The continuum disordered pinning model.

Area of Science:

• Probability Theory
• Stochastic Processes
• Statistical Mechanics

Background:

• Renewal processes with polynomial tails possess a non-trivial scaling limit, the alpha-stable regenerative set.
• Disordered pinning models are Gibbs transformations of renewal processes in an i.i.d. random environment.

Purpose of the Study:

• To investigate the scaling limit of disordered pinning models for alpha=2.
• To introduce and characterize the continuum disordered pinning model (CDPM).

Main Methods:

• Analysis of Gibbs transformations of renewal processes in a random environment.
• Study of scaling limits and universal properties of stochastic models.

Main Results:

• For alpha=2, disordered pinning models exhibit a universal scaling limit, the continuum disordered pinning model (CDPM).
• The CDPM is a random closed subset of R^2 in a white noise random environment.
• While fixed properties (e.g., Hausdorff dimension) are preserved, the law of the CDPM is singular with respect to the alpha-stable regenerative set.

Conclusions:

• The existence of the CDPM demonstrates disorder relevance for pinning models when alpha=2.
• The CDPM represents a novel disordered continuum model with unique statistical properties.