large deviations for random walks in a random environment we prove two large deviation principles for random walks in a random environment one for quenched walks in general ergodic environments and the other for averaged walks in a product environment 2003 wiley periodicals inc

competition interfaces and second class particles the one dimensional nearest neighbor totally asymmetric simple exclusion process can be constructed in the same space as a last passage percolation model in z 2 we show that the trajectory of a second class particle in the exclusion process can be linearly mapped into the competition interface between two growing clusters in the last passage percolation model using technology built up for geodesics in percolation we show that the competition interface converges almost surely to an asymptotic random direction as a consequence we get a new proof for the strong law of large numbers for the second class particle in the rarefaction fan and describe the distribution of the asymptotic angle of the competition interface

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

bounds of the number of rational maps between varieties of general type we give a bound for the number of rational maps between algebraic varieties of general type under mild hypothesis on the canonical map we use an idea inspired by tanabe s work instead of attaching a morphism of hodge structures to a rational map we simply associate to it a piece of the integral hodge lattice this procedure does not give an injective map but by means of a geometric argument we can estimate the number of maps with the same image

non equilibrium behaviour of a many particle process density profile and local equilibria one considers a simple exclusion particle jump process on where the underlying one particle motion is a degenerate random walk that moves only to the right one starts with the configuration in which the left halfline is completely occupied and the right one free it is shown that the number of particles at time t between site u t and v t divided by t converges a s to int limits u nu f w dw where f might be called the density profile it is explicitely determined and shown to be an affine function secondly we prove that the distribution of the process looked at by an observer travelling at constant speed u converges weakly to the bernoulli measure with density f u as the time tends to infinity

geodesics and spanning trees for euclidean first passage percolation the metric d q q on the set q of particle locations of a homogeneous poisson process on r d defined as the infimum of i q i q i 1 1 over sequences in q starting with q and ending with q where denotes euclidean distance has nontrivial geodesics when a 1 the cases 1 are the euclidean first passage percolation fpp models introduced earlier by the authors while the geodesics in the case are exactly the paths from the euclidean minimal spanning trees forests of aldous and steele we compare and contrast results and conjectures for these two situations new results for 1 and any d include inequalities on the fluctuation exponents for the metric x 1 2 and for the geodesies 3 4 in strong enough versions to yield conclusions not yet obtained for lattice fpp almost surely every semiinfinite geodesic has an asymptotic direction and every direction has a semiinfinite geodesic from every q for d 2 and 2 further results follow concerning spanning trees of semiinfinite geodesics and related random surfaces

hammersley s interacting particle process and longest increasing subsequences in a famous paper 8 hammersley investigated the lengthl n of the longest increasing subsequence of a randomn permutation implicit in that paper is a certain one dimensional continuous space interacting particle process by studying a hydrodynamical limit for hammersley s process we show by fairly soft arguments that limn 1 2 el n 2 this is a known result but previous proofs 14 11 relied on hard analysis of combinatorial asymptotics

on the partition function of a directed polymer in a gaussian random environment the purpose of this work is the study of the partition function of a dimensional lattice directed polymer in a gaussian random environment being the inverse of temperature in the low dimensional cases we prove that for all the renormalized partition function converges to 0 and the correlation of two independent configurations does not converge to 0 in the high dimensional case a lower tail of has been obtained for small furthermore we express some thermodynamic quantities in terms of the path measure alone

differentiability at the edge of the percolation cone and related results in first passage percolation we study first passage percolation in two dimensions using measures on passage times with b inf supp 0 and mu b p geq vec p c the threshold for oriented percolation we first show that for each such the boundary of the limit shape for is differentiable at the endpoints of flat edges in the so called percolation cone we then conclude that the limit shape must be non polygonal for all of these measures furthermore the associated richardson type growth model admits infinite coexistence and if is not purely atomic the graph of infection has infinitely many ends we go on to show that lower bounds for fluctuations of the passage time given by newman piza extend to these measures we establish a lower bound for the variance of the passage time to distance n of order log n in any direction outside the percolation cone under a condition of finite exponential moments for this result confirms a prediction of newman and piza ann probab 23 977 1005 1995 and zhang ann probab 36 331 362 2008 under the assumption of finite radius of curvature for the limit shape in these directions we obtain a power law lower bound for the variance and an inequality between the exponents and

