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The critical disordered pinning measure

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  • Ran Wei
  • Jinjiong Yu

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.

Suggested Citation

  • Ran Wei & Jinjiong Yu, 2024. "The critical disordered pinning measure," Papers 2402.17642, arXiv.org, revised Mar 2024.
    • Handle: RePEc:arx:papers:2402.17642
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    References listed on IDEAS

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    1. Eyal Neuman & Mathieu Rosenbaum, 2017. "Fractional Brownian motion with zero Hurst parameter: a rough volatility viewpoint," Papers 1711.00427, arXiv.org, revised May 2018.
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