We study a model of directed polymers in a random environment, in which i.i.d. disorder values are assigned to the sites of the (d+1)-dimensional integer lattice, and the energy associated to a given space-time path of length n is minus the sum of the disorder values along that path. For the associated partition function Zn, under mild conditions on the disorder variables, we establish exponential concentration of log Zn about its mean on the subgaussian scale . This is used to show that (E log Zn)/n converges to the free energy at a rate which is also subgaussian, specifically O( log log n). Work joint with N.Zygouras.
Fluctuations for a random growth model.
We consider internal diffusion limited aggregation in dimension larger than or equal to two. This is a random growth model, where random walks start at the origin of the d-dimensional lattice, one at a time, and stop moving when reaching a site not occupied by previous walks. It is known that the asymptotic shape of the cluster is a sphere. The worse fluctuations is then measured in terms of inner (resp. outer) radius of a ball contained in (resp. containing) the cluster. When dimension is three or more, we show that the inner and outer radii fluctuations have matching upper and lower bounds of order of the square root of the logarithm of the radius. This is a joint work with Alexandre Gaudillière
An iterative construction of solutions of the TAP equations.
The TAP equations (Thouless-Anderson-Palmer) are supposed to describe the "pure states" in the Sherrington-Kirpatrick model of spin glasses. A proof of them has been given by Talagrand in the high-temperature regime which is supposed but not proved to be the region above the de Almayda-Thouless line. We propose an iterative construction of solutions which is shown to converge up to, and including, the de Almayda- Thouless line.
So far, localization statements for directed polymers in random medium deal with the location of the endpoint ("favourite site" for polymer). We introduce a pathwise property, roughly: there exists a "favorite path" depending on the environment as well as the model parameters and time horizon, such that the polymer path has a significant overlap with the favorite path. In a joint work with Mike Cranston, we establish this property in the parabolic Anderson model. We also obtain complete localization, i.e., the overlap tends to its maximal value 1 as the product (diffusivity x temperature2) tends to 0.
In this talk, I will discuss joint work with Ben Hambly (Oxford) regarding the high frequency asymptotics of the eigenvalue counting function associated with the natural Dirichlet form on alpha-stable trees, which lead in turn to short-time heat kernel asymptotics for these random structures. In particular, I will explain how our results are proved using self-similar fractal arguments that involve decomposing the relevant tree into three pieces in the alpha=2 case and a countable number of pieces otherwise. The talk will also include a brief discussion of how these ideas can be adapted to the scaling limit of the critical random graph.
Finite time corrections in KPZ growth models
We consider some models in the Kardar-Parisi-Zhang universality class for which, in the limit of large time t, universality of fluctuations has been previously obtained. We focus on the convergence to the limiting distributions and in particular on the (non-universal) first order corrections.
Random walks in random environment on a strip in sub-diffusive regimes
This is a joint work with D. Dolgopyat. We consider a random walk (RW) in random environment on a strip in a sud-diffusive regime. We show that the time the walk spends in a box of length N (equivalently, the hitting time for N) can asymptotically be presented as linear combination of i.i.d. exponential random variables with coefficients forming a Poisson process with explicitly given density. RWs with bounded jumps on a line can be viewed as a particular case of this model. The latter in turn is a natural generalisation of the classical simple RW (with jumps to nearest neighbours) and was first considered in the classical paper going back to 1975 (Solomon, Kesten-Kozlov-Spitzer). In the sub-diffusive regimes, only very partial results and only in annealed models were known about the RWs with bounded jumps so far.
A CLT for stretched polymers at weak disorder
We show that -step stretched polymers on in a random i.i.d. potential obey a CLT at sufficiently weak disorder in any dimension . Joint work with Yvan Velenik
Rate of convergence estimates for the zero dissipation limit in Abelian sandpiles
We consider a continuous height version of the Abelian sandpile model with small amount of bulk dissipation gamma -> 0 on each toppling, in dimensions d = 2, 3. In the limit gamma -> 0, we give a power law upper bound, based on coupling, on the rate at which the stationary measure converges to the discrete critical sandpile measure. The proofs are based on a coding of the stationary measure by weighted spanning trees, and an analysis of the latter via Wilson's algorithm. In the course of the proof, we prove an estimate on coupling a geometrically killed loop-erased random walk to an unkilled loop-erased random walk.
Superdiffusivity for directed polymer and related models have been extensively studied by physicists and mathematicians in recent years, with a special focus on the two dimensional case. Physicists have predicted that transversal fluctuation of trajectories should be of order Lξ where L is the size of the system where ξ=2/3. These predictions have been partly confirmed by mathematicians with bounds on the exponent 3/5≤ξ≤3/4 (Wühtrich, Petermann, Méjane...) for quite general models or by getting exactly ξ=2/3 (Johansson, Seppalainen...) for very particular models for which exact computation are possible. Very few is known concerning the value ξ for higher dimension in the low-temperature regime even at a conjectural point of view. In this talk we deal with the much less studied case where the environment present longe-range slowly decaying correlation, either transversal or isotropic (in arbitrary dimension), and discuss the effect that it has on superdiffusivity properties. In some particular cases, we are able to get the exact value of the volume exponent by using arguments involving only energy vs. entropy competition.
