In this paper we look at the pinning of a directed polymer by a onedimensional linear interface carrying random charges. There are two phases, localized and delocalized, depending on the inverse temperature and on the disorder bias. Using quenched and annealed large deviation principles for the empirical process of words drawn from a random letter sequence according to a random renewal process [Birkner, Greven and den Hollander, Probab. Theory Related Fields 148 (2010) 403-456], we derive variational formulas for the quenched, respectively, annealed critical curve separating the two phases. These variational formulas are used to obtain a necessary and sufficient criterion, stated in terms of relative entropies, for the two critical curves to be different at a given inverse temperature, a property referred to as relevance of the disorder. This criterion in turn is used to show that the regimes of relevant and irrelevant disorder are separated by a unique inverse critical temperature. Subsequently, upper and lower bounds are derived for the inverse critical temperature, from which sufficient conditions under which it is strictly positive, respectively, finite are obtained. The former condition is believed to be necessary as well, a problem that we will address in a forthcoming paper.Random pinning has been studied extensively in the literature. The present paper opens up a window with a variational view. Our variational formulas for the quenched and the annealed critical curve are new and provide valuable insight into the nature of the phase transition. Our results on the inverse critical temperature drawn from these variational formulas are not new, but they offer an alternative approach, that is, flexible enough to be extended to other models of random polymers with disorder. quenched vs. annealed large deviation principle, quenched vs. annealed critical curve, relevant vs. irrelevant disorder, critical temperature. que c (0) = 0; see Appendix A. It was shown in Alexander and Sidoravicius [2] that h que c (β) > 0 for β ∈ (0, ∞). Therefore we have the qualitative picture drawn in Figure 2. We further remark that lim β→∞ h que c (β)/β is finite if and only if supp(μ 0 ) is bounded from above.The mean value of the disorder is E(βω 0 −h) = −h. Thus, we see from Figure 2 that for the random pinning model localization may even occur for moderately negative mean values of the disorder, contrary to what happens for the homogeneous pinning model, where localization occurs only for a strictly positive parameter; see Appendix A. In other words, even a globally repulsive random interface can pin the polymer: all that the polymer needs to do is to hit some positive values of the disorder and avoid the negative values of the disorder.

We study the singular values of certain triangular random matrices. When their elements are i.i.d. standard complex Gaussian random variables, the squares of the singular values form a biorthogonal ensemble, and with an appropriate change in the distribution of the diagonal elements, they give the biorthogonal Laguerre ensemble. For triangular Wigner matrices, we give alternative proofs for the convergence of the empirical distribution of the appropriately scaled squares of the singular eigenvalues to a distribution with support [0, e], as well as for the almost sure convergence of the rescaled largest singular eigenvalue to √ e under the additional assumption of mean zero and finite fourth moment for the law of the matrix elements. n i=1 δ λ n,m i 1

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According to a theorem of S. Schumacher and T. Brox, for a diffusion X in a Brownian environment it holds that (Xt − b log t )/ log 2 t → 0 in probability, as t → ∞, where b is a stochastic process having an explicit description and depending only on the environment. We compute the distribution of the number of sign changes for b on an interval [1, x] and study some of the consequences of the computation; in particular we get the probability of b keeping the same sign on that interval. These results have been announced in 1999 in a non-rigorous paper by P. Le Doussal, C. Monthus, and D. Fisher and were treated with a Renormalization Group analysis. We prove that this analysis can be made rigorous using a path decomposition for the Brownian environment and renewal theory. Finally, we comment on the information these results give about the behavior of the diffusion.