Random walk with long-range interaction with a barrier and its dual: Exact results

Abstract

We consider the random walk on Z + = {0, 1, ...} , with up and down transition probabilities given the chain is in state x ∈ {1, 2, ...}: (1) px = 1 2 " 1 − δ 2x + δ « and qx = 1 2 " 1 + δ 2x + δ «. Here δ ≥ −1 is a real tuning parameter. We assume that this random walk is reflected at the origin. For δ > 0, the walker is attracted to the origin: The strength of the attraction goes like δ 2x for large x and so is long-ranged. For δ < 0, the walker is repelled from the origin. This chain is irreducible and periodic; it is always recurrent, either positive or null recurrent. Using Karlin-McGregor's spectral representations in terms of orthogonal polynomials and first associated orthogonal polynomials, exact expressions are obtained for first-return time probabilities to the origin (excursion length), eventual return (contact) probability, excursion height and spatial moments of the walker. All exhibit power-law decay in some range of the parameter δ. In the study, an important role is played by the Wall duality relation for birth and death chains with reflecting barrier. Some qualitative aspects of the dual random walk (obtained by interchanging px and qx) are therefore also included.

Excursions and path functionals for stochastic processes with asymptotically zero drifts

“…In these very special cases, Fal' [14] gives asymptotics for excursion times and the number of excursions (cf our Theorem 2.4), while several authors [16,18,37,38,40] give iterated-logarithm type upper bounds in the diffusive case (cf our Theorem 2.6(i)). 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.…”

“…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.…”

“…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. Note that for δ ∈ (1, 2), 1 1+δ > 1 − δ 2 > 2−δ 1+δ .…”

“…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].…”