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From disorder relevance to the 2d Stochastic Heat Equation.

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We consider statistical mechanics models defined on a lattice, such as pinning models, directed polymers, random field Ising model, in which disorder acts as an external random field. Such models are called disorder relevant, if arbitrar- ily weak disorder changes the qualitative properties of the model. Via a Lindeberg principle for multilinear polynomials we show that disorder relevance manifests it- self through the existence of a disordered high-temperature limit for the partition function, which is given in terms of Wiener chaos and is model specific. When disorder becomes marginally relevant a fundamentally new structure emerges, which leads to a universal scaling limit for all different (currently of directed poly- mer type) models that fall in this class. A notable such representative is the two dimensional SHE with multipicative space-time white noise (which in the SPDE language is characterised as “critical”). In this case certain analogies with Gaussian Multiplicative Chaos and log-correlated Gaussian fields appear. Based on joint works with Francesco Caravenna and Rongfeng Sun.

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