Abstracts
Fluctuations and large deviations for polymer models (slides)
Timo Seppäläinen - University of Wisconsin
This talk describes results on large deviations and fluctuations of random walks in random potentials, also called polymer models. We present a variational formula for the limiting free energy that involves a natural entropy for the model. For some specific directed 1+1 dimensional models we can compute rigorously fluctuation exponents for the free energy and the polymer path. The values of the exponents are the ones conjectured in the theoretical physics literature. We describe in particular a lattice polymer with log-gamma distributed weights, and the Hopf-Cole solution of the KPZ (Kardar-Parisi-Zhang) equation.
Parts of this talk are joint work with Marton Balazs, Jeremy Quastel, Firas Rassoul-Agha, Benedek Valko, and Atilla Yilmaz.
Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
Ivan Corwin - Courant Institute (NYU)
We consider the solution of the stochastic heat equation with multiplicative noise and delta function initial condition whose logarithm, with appropriate normalizations, is the free energy of the continuum directed polymer, or the solution of the Kardar-Parisi-Zhang equation with narrow wedge initial conditions. We prove explicit formulas for the one-dimensional marginal distributions -- the crossover distributions -- which interpolate between a standard Gaussian distribution (small time) and the GUE Tracy-Widom distribution (large time). The proof is via a rigorous steepest descent analysis of the Tracy-Widom formula for the asymmetric simple exclusion with anti-shock initial data, which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit. The limit also describes the crossover behaviour between the symmetric and asymmetric exclusion processes.
Phase transition for the speed of the biased random walk on the supercritical percolation cluster (slides)
Alexander Fribergh - Courant Institute (NYU)
We prove the sharpness of the phase transition for speed in the biased random walk on the supercritical percolation cluster on $\mathbb{Z}^d$. That is, for each $d\geq 2$, and for any supercritical parameter $p > p_c$, we prove the existence of a critical strength for the bias, such that, below this value, the speed is positive, and, above the value, it is zero. We identify the value of the critical bias explicitly, and, in the sub-ballistic regime, we find the polynomial order of the distance moved by the particle. Each of these conclusions is obtained by investigating the geometry of the traps that are most effective at delaying the walk. A key element in proving our results is to understand that, on large scales, the particle trajectory is essentially one-dimensional; we prove such a dynamic renormalization statement in a much stronger form than was previously known.
Survival probability of a random walk among independent random walks
Alejandro F. Ramírez - PUC, Santiago
Consider a simple random walk on the hyper-cubic lattice which is killed when it jumps to a site where a trap is present. The traps move themselves as simple random walks and have an initial law which is product Poisson. We prove that the average survival probability decays asymptotically for large $t$ as $e^{-\lambda_1\sqrt{t}}$ for $d = 1$, as $e^{-\lambda_2{t/log t}}$ for $d = 2$, and as $e^{-\lambda_d t}$ for $d\ge 3$, where $\lambda_d>0$ for $d\ge 1$ are constants while $\lambda_1$ and $\lambda_2$ can be identified explicitly. These results are based on works published in 2004 by Moureau, Oshanin, Bénichou and Coppey and use the so called Pascal principle. In addition, we show that the quenched survival probability decays asymptotically as $e^{-\tilde\lambda_dt}$, where $\tilde\lambda_d > 0$ for $d\ge 1$ are constants. This result requires the use of large deviation methods previously developed by Varadhan in 2004 for random walk in random environment and a percolation argument developed by Kesten and Sidoravicius.
This is a joint work with A. Drewitz, J. Gärtner and R. Sun.
Crossing velocities for annealed random walks in a random potential (slides)
Elena Kosygina - Baruch College (CUNY)
We consider random walks in an i.i.d. non-negative potential on the $d$-dimensional integer lattice. The walks are conditioned to hit a remote location and are studied under the annealed path measure. We show that the expected time to reach $y$ increases only linearly in $|y|$ so that the walk is ballistic in all dimensions. In dimension one we prove the existence of the asymptotic speed as $y$ goes to infinity.
Joint work with Thomas Mountford (EPFL, Lausanne).
Vacant set of random walk on random graphs (slides)
Jiri Cerny - ETH, Zurich
The vacant set is the set of vertices not visited by a random walk on a graph before a given time $T$. In the talk, I will discuss properties of this random subset of the graph, the phase transition conjectured in its connectivity properties (in the `thermodynamic limit' when $|G|$ and $T$ grow simultaneously), and the relation of the problem to the random interlacement percolation. I will then concentrate on the case when $G$ is a large-girth expander or a random regular graph, where the conjectured phase transition (and much more) can be proved.
Random walk on Galton-Watson trees with random conductances
Nina Gantert - Münster, Germany
We consider the random conductance model, where the underlying graph is an infinite Galton-Watson tree. The conductances are independent, their law may depend on the degree of the incident vertices. We show that, if the mean conductance is finite, the random walk has a deterministic, strictly positive speed $v$. We give a formula for $v$ in terms of certain effective conductances. If the conductances share the same expectation, the speed is not larger than the speed of simple random walk on the Galton-Watson tree. The proof relies on finding a reversible measure for the environment, seen from the particle.
The talk is based on joint work with Sebastian Mueller, Serguei Popov and Marina Vachkovskaia.
Title: TBA
Omer Angel - University of British Columbia
TBA
On the association and Central Limit Theorem for solutions of the Parabolic Anderson Model
Mike Cranston - University of California, Irvine
We consider large scale behavior of the solution set $\{u(t,x):\,x\in {\bf{Z^d}}\}$ of the parabolic Anderson equation \begin{eqnarray*} u(t,x)=1+\kappa\int_0^t\Delta u(s,x)ds+\int_0^tu(s,x)\circ dW_x(s),\qquad x\in{\bf{Z^d}},\,\,t\ge0, \end{eqnarray*} where $\{W_x:x\in {\bf{Z^d}}\}$ is a field of $iid$ standard, one-dimensional Brownian motions, $\Delta$ is the discrete Laplacian and $\kappa>0$ and $\circ dW_x(t)$ denotes the Stratonovitch differential of $W_x(t).$ We establish that the properly normalized sum, $\sum_{x\in \Lambda_L}u(t,x),$ over spatially growing boxes $\Lambda_L=\{x\in{\bf{Z^d}}:||x|| < L \}$ has an asymptotically normal distribution if the box $\Lambda_L$ grows sufficiently quickly with $t$ and provided $\kappa$ is sufficiently small depending on dimension. The asymptotic distribution of properly normalized sums over spatially growing disjoint boxes $\Lambda^1_L,\Lambda^2_L$ is asymptotically independent. Thus, on sufficiently large scales the field of solutions averaged over disjoint large boxes looks like an iid Gaussian field. We identify the variance of this Gaussian distribution in terms of the eigenfunction of the positive eigenvalue of the operator $2\kappa \Delta+\delta_0.$
From random interlacements to coordinate percolation
Vladas Sidoravicius - CWI, Amsterdam and IMPA, Rio de Janeiro
In this talk I will present some new results related to decay of correlations in dependent percolation models, such as Random Interlacements, introduced by A.-S. Sznitman, and so called Coordinate Percolation, invented by P. Winkler, and explore connections between these models. Parts of this talk are joint works with M. Hilario, A.-S. Sznitman and A. Teixeira, and H. Kesten, B.N.B. Lima and M.E.Vares.