Random Walk Weakly Attracted to a Wall

Abstract

We consider a random walk X n in Z + , starting at X 0 = x ≥ 0, with transition probabilitiesand X n+1 = 1 whenever X n = 0. We prove EX n ∼ const. n 1− δ 2 as n ր ∞ when δ ∈ (1, 2). The proof is based upon the Karlin-McGregor spectral representation, which is made explicit for this random walk.

Excursions and path functionals for stochastic processes with asymptotically zero drifts

“…Huillet [23] gives sharper versions of our Theorems 2.2, 2.3, and 2.6 in this special case: see Propositions 2, 9, 10, and 11 of [23]. The main result of [12] (see also Proposition 15 of [23]) is that, for δ ∈ (1, 2), E[X t ] ∼ K δ t 1− δ 2 , being one possible measure of the spatial extent of the polymer. Perhaps more natural (certainly more readily interpreted in terms of path properties) are the quantities max 1≤s≤t X s and t −1 t s=1 X s that we study in the present paper; their scaling exponents for the case δ ∈ (1, 2) are 1 1+δ (our Theorem 2.6, or Proposition 10 of [23]) and 2−δ 1+δ (our Corollary 2.1) respectively.…”

“…Much of the existing work is restricted to nearest-neighbour random walks on Z + , where explicit calculations are facilitated by reversibility and associated algebraic structure (such as Karlin-McGregor theory [26]); see e.g. [1,12,23] for models inspired directly by random polymers, and e.g. [11,18,40] for related work.…”

“…Alexander [1] calls such nearest-neighbour random walks with drift O(1/x) at x 'Bessel-like', and gives sharp results on the asymptotics of return times, among other things. There seems to have as yet been no success in applying the methods of [1,12,23] beyond the nearest-neighbour setting. Our basic analytical method will be built on the fact that f γ,ν (X t ) is a submartingale or supermartingale, for X t outside some bounded set, for appropriate γ and ν.…”

“…In the last decade or so, significant interest in processes with asymptotically zero drifts has come from a community of probabilists and statistical physicists from the point of view of modelling the configurations of polymers and interfaces. A now standard approach in this field is to take as an underlying model a nearest-neighbour random walk with an asymptotically zero drift: see for example [1,12,23]. Such nearest-neighbour models are amenable to explicit calculation, often via intricate algebraic methods such as Karlin-McGregor spectral theory and orthogonal polynomials [26]; other recent work on these models, not directly motivated by polymer models, includes for example [11,18,29,40].…”

Rate of escape and central limit theorem for the supercritical Lamperti problem

“…[5,6]), particularly in the context of modelling random polymers (see e.g. [3,9]). The study of continuous-time analogues of the general Lamperti problem seems to have begun only recently: see e.g.…”