Universality in marginally relevant disordered systems

TL;DR

This paper demonstrates that various marginally relevant disordered systems, including directed polymers and pinning models, converge to a universal log-normal limit in a weak disorder regime, revealing shared chaos structures.

Contribution

It establishes a universal limit for different disordered models and links their behavior to the two-dimensional Stochastic Heat Equation, using the Fourth Moment Theorem.

Findings

Convergence to a universal log-normal random field in the weak disorder limit.

Shared chaos structure among different marginally relevant disordered systems.

Connection between disordered models and the two-dimensional Stochastic Heat Equation.

Abstract

We consider disordered systems of directed polymer type, for which disorder is so-called marginally relevant. These include the usual (short-range) directed polymer model in dimension (2+1), the long-range directed polymer model with Cauchy tails in dimension (1+1) and the disordered pinning model with tail exponent 1/2. We show that in a suitable weak disorder and continuum limit, the partition functions of these different models converge to a universal limit: a log-normal random field with a multi-scale correlation structure, which undergoes a phase transition as the disorder strength varies. As a by-product, we show that the solution of the two-dimensional Stochastic Heat Equation, suitably regularized, converges to the same limit. The proof, which uses the celebrated Fourth Moment Theorem, reveals an interesting chaos structure shared by all models in the above class.