The critical disordered pinning measure

TL;DR

This work analyzes a marginally relevant disordered pinning model at criticality and proves the existence of a universal limiting random measure, the CDPM, in the critical window. It develops a robust framework combining polynomial chaos, coarse graining, and Dickman-subordinator renewal analysis to show convergence of time-to-time and space-to-space measures, and links these discrete limits to continuous SPDEs via critical SVE and SHE correspondences. A key contribution is the equivalence between convergences of pinning and directed polymer measures, enabling transfer of results to the continuous setting and to rough volatility models. The results illuminate universal scaling behavior in marginally relevant disordered systems and provide a rigorous bridge between discrete pinning, directed polymers, and critical SPDEs, with potential applications to stochastic volatility and interface models.

Abstract

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

Theorems & Definitions (53)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8
• Remark 1.9
• Theorem 1.10