The critical disordered pinning measure

TL;DR

This paper investigates a disordered pinning model with finite moments, demonstrating convergence of partition functions to a unique critical measure and connecting it to critical stochastic equations relevant in volatility and heat transfer models.

Contribution

It introduces the critical disordered pinning measure as a limit in the critical window and links it to critical stochastic Volterra and heat equations.

Findings

Partition functions converge to a unique limiting measure.

Results apply to models related to rough volatility and stochastic heat equations.

Establishes a phase transition in the intermediate disorder regime.

Abstract

In this paper, we study a disordered pinning model induced by a random walk whose increments have a finite $(2+\kappa)$-th moment for some $\kappa>0$. 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.