morphisms and period matrices abstract bounding the number of morphisms between compact riemann surfaces is a long standing problem coming from complex and algebraic geometry we show that linear algebra techniques allow to improve the known results when we assume a kind of condition number bound for the period matrix

quenched free energy and large deviations for random walks in random potentials we study quenched distributions on random walks in a random potential on integer lattices of arbitrary dimension and with an arbitrary finite set of admissible steps the potential can be unbounded and can depend on a few steps of the walk directed undirected and stretched polymers as well as random walk in random environment are covered the restriction needed is on the moment of the potential in relation to the degree of mixing of the ergodic environment we derive two variational formulas for the limiting quenched free energy and prove a process level quenched large deviation principle ldp for the empirical measure as a corollary we obtain ldps for types of random walks in random environments not covered by earlier results 2012 wiley periodicals inc

large deviations for lattice systems i parametrized independent fields we prove large deviation theorems for empirical measures of independent random fields whose distributions depend measurably on an auxiliary parameter this dependence respects the action of the shift group and a large deviation principle holds whenever a certain ergodicity condition is satisfied we also investigate the entropy functions for these processes especially in relation to the usual relative entropy

busemann functions and the speed of a second class particle in the rarefaction fan in this paper we will show how the results found in probab theory related fields 154 2012 89 125 about the busemann functions in last passage percolation can be used to calculate the asymptotic distribution of the speed of a single second class particle starting from an arbitrary deterministic configuration which has a rarefaction fan in either the totally asymetric exclusion process or the hammersley interacting particle process the method will be to use the well known last passage percolation description of the exclusion process and of the hammersley process and then the well known connection between second class particles and competition interfaces

space time stationary solutions for the burgers equation we construct space time stationary solutions of the 1d burgers equation with random forcing in the absence of periodicity or any other compactness assumptions more precisely for the forcing given by a homogeneous poissonian point field in space time we prove that there is a unique global solution with any prescribed average ve locity these global solutions serve as one point random attractors for the infinite dimensional dynamical system associated to solutions to the cauchy problem the probability distribution of the global solutions de fines a stationary distribution for the corresponding markov process we describe a broad class of initial cauchy data for which the distribution of the markov process converges to the above stationary distribution our construction of the global solutions is based on a study of the field of action minimizing curves we prove that for an arbitrary value of the average velocity with probability 1 there exists a unique field of action minimizing curves initiated at all of the poissonian points moreover action minimizing curves corresponding to different starting points merge with each other in finite time

variational formulas and cocycle solutions for directed polymer and percolation models we discuss variational formulas for the law of large numbers limits of certain models of motion in a random medium namely the limiting time constant for last passage percolation and the limiting free energy for directed polymers the results are valid for models in arbitrary dimension steps of the admissible paths can be general the environment process is ergodic under spatial translations and the potential accumulated along a path can depend on the environment and the next step of the path the variational formulas come in two types one minimizes over gradient like cocycles and another one maximizes over invariant measures on the space of environments and paths minimizing cocycles can be obtained from busemann functions when these can be proved to exist the results are illustrated through 1 1 dimensional exactly solvable examples periodic examples and polymers in weak disorder

geodesics in two dimensional first passage percolation we consider standard first passage percolation on z 2 geodesics are nearest neighbor paths in z 2 each of whose segments is time minimizing we prove part of the conjecture that doubly infinite geodesics do not exist our main tool is a result of independent interest about the coalescing of semi infinite geodesics