Strongly correlated stochastic systems

TL;DR

This thesis introduces universal analytical tools for strongly correlated stochastic systems, focusing on extreme value and gap statistics, especially in systems with stochastic resetting that induce long-range correlations.

Contribution

It develops a universal framework for conditionally independent variables and derives closed-form expressions for various models with resetting, advancing understanding of non-equilibrium steady states.

Findings

Resetting induces analytically tractable non-equilibrium steady states.

Universal formulas derived for diverse models like Brownian motion and Levy flights.

Resetting can enhance or impair search efficiency, with proposed improved protocols.

Abstract

This thesis develops exact analytical tools to study strongly correlated stochastic systems, with a focus on extreme value statistics, gap statistics, and full counting statistics in multi-particle processes. A central contribution is the universal characterization of conditionally independent identically distributed variables-random variables that become independent upon conditioning on latent parameters. This structure arises naturally in systems with stochastic resetting, a mechanism that generates strong long-range correlations while retaining analytical tractability. Using this framework, we derive universal closed-form expressions for several observables across diverse models, including Brownian motion, Levy flights, ballistic particles, and Dyson Brownian motion, under various resetting protocols. In particular, we demonstrate that resetting induces analytically tractable non-equilibrium steady states. Theoretical predictions are supported by numerical comparisons and experimental comparisons in systems such as diffusive particles in switching harmonic traps. Applications to search optimization are also explored, identifying regimes where resetting enhances or impairs first-passage efficiency, and proposing rescaling-based protocols that outperform traditional resetting.