Aging Record Statistics in Saturating Self-Interacting Random Walks

TL;DR

This paper derives the exact asymptotic distribution of record ages in saturating self-interacting random walks, revealing how memory effects induce aging phenomena in non-Markovian processes.

Contribution

It provides the first exact asymptotic distribution of record ages in a broad class of non-Markovian processes, elucidating aging dynamics with saturating self-interaction.

Findings

Identifies two asymptotic regimes: short-time governed by explored region geometry, long-time dominated by memory effects.

Shows memory effects induce aging, causing dependence of record age on the record index k.

Extends understanding of non-Markovian dynamics beyond scaling theory with exact results.

Abstract

The record age tau_k, defined as the time between the k-th and k+1-st record-breaking events, is a central observable of extreme-value statistics. In Markovian processes, the absence of memory makes tau_k independent of k. How memory breaks this invariance and induces aging, meaning a dependence of tau_k on k, remains a fundamental question, closely connected to widely observed aging phenomena in non-Markovian dynamics. In this Letter, we derive the exact asymptotic distribution of tau_k for saturating self-interacting random walks, a broad class of non-Markovian processes. We uncover two asymptotic regimes, in agreement with recent scaling predictions: at short times (tau much smaller than k squared), record statistics are governed by the geometry of the explored region, while at long times (tau much larger than k squared), memory effects become subdominant and reduce to nontrivial prefactor corrections. Our exact result provides a rare analytic window beyond scaling theory and extends to a framework that fully quantifies aging dynamics in the presence of saturating self-interaction.