  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.
