Abstract. 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 p2`1q, the long-range directed polymer model with Cauchy tails in dimension p1`1q 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 multiscale 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.

Abstract. Inspired by recent work of Alberts, Khanin and Quastel [AKQ14a], we formulate general conditions ensuring that a sequence of multi-linear polynomials of independent random variables (called polynomial chaos expansions) converges to a limiting random variable, given by a Wiener chaos expansion over the d-dimensional white noise. A key ingredient in our approach is a Lindeberg principle for polynomial chaos expansions, which extends earlier work of Mossel, O'Donnell and Oleszkiewicz [MOO10]. These results provide a unified framework to study the continuum and weak disorder scaling limits of statistical mechanics systems that are disorder relevant, including the disordered pinning model, the (long-range) directed polymer model in dimension 1 + 1, and the two-dimensional random field Ising model. This gives a new perspective in the study of disorder relevance, and leads to interesting new continuum models that warrant further studies.

We consider a random field ϕ : {1, . . . , N } → R as a model for a linear chain attracted to the defect line ϕ = 0, that is, the x-axis. The free law of the field is specified by the density exp(− i V (∆ϕi)) with respect to the Lebesgue measure on R N , where ∆ is the discrete Laplacian and we allow for a very large class of potentials V (·). The interaction with the defect line is introduced by giving the field a reward ε ≥ 0 each time it touches the x-axis. We call this model the pinning model. We consider a second model, the wetting model, in which, in addition to the pinning reward, the field is also constrained to stay nonnegative.We show that both models undergo a phase transition as the intensity ε of the pinning reward varies: both in the pinning (a = p) and in the wetting (a = w) case, there exists a critical value ε a c such that when ε > ε a c the field touches the defect line a positive fraction of times (localization), while this does not happen for ε < ε a c (delocalization). The two critical values are nontrivial and distinct: 0 < ε p c < ε w c < ∞, and they are the only nonanalyticity points of the respective free energies. For the pinning model the transition is of second order, hence the field at ε = ε p c is delocalized. On the other hand, the transition in the wetting model is of first order and for ε = ε w c the field is localized. The core of our approach is a Markov renewal theory description of the field.if J k = 0, k a,ε J k−1 ,J k (n)1 (n≥2) m≥2 k a,ε J k−1 ,J k (m), if J k = 0, J k = ∞,

Let $\{S_n\}$ be a random walk in the domain of attraction of a stable law $\mathcal{Y}$, i.e. there exists a sequence of positive real numbers $(a_n)$ such that $S_n/a_n$ converges in law to $\mathcal{Y}$. Our main result is that the rescaled process $(S_{\lfloor nt\rfloor}/a_n, t\ge 0)$, when conditioned to stay positive, converges in law (in the functional sense) towards the corresponding stable L\'{e}vy process conditioned to stay positive. Under some additional assumptions, we also prove a related invariance principle for the random walk killed at its first entrance in the negative half-line and conditioned to die at zero.Comment: Published in at http://dx.doi.org/10.1214/07-AIHP119 the Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques (http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics (http://www.imstat.org

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