The critical disordered pinning measure

Ran Wei and Jinjiong Yu

Papers from arXiv.org

Abstract: In this paper, we study a disordered pinning model induced by a random walk whose increments have a finite fourth moment and vanishing first and third moments. It is known that this model is marginally relevant, and moreover, it undergoes a phase transition in an intermediate disorder regime. We show that, in the critical window, the point-to-point partition functions converge to a unique limiting random measure, which we call the critical disordered pinning measure. We also obtain an analogous result for a continuous counterpart to the pinning model, which is closely related to two other models: one is a critical stochastic Volterra equation that gives rise to a rough volatility model, and the other is a critical stochastic heat equation with multiplicative noise that is white in time and delta in space.

Date: 2024-02, Revised 2024-03
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