## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 27, Number 5 (2017), 3050-3112.

### Universality in marginally relevant disordered systems

Francesco Caravenna, Rongfeng Sun, and Nikos Zygouras

#### Abstract

We consider disordered systems of a 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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 27, Number 5 (2017), 3050-3112.

**Dates**

Received: August 2016

Revised: December 2016

First available in Project Euclid: 3 November 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1509696041

**Digital Object Identifier**

doi:10.1214/17-AAP1276

**Mathematical Reviews number (MathSciNet)**

MR3719953

**Zentralblatt MATH identifier**

06822212

**Subjects**

Primary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Secondary: 82D60: Polymers 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

**Keywords**

Directed polymer pinning model polynomial chaos disordered system fourth moment theorem marginal disorder relevance stochastic heat equation

#### Citation

Caravenna, Francesco; Sun, Rongfeng; Zygouras, Nikos. Universality in marginally relevant disordered systems. Ann. Appl. Probab. 27 (2017), no. 5, 3050--3112. doi:10.1214/17-AAP1276. https://projecteuclid.org/euclid.aoap/1509696041