long time behaviour of random walks on the integer lattice

Bartosz Trojan|arXiv (Cornell University)|Dec 30, 2015

Stochastic processes and statistical mechanics7 references3 citations

TL;DR

This paper establishes a new asymptotic formula for the transition function $ p(n;x) $ of irreducible finite-range random walks on the $ d $-dimensional integer lattice, valid for $ |x|_1 = o(n) $, using a refined Fourier inversion and Laplace method with geometric control of degenerating quadratic forms. The key contribution is a non-Gaussian, exponentially decaying approximation that extends the region of validity beyond previous results, particularly for large $ |x|_2 $, by introducing a convex rate function $ \phi(\delta) $ with explicit error bounds.

ABSTRACT

We consider an irreducible finite range random walk on the $d$-dimensional integer lattice and study asymptotic behaviour of its transition function $p(n; x)$. In particular, for simple random walk our asymptotic formula is valid as long as $n (n - |x|_1)^{-2}$ tends to zero.

Motivation & Objective

• To extend the region of validity for asymptotic approximations of the transition function $ p(n;x) $ of random walks on $ \mathbb{Z}^d $, beyond the standard local limit theorem.
• To resolve the limitation of prior asymptotic formulas that break down when $ |x|_2 \gtrsim n^{3/4} $, by developing a formula valid for $ |x|_1 = o(n) $.
• To provide a non-Gaussian, exponentially decaying approximation for $ p(n;x) $ that captures the correct decay rate even in the large deviation regime.
• To generalize the result beyond simple random walk to any irreducible finite-range random walk with zero or non-zero mean.

Proposed method

• Use of the Fourier inversion formula to express $ p(n;x) $ as an oscillatory integral over the torus $ \mathbb{T}^d $.
• Splitting the integral into two parts: one near the origin (analyzed via the Laplace method), and one away from it (controlled via geometric arguments).
• Introduction of the convex rate function $ \phi(\delta) = \max_x \{ \langle x, \delta \rangle - \log \kappa(x) \} $, where $ \kappa(x) = \frac{1}{d}(\cosh x_1 + \cdots + \cosh x_d) $, to capture the large deviation behavior.
• Explicit analysis of the degeneration of the quadratic form $ B_s $ as $ |\delta|_1 \to 1 $, using the implicit mapping $ s = \nabla\phi(\delta) $.
• Derivation of error bounds involving $ n^{-1}(1 - |\delta|_1)^{-2} $, leading to a uniform asymptotic expansion with controlled error terms.
• Application of the method to a random walk on the triangular lattice via lattice embedding and transformation to $ \mathbb{Z}^2 $, confirming the formula's robustness.

Experimental results

Research questions

• RQ1 Can an asymptotic formula for $ p(n;x) $ be derived that remains valid for $ |x|_1 = o(n) $, extending beyond the $ |x|_2 = o(n^{3/4}) $ regime of prior results?
• RQ2 How can the error term in the local limit theorem be controlled when $ |x|_2 \gg \sqrt{n} $, especially in the large deviation regime?
• RQ3 What is the correct non-Gaussian correction to the Gaussian approximation in the transition density of finite-range random walks on $ \mathbb{Z}^d $?
• RQ4 Is the asymptotic formula robust under lattice transformations, such as embedding a triangular lattice walk into $ \mathbb{Z}^2 $?

Key findings

• The asymptotic formula $ p(n;x) = (2\pi n)^{-d/2}(\det B_s)^{-1/2}e^{-n\phi(\delta)}(2 + \mathcal{O}(n^{-1}(1 - |\delta|_1)^{-2})) $ holds uniformly for $ |x|_1 \equiv n \pmod{2} $, with $ \delta = x/n $, and is valid as long as $ n(1 - |\delta|_1)^2 \to \infty $.
• For $ |\delta|_1 \leq 1 - \epsilon $, the formula simplifies to $ p(n;x) = d^{d/2}(2\pi n)^{-d/2}e^{-n\phi(\delta)}(2 + \mathcal{O}(|\delta|_1) + \mathcal{O}(n^{-1})) $, valid uniformly for $ |x|_1 = o(n) $.
• The rate function $ \phi(\delta) $ is strictly convex and comparable to $ |\delta|_2^2 $, but cannot be replaced by a quadratic form without introducing additional error, highlighting the non-Gaussian nature of the correction.
• The method controls the degeneration of the Hessian $ B_s $ as $ |\delta|_1 \to 1 $, where $ |s|_2 \to \infty $, by analyzing the implicit mapping $ s = \nabla\phi(\delta) $.
• The formula is applied to a random walk on the triangular lattice, confirming that the asymptotic holds after lattice transformation, with $ \tilde{q}(n;x) \sim (2\pi n)^{-1}e^{-n\phi(\delta)}(3\sqrt{3} + \mathcal{O}(|\delta|_1) + \mathcal{O}(n^{-1})) $.
• The error term $ \mathcal{O}(|\delta|_1) $ in the final asymptotic is uniform in $ n $, and the formula remains valid in the large deviation regime $ |x|_1 = o(n) $, significantly extending the scope of prior results.