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

Universality in marginally relevant disordered systems