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

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) p x = 1 2 ( 1 − δ 2 x + δ ) and q x = 1 2 ( 1 + δ 2 x + 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 2 x 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. 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 p x and q x ) are therefore also included.