Critical Spectral Invariants in Random Walks with Geometric Resetting

TL;DR

This paper analyzes how geometric resetting affects the gambler's ruin problem, revealing spectral invariants and critical behaviors in absorption probabilities within confined domains.

Contribution

It introduces a spectral framework for analyzing resetting effects on ruin probabilities, uncovering invariance properties and critical points in the classical problem.

Findings

Spectral representation diagonalizes the transition operator.

Ruin probabilities exhibit geometric invariance for even domain sizes.

Explicit closed-form expressions for ruin probabilities under resetting.

Abstract

Stochastic resetting -- the intermittent restart of random processes -- has profoundly reshaped first-passage theory, providing a mechanism to control and optimize completion times. While the influence of resetting on mean first-passage times is now well understood, its impact on absorption probabilities in confined domains remains comparatively unexplored. We present a complete analysis of the classical gambler's ruin problem under geometric resetting. At each time step, the walker is reset to its initial position with probability gamma, or otherwise performs a biased nearest-neighbor step. Our approach proceeds in three stages. First, we derive a renewal equation for the ruin probability q_z(gamma) by conditioning on the first step. Second, we develop a spectral representation on a weighted Hilbert space that diagonalizes the transition operator and yields explicit closed-form expressions. Third, this representation enables a precise critical-point analysis in state space. Our central result is a striking geometric invariance: when the domain size a is even, the ruin