Julien Poisat

Quenched central limit theorems for random walks in random scenery

Random walks in random scenery are processes defined by Zn:=∑k=1nωSk where S:=(Sk,k≥0) is a random walk evolving in Zd and ω:=(ωx,x∈Zd) 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 (Zn)n≥1 is proved to converge in distribution to a centered Gaussian law with explicit variance.

On quenched and annealed critical curves of random pinning model with finite range correlations

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

Random pinning model with finite range correlations: Disorder relevant regime

q=1

Annealed scaling for a charged polymer

,

Annealed scaling for a charged polymer in dimensions two and higher

2

The Critical Curves of the Random Pinning and Copolymer Models at Weak Coupling

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.

On the critical curves of the pinning and copolymer models in correlated Gaussian environment

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.

Asymptotics of the critical time in Wiener sausage percolation with a small radius

This paper studies an undirected polymer chain living on the one-dimensional integer lattice and carrying i.i.d. random charges. Each self-intersection of the polymer chain contributes to the interaction Hamiltonian an energy 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 focus is on the annealed free energy per monomer in the limit as the length of the polymer chain tends to infinity. We derive a spectral representation for the free energy and use this to prove that there is a critical curve in the parameter plane of charge bias versus inverse temperature separating a ballistic phase from a subballistic phase. We show that the phase transition is 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 flat pieces, which correspond to an inhomogeneous strategy for the polymer to realise a large deviation. The large deviation principles in turn lead to laws of large numbers and central limit theorems. We identify the scaling behaviour of the critical curve for small and for large charge bias. In addition, we identify the scaling behaviour of the free energy for small charge bias and small inverse temperature. Both are linked to an associated Sturm-Liouville eigenvalue problem. A key tool in our analysis is the Ray-Knight formula for the local times of the one-dimensional simple random walk. This formula is exploited to derive a closed form expression for the generating function of the annealed partition function, and for several related quantities. This expression in turn serves as the starting point for the derivation of the spectral representation for the free energy, and for the scaling theorems. What happens for the quenched free energy per monomer remains open. We state two modest results and raise a few questions.

A quenched functional central limit theorem for planar random walks in random sceneries

This paper considers an undirected polymer chain on ℤ^d, d ≥ 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 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. We show that in a subset of the collapsed phase the polymer chain is 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. The scaling of the critical curve for small charge bias and the scaling of the annealed free energy for small inverse temperature are both anomalous. 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. 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.