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October 2017 Universality in marginally relevant disordered systems
Francesco Caravenna, Rongfeng Sun, Nikos Zygouras
Ann. Appl. Probab. 27(5): 3050-3112 (October 2017). DOI: 10.1214/17-AAP1276


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.


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Francesco Caravenna. Rongfeng Sun. Nikos Zygouras. "Universality in marginally relevant disordered systems." Ann. Appl. Probab. 27 (5) 3050 - 3112, October 2017.


Received: 1 August 2016; Revised: 1 December 2016; Published: October 2017
First available in Project Euclid: 3 November 2017

zbMATH: 06822212
MathSciNet: MR3719953
Digital Object Identifier: 10.1214/17-AAP1276

Primary: 82B44
Secondary: 60K35, 82D60

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.27 • No. 5 • October 2017
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