Critical Spectral Invariants in Random Walks with Geometric Resetting
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918409293266944 |
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| author | Coso, Juan Antonio Vega |
| author_facet | Coso, Juan Antonio Vega |
| contents | 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 |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24803 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Critical Spectral Invariants in Random Walks with Geometric Resetting Coso, Juan Antonio Vega Probability 60G55, 60K05 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 |
| title | Critical Spectral Invariants in Random Walks with Geometric Resetting |
| topic | Probability 60G55, 60K05 |
| url | https://arxiv.org/abs/2603.24803 |