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We consider a one-dimensional simple random walk surviving among a field of static soft traps : each time it meets a trap the walk is killed with probability 1--e --$\beta$ , where $\beta$ is a positive and fixed parameter. The positions of the traps are sampled independently from the walk and according to a renewal process. The increments between consecutive traps, or gaps, are assumed to have a power-law decaying tail with exponent $\gamma$ > 0. We prove convergence in law for the properly rescaled logarithm of the quenched survival probability as time goes to infinity. The normalization exponent is $\gamma$/($\gamma$ + 2), while the limiting law writes as a variational formula with both universal and non-universal features. The latter involves (i) a Poisson point process that emerges as the universal scaling limit of the properly rescaled gaps and (ii) a function of the parameter $\beta$ that we call asymptotic cost of crossing per trap and that may, in principle, depend on the details of the gap distribution. Our proof suggests a confinement strategy of the walk in a single large gap. This model may also be seen as a (1 + 1)-directed polymer among many repulsive interfaces, in which case $\beta$ corresponds to the strength of repulsion, the survival probability to the partition function and its logarithm to the finite-volume free energy. Along the way we prove a stochastic monotonicity property for the hitting time of the killed random walk with respect to the non-killed one, that could be of interest in other contexts, see Proposition 3.5., Comment: Version 2: shorter proof of Proposition 3.5

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.

This paper considers an undirected polymer chain on $\mathbb{Z}^d$, $d \geq 2$, with i.i.d.\ random charges attached to its constituent monomers. Each self-intersection of the polymer chain contributes an energy to the interaction Hamiltonian that is equal to the product of the charges of the two monomers that meet. The joint probability distribution for the polymer chain and the charges is given by the Gibbs distribution associated with the interaction Hamiltonian. The object of interest is the \emph{annealed free energy} per monomer in the limit as the length $n$ of the polymer chain tends to infinity. We show that there is a critical curve in the parameter plane spanned by the charge bias and the inverse temperature separating an \emph{extended phase} from a \emph{collapsed phase}. We derive the scaling of the critical curve for small and for large charge bias and the scaling of the annealed free energy for small inverse temperature, which are both anomalous. We show that in a subset of the collapsed phase the polymer chain is \emph{subdiffusive}, namely, on scale $(n/\log n)^{1/(d+2)}$ it moves like a Brownian motion conditioned to stay inside a ball with a deterministic radius and a randomly shifted center. We expect this scaling to hold throughout the collapsed phase. We further expect that in the extended phase the polymer chain scales like a weakly self-avoiding walk. Proofs are based on a detailed analysis for simple random walk of the downward large deviations of the self-intersection local time and the upward large deviations of the range. Part of our scaling results are rough, and we formulate conjectures under which they can be sharpened. The existence of the free energy remains an open problem, which we are able to settle in a subset of the collapsed phase for a subclass of charge distributions., Comment: 35 pages, 6 figures

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

We investigate the disordered copolymer and pinning models, in the case of a correlated Gaussian environment with summable correlations, and when the return distribution of the underlying renewal process has a polynomial tail. As far as the copolymer model is concerned, we prove disorder relevance both in terms of critical points and critical exponents, in the case of non-negative correlations. When some of the correlations are negative, even the annealed model becomes non-trivial. Moreover, when the return distribution has a finite mean, we are able to compute the weak coupling limit of the critical curves for both models, with no restriction on the correlations other than summability. This generalizes the result of Berger, Caravenna, Poisat, Sun and Zygouras \cite{cf:BCPSZ} to the correlated case. Interestingly, in the copolymer model, the weak coupling limit of the critical curve turns out to be the maximum of two quantities: one generalizing the limit found in the IID case \cite{cf:BCPSZ}, the other one generalizing the so-called Monthus bound., Comment: 35 pages

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.

When an i.i.d.\ sequence of letters is cut into words according to i.i.d.\ renewal times, an i.i.d.\ sequence of words is obtained. In the \emph{annealed} LDP (large deviation principle) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t.\ the reference law of words. In Birkner, Greven and den Hollander \cite{BGdH10} the \emph{quenched} LDP (= conditional on a typical letter sequence) was derived for the case where the renewal times have an \emph{algebraic} tail. The rate function turned out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t.\ the reference law of letters, obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. The purpose of the present paper is to extend both LDP's to letter sequences that are not i.i.d. It is shown that both LDP's carry over when the letter sequence satisfies a mixing condition called \emph{summable variation}. The rate functions are again given by specific relative entropies w.r.t.\ the reference law of words, respectively, letters. But since neither of these reference laws is i.i.d., several approximation arguments are needed to obtain the extension., Comment: 15 pages. Corrections in Proposition 3.1 and Lemma 4.3, new Lemma 5.3, additional explanations in Section 4.4 and other minor modifications

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.

This paper focuses on directed polymers pinned at a disordered and correlated interface. We assume that the disorder sequence is a $q$-order moving average and show that the critical curve of the annealed model can be expressed in terms of the Perron–Frobenius eigenvalue of an explicit transfer matrix, which generalizes the annealed bound of the critical curve for i.i.d. disorder. We provide explicit values of the annealed critical curve for $q=1$ and $q=2$ and a weak disorder asymptotic in the general case. Following the renewal theory approach of pinning, the processes arising in the study of the annealed model are particular Markov renewal processes. We consider the intersection of two replicas of this process to prove a result of disorder irrelevance (i.e. quenched and annealed critical curves as well as exponents coincide) via the method of second moment.