Search

We investigate the effect of correlated disorder on the localization transition undergone by a renewal sequence with loop exponent $\alpha$ > 0, when the correlated sequence is given by another independent renewal set with loop exponent $\alpha$ > 0. Using the renewal structure of the disorder sequence, we compute the annealed critical point and exponent. Then, using a smoothing inequality for the quenched free energy and second moment estimates for the quenched partition function, combined with decoupling inequalities, we prove that in the case $\alpha$ > 2 (summable correlations), disorder is irrelevant if $\alpha$ < 1/2 and relevant if $\alpha$ > 1/2, which extends the Harris criterion for independent disorder. The case $\alpha$ $\in$ (1, 2) (non-summable correlations) remains largely open, but we are able to prove that disorder is relevant for $\alpha$ > 1/ $\alpha$, a condition that is expected to be non-optimal. Predictions on the criterion for disorder relevance in this case are discussed. Finally, the case $\alpha$ $\in$ (0, 1) is somewhat special but treated for completeness: in this case, disorder has no effect on the quenched free energy, but the annealed model exhibits a phase transition.

International audience; We consider a continuum percolation model on $\R^d$, where $d\geq 4$.The occupied set is given by the union of independent Wiener sausages with radius $r$ running up to time $t$ and whoseinitial points are distributed according to a homogeneous Poisson point process.It was established in a previous work by Erhard, Mart\'{i}nez and Poisat~\cite{EMP13} that (1) if $r$ is small enough there is a non-trivial percolation transitionin $t$ occuring at a critical time $t_c(r)$ and (2) in the supercritical regime the unbounded cluster is unique. In this paper we investigate the asymptotic behaviour of the critical time when the radius $r$ converges to $0$. The latter does not seem to be deducible from simple scaling arguments. We prove that for $d\geq 4$, there is a positive constant $c$ such that$c^{-1}\sqrt{\log(1/r)}\leq t_c(r)\leq c\sqrt{\log(1/r)}$ when $d=4$ and $c^{-1}r^{(4-d)/2}\leq t_c(r) \leq c\ r^{(4-d)/2}$ when $d\geq 5$, as $r$ converges to $0$. We derive along the way moment estimates on the capacity of Wiener sausages, which may be of independent interest.

We consider a continuum percolation model on $$\mathbb {R}^d$$ , $$d\ge 1$$ . For $$t,\lambda \in (0,\infty )$$ and $$d\in \{1,2,3\}$$ , the occupied set is given by the union of independent Brownian paths running up to time t whose initial points form a Poisson point process with intensity $$\lambda >0$$ . When $$d\ge 4$$ , the Brownian paths are replaced by Wiener sausages with radius $$r>0$$ . We establish that, for $$d=1$$ and all choices of t, no percolation occurs, whereas for $$d\ge 2$$ , there is a non-trivial percolation transition in t, provided $$\lambda $$ and r are chosen properly. The last statement means that $$\lambda $$ has to be chosen to be strictly smaller than the critical percolation parameter for the occupied set at time zero (which is infinite when $$d\in \{2,3\}$$ , but finite and dependent on r when $$d\ge 4$$ ). We further show that for all $$d\ge 2$$ , the unbounded cluster in the supercritical phase is unique. Along the way a finite box criterion for non-percolation in the Boolean model is extended to radius distributions with an exponential tail. This may be of independent interest. The present paper settles the basic properties of the model and should be viewed as a springboard for finer results.

We study an undirected polymer chain living on the 1-dimensional integer lattice and carrying i.i.d.\ random charges. Each self-intersection of the polymer contributes to the Hamiltonian an energy that is equal to the product of the charges of the two monomers that meet. The joint law for the polymer chain and the charges is given by the Gibbs distribution associated with the Hamiltonian. The focus is on the \emph{annealed free energy} per monomer in the limit as polymer length diverges. We derive a \emph{spectral representation} for the free energy and we exhibit a critical curve in the parameter plane of charge bias versus inverse temperature separating a \emph{ballistic phase} from a \emph{subballistic phase}. The phase transition is \emph{first order}. We prove large deviation principles for the laws of the empirical speed and the empirical charge, and derive a spectral representation for the associated rate functions. Interestingly, in both phases both rate functions exhibit \emph{flat pieces}, which correspond to an inhomogeneous strategy for the polymer to realise a large deviation. The large deviation principles lead to laws of large numbers and central limit theorems. We identify the scaling behavior of the critical curve for small and large charge bias and also the scaling behavior of the free energy for small charge bias and small inverse temperature. Both are linked to an associated \emph{Sturm-Liouville eigenvalue problem}. A key tool in our analysis is the Ray-Knight formula for the local times of 1-dimensional simple random walk. It is used to derive a \emph{closed form expression} for the generating function of the annealed partition function and to derive the spectral representation for the free energy. What happens for the \emph{quenched free energy} remains open. We state two modest results and raise a few questions., Accepted for publication in Mathematical Physics, Analysis and Geometry, 76 pages, 13 figures

We study random pinning and copolymer models, when the return distribution of the underlying renewal process has a polynomial tail with finite mean. We compute the asymptotic behavior of the critical curves of the models in the weak coupling regime, showing that it is universal. This proves a conjecture of Bolthausen, den Hollander and Opoku for copolymer models (ref. [8]), which we also extend to pinning models., Added a heuristic explanation, updated references, and other minor changes; version to appear on CMP

Random walks in random sceneries (RWRS) are simple examples of stochastic processes in disordered media. They were introduced at the end of the 70's by Kesten-Spitzer and Borodin, motivated by the construction of new self-similar processes with stationary increments. Two sources of randomness enter in their definition: a random field $\xi = (\xi_x)_{x \in \Z^d}$ of i.i.d.\ random variables, which is called the \emph{random scenery}, and a random walk $S = (S_n)_{n \in \N}$ evolving in $\Z^d$, independent from the scenery. The RWRS $Z = (Z_n)_{n \in \N}$ is then defined as the accumulated scenery along the trajectory of the random walk, i.e., $Z_n := \sum_{k=1}^n \xi_{S_k}$. The law of $Z$ under the joint law of $\xi$ and $S$ is called "annealed", and the conditional law given $\xi$ is called "quenched". Recently, central limit theorems under the quenched law were proved for $Z$ by the first two authors for a class of transient random walks including walks with finite variance in dimension $d \ge 3$. In this paper we extend their results to dimension $d=2$., Comment: 11 pages; weaker condition on the moment of the scenery

Random walks in random scenery are processes defined by Z n : = ∑ k = 1 n ω S k where S : = ( S k , k ≥ 0 ) is a random walk evolving in Z d and ω : = ( ω x , x ∈ Z d ) is a sequence of i.i.d. real random variables. Under suitable assumptions on the random walk S and the random scenery ω , almost surely with respect to ω , the correctly renormalized sequence ( Z n ) n ≥ 1 is proved to converge in distribution to a centered Gaussian law with explicit variance.

The purpose of this paper is to show how one can extend some results on disorder relevance obtained for the random pinning model with i.i.d disorder to the model with finite range correlated disorder. In a previous work, the annealed critical curve of the latter model was computed, and equality of quenched and annealed critical points, as well as exponents, was proved under some conditions on the return exponent of the interarrival times. Here we complete this work by looking at the disorder relevant regime, where annealed and quenched critical points differ. All these results show that the Harris criterion, which was proved to be correct in the i.i.d case, remains valid in our setup. We strongly use Markov renewal constructions that were introduced in the solving of the annealed model.