Critical Spectral Invariants in Random Walks with Geometric 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

Figures (8)

• Figure 1: Ruin probability for biased walk ($p=0.6$, $a=5$) as a function of initial position $z$ for various reset rates $\gamma$. The critical position where the effect of resetting changes sign is approximately $z^* \approx 2.5$.
• Figure 2: Ruin probability for symmetric walk ($p=0.5$, $a=5$). The midpoint $z^*=2.5$ remains at $q_{z^*}(\gamma) = 0.5$ for all $\gamma$, demonstrating perfect midpoint invariance.
• Figure 3: Derivative $\partial q_z(\gamma) / \partial \gamma$ as a function of initial position $z$ for $a = 10$. The critical point $z^\dagger = 5 = a/2$ (marked with circle) is universal: independent of both the bias parameter $p$ and the reset parameter $\gamma$. For all $p \in (0,1)$, resetting increases ruin probability when $z < 5$ and decreases it when $z > 5$. The symmetric case $p = 0.5$ (blue solid lines) shows perfect antisymmetry about $z = 5$, while the asymmetric cases ($p=0.4$ green densely dashed, $p=0.6$ red dashed) show magnitude and asymmetry differences.
• Figure 4: Derivative $\partial q_z(\gamma) / \partial \gamma$ as a function of initial position $z$ for $a = 11$ (odd). Unlike the even case (Figure \ref{['fig:critical_point']}), the sign-change point is not universal and depends on the bias parameter $p$. The theoretical midpoint $a/2 = 5.5$ (orange dashed line) is not an integer position. For the symmetric case $p = 0.5$ (blue curves), the sign change occurs between $z=5$ and $z=6$, with both positions showing small values $|h_{z}(\gamma)| \sim O(1/a)$, confirming approximate invariance. For biased walks ($p \neq 0.5$), the critical point shifts: $p = 0.4$ (green densely dashed) shows stronger asymmetry toward larger $z$, while $p = 0.6$ (red dashed) shifts the balance toward smaller $z$. The effect magnitude still decreases with $\gamma$, but the lack of exact symmetry protection means no single position exhibits perfect invariance.
• Figure 5: Symmetric case ($p = q = 0.5$) for $a = 10$. Universal antisymmetry about $z = 5$ with $\gamma$-dependent magnitude.

Theorems & Definitions (7)

• proof
• proof : Proof sketch
• proof
• Lemma 1: Boundary behavior
• proof
• Lemma 2: Existence of sign change
• proof