Last passage percolation in one and two dimensions
I'll discuss a long-range last-passage percolation model on Z, in which every edge carries an i.i.d. weight. We look at the behaviour of the maximum weight of a directed path between the points 0 and . This last-passage time shows two very different types of behaviour, depending on the tail of the weight distribution. If the distribution has finite variance, the process has a certain regenerative structure, and a law of large numbers and fluctuation results can be shown. However, if the variance is infinite, there are scaling limits and asymptotic distributions expressed in terms of a "continuous last-passage percolation" model on [0,1]; these are related to corresponding results for two-dimensional nearest-neighbour last-passage percolation with heavy-tailed weights. There are partial results about geodesics for the model, but many questions remain open; there are also interesting open questions concerning related models of tilings and stable allocations in two or more dimensions. This includes joint work with Sergey Foss and Philipp Schmidt.
We study a dynamical random network model in which at every construction step a new vertex is introduced and attached to every existing vertex independently with a probability proportional to a concave function of its current degree. Such a network is called robust if it survives percolation with any positive retention parameter. We characterise the robust networks and explicitly determine the critical percolation threshold in the case of nonrobust networks. The talk is based on joint work with Steffen Dereich (Marburg).
Lyapunov exponents limits for Random walks in Random Potentials
We consider random walks in under random potential \lambda where the random variables for are i.i.d. and examine the behaviour of the quenched and annealed exponents as parameter \lambda tends to zero. Joint with Jean-Christophe Mourat.
Some aspects of Random walks in random Dirichlet environments
Dirichlet environments form a familly of (non-reversible) i.i.d. environments where at each site the transition probabilities are chosen according to a Dirichlet law. I will review some of the results proved on this model concerning the environment viewed from the particule and transience and directional transience. I will try to explain why this familly seems to be simpler to analyse than general environments and will give some open questions.
Connectivity properties of random interlacements
In this talk, we consider the interlacement Poisson point process on the space of doubly-infinite -valued trajectories modulo time-shift, tending to infinity at positive and negative infinite times, introduced by Sznitman. The set of edges traversed by at least one of these trajectories induces the random interlacement graph (at level u), an infinite connected subgraph of . We will summarize recent results about connectivity properties of the random interlacement graph including the transience of the graph and the non-trivial Bernoulli phase transition. Our results are valid for dimensions and levels u>0. The talk is based on joint work with Balazs Rath (ETH, Zurich).
A replica analysis of the one-dimensional KPZ equation
Since the idenfitication of the height distribution for the KPZ equation for the narrow wedge initial condition, there have been active research on the topic. One useful way to treat the problem is to use the replica for the partition function of the directed polymer. In this presentation, we explain how it works for a half-Brownian motion initial condition.
Among 1+1 dimensional directed lattice polymers, log- gamma distributed weights are a special case that is amenable to various useful exact calculations (an ``exactly solvable'' case). This talk discusses various aspects of the log-gamma model: bounds for fluctuation exponents, large deviation rate functions, and an approach to analyzing the model through a geometric version of the Robinson- Schensted-Knuth correspondence. Parts of this talk are joint work with Ivan Corwin, Nicos Georgiou, Neil O'Connell and Nikos Zygouras.
From Random Interlacements to Coordinate Percolation and back
We consider a process , introduced by Durrett and Rogers in 1992 in order to model the shape of a growing polymer; it undergoes a drift which depends on its past trajectory, and a Brownian increment. Our work concerns two conjectures by these authors (1992), concerning repulsive interaction functions in dimension ( for all in , ). We showed the first one with T. Mountford (AIHP, 2008, AIHP Prize 2009), for certain functions with heavy tails, leading to transience to or with probability . We partially proved the second one with B. T\'oth and B. Valk\'o (to appear in Ann. Prob. 2011), for rapidly decreasing functions , through a study of the local time environment viewed from the particule: we explicitly display an associated invariant measure, which enables us to prove under certain initial conditions that a.s., that the process is at least diffusive asymptotically and superdiffusive under certain assumptions.
The stochastic heat equation and the Karlin-McGregor formula
The stochastic heat equation with multiplicative space-time white noise is a continuous analogue of a random polymer model which is itself a finite temperature version of last passage percolation. For the latter it is known that the RSK correspondence provides an important link with random matrix theory. The talk will be about extending the SHE to a multilayer process which corresponds to looking for the analogue of all the eigenvalues of the random matrix ensemble rather than just the largest. It is based on joint work with Neil O'Connell.
We consider a two-dimensional infinitesimal continuum percolation model with columnar dependence. It is related to several other models in probability including Lipschitz percolation, oriented percolation, first-passage percolation, Poisson matchings, coverings of the circle by random arcs and Brownian motion. Several open questions are posed.