Hubert Lacoin

A TWO CITIES THEOREM FOR THE PARABOLIC ANDERSON MODEL

The parabolic Anderson problem is the Cauchy problem for the heat equation partial derivative(t)u(t, z) = Delta u(t,z) + xi(z)u(t,z) on (0,infinity) x Z(d) with random potential (xi(z): z is an element of Z(d)). We consider independent and identically distributed potentials, such that the distribution function of (z) converges polynomially at infinity. If u is initially localized in the origin, that is, if u(0, z) = 1(0)(z), we show that, as time goes to infinity, the solution is completely localized in two points almost surely and in one point with high probability. We also identify the asymptotic behavior of the concentration sites in terms of a weak limit theorem.

A Scaling Limit Theorem for the Parabolic Anderson Model with Exponential Potential

The parabolic Anderson problem is the Cauchy problem for the heat equation with random potential and localized initial condition. In this paper, we consider potentials which are constant in time and independent exponentially distributed in space. We study the growth rate of the total mass of the solution in terms of weak and almost sure limit theorems, and the spatial spread of the mass in terms of a scaling limit theorem. The latter result shows that in this case, just like in the case of heavy tailed potentials, the mass gets trapped in a single relevant island with high probability.

Copolymers at Selective Interfaces: New Bounds on the Phase Diagram

AbstractWe investigate the phase diagram of disordered copolymers at the interface between two selective solvents, and in particular its weak-coupling behavior, encoded in the slope mc of the critical line at the origin. We focus on the directed walk case, which has turned out to be, in spite of the apparent simplicity, extremely challenging. In mathematical terms, the partition function of such a model does not depend on all the details of the Markov chain that models the polymer, but only on the time elapsed between successive returns to zero and on whether the walk is in the upper or lower half plane between such returns. This observation leads to a natural generalization of the model, in terms of arbitrary laws of return times: the most interesting case being the one of return times with power law tails (with exponent 1+α, α=1/2 in the case of the symmetric random walk). The main results we present here are: (1)the improvement of the known result 1/(1+α)≤mc≤1, as soon as α>1 for what concerns the upper bound, and down to α≈0.65 for the lower bound.(2)a proof of the fact that the critical curve lies strictly below the critical curve of the annealed model for every non-zero value of the coupling parameter. We also provide an argument that rigorously shows the strong dependence of the phase diagram on the details of the return probability (and not only on the tail behavior). Lower bounds are obtained by exhibiting a new localization strategy, while upper bounds are based on estimates of non-integer moments of the partition function.

Influence of spatial correlation for directed polymers

In this paper, we study a model of a Brownian polymer in ℝ + x ℝ d , introduced by Rovira and Tindel [J. Funct. Anal. 222 (2005) 178-201]. Our investigation focuses mainly on the effect of strong spatial correlation in the environment in that model in terms of free energy, fluctuation exponent and volume exponent. In particular, we prove that under some assumptions, very strong disorder and superdiffusivity hold at all temperatures when d ≥ 3 and provide a novel approach to Petermanns superdiffusivity result in dimension one [Superdiffusivity of directed polymers in random environment (2000) Ph.D. thesis]. We also derive results for a Brownian model of pinning in a nonrandom potential with power-law decay at infinity.

Mixing time and cutoff for the adjacent transposition shuffle and the simple exclusion

In this paper, we investigate the mixing time of the adjacent transposition shuffle for a deck of cards. We prove that around time N^2\log N/(2\pi^2), the total-variation distance to equilibrium of the deck distribution drops abruptly from 1 to 0, and that the separation distance has a similar behavior but with a transition occurring at time (N^2\log N)/\pi^2. This solves a conjecture formulated by David Wilson. We present also similar results for the exclusion process on a segment of length N with k particles.

The Effect of Disorder on the Free-Energy for the Random Walk Pinning Model: Smoothing of the Phase Transition and Low Temperature Asymptotics

We consider the continuous time version of the Random Walk Pinning Model (RWPM), studied in (Berger and Toninelli (Electron. J. Probab., to appear) and Birkner and Sun (Ann. Inst. Henri Poincaré Probab. Stat. 46:414–441, 2010; arXiv:0912.1663). Given a fixed realization of a random walk Y on ℤd with jump rate ρ (that plays the role of the random medium), we modify the law of a random walk X on ℤd with jump rate 1 by reweighting the paths, giving an energy reward proportional to the intersection time

A Smoothing Inequality for Hierarchical Pinning Models

L_{t}(X,Y)=\int_{0}^{t} \mathbf {1}_{X_{s}=Y_{s}}\,\mathrm {d}s

The high-temperature behavior for the directed polymer in dimension 1+2

: the weight of the path under the new measure is exp (βLt(X,Y)), β∈ℝ. As β increases, the system exhibits a delocalization/localization transition: there is a critical value βc, such that if β>βc the two walks stick together for almost-all Y realizations. A natural question is that of disorder relevance, that is whether the quenched and annealed systems have the same behavior. In this paper we investigate how the disorder modifies the shape of the free energy curve: (1) We prove that, in dimension d≥3, the presence of disorder makes the phase transition at least of second order. This, in dimension d≥4, contrasts with the fact that the phase transition of the annealed system is of first order. (2) In any dimension, we prove that disorder modifies the low temperature asymptotic of the free energy.

Total variation and separation cutoffs are not equivalent and neither one implies the other

We consider a hierarchical pinning model introduced by B. Derrida, V. Hakim and J. Vannimenus in [3], which undergoes a localization/delocalization phase transition. This depends on a parameter B > 2, related to the geometry of the hierarchical lattice. We prove that the phase transition is of second order in presence of disorder. This implies that disorder smoothes the transition in the so-called relevant disorder case, i.e., \( B > B_c = 2 + \sqrt 2 \).

The Simple Exclusion Process on the Circle has a diffusive Cutoff Window

We investigate the high-temperature behavior of the directed polymer model in dimension