the measurable kesten theorem we give an explicit bound on the spectral radius in terms of the densities of short cycles in finite d regular graphs it follows that the a finite d regular ramanujan graph g contains a negligible number of cycles of size less than c log log g we prove that infinite d regular ramanujan unimodular random graphs are trees through benjamini schramm convergence this leads to the following rigidity result if most eigenvalues of a d regular finite graph g fall in the alon boppana region then the eigenvalue distribution of g is close to the spectral measure of the d regular tree in particular g contains few short cycles in contrast we show that d regular unimodular random graphs with maximal growth are not necessarily trees institute of mathematical statistics 2016

multiplicity results of periodic solutions for a class of second order delay differential systems abstract this paper is concerned with the periodic boundary value problem 1 u t u t f t u t a e t 0 2 u 0 u 2 u 0 u 2 0 where 0 is a given constant r is a parameter the variational principle is given and some multiplicity results of periodic solutions of 1 are obtained directly via variational methods

quenched large deviations for simple random walks on percolation clusters including long range correlations we prove a quenched large deviation principle ldp for a simple random walk on a supercritical percolation cluster srwpc on mathbb z d d geq 2 the models under interest include classical bernoulli bond and site percolation as well as models that exhibit long range correlations like the random cluster model the random interlacement and the vacant set of random interlacements for d geq 3 and the level sets of the gaussian free field d geq 3 inspired by the methods developed by kosygina et al commun pure appl math 59 1489 1521 2006 for proving quenched ldp for elliptic diffusions with a random drift and by yilmaz commun pure appl math 62 8 1033 1075 2009 and rosenbluth quenched large deviations for multidimensional random walks in a random environment a variational formula ph d thesis nyu arxiv 0804 1444v1 for similar results regarding elliptic random walks in random environment we take the point of view of the moving particle and prove a large deviation principle for the quenched distribution of the pair empirical measures of the environment markov chain in the non elliptic case of srwpc via a contraction principle this reduces easily to a quenched ldp for the distribution of the mean velocity of the random walk and both rate functions admit explicit variational formulas the main difficulty in our set up lies in the inherent non ellipticity as well as the lack of translation invariance stemming from conditioning on the fact that the origin belongs to the infinite cluster we develop a unifying approach for proving quenched large deviations for srwpc based on exploiting coercivity properties of the relative entropies in the context of convex variational analysis combined with input from ergodic theory and invoking geometric properties of the supercritical percolation cluster

random walks on supercritical percolation clusters we obtain gaussian upper and lower bounds on the transition density qt x y of the continuous time simple random walk on a supercritical percolation cluster c1 in the euclidean lattice the bounds analogous to aronsen s bounds for uniformly elliptic divergence form diusions hold with constants ci depending only on p the percolation probability and d the irregular nature of the medium means that the bound for qt x only holds for t sx where the constant sx depends on the percolation congura tion

the growth of the infinite long range percolation cluster we consider long range percolation on z d where the probability that two vertices at distance r are connected by an edge is given by p r 1 exp lambda r is an element of 0 1 and the presence or absence of different edges are independent here lambda r is a strictly positive nonincreasing regularly varying function we investigate the asymptotic growth of the size of the k ball around the origin vertical bar b k vertical bar that is the number of vertices that ale within graphdistance k of the origin for k infinity for different lambda r we show that conditioned on the origin being in the unique infinite cluster nonempty classes of nonmereasimi regularly varying lambda r exist for which respectively vertical bar b k vertical bar 1 k infinity almost surely there exist 1 infinity p a 1 1 almost surely this result can be applied to spatial sir epidemics in particular regimes are identified for which the basic reproduction number r 0 which is an important quantity for epidemics in unstructured populations has a useful counterpart in spatial epidemics

some new results for charlier and meixner polynomials abstract in bavinck 1996 we proved the following general result for laguerre polynomials for all x c k j i k s l i 1 i k x l j k j x i 2s j x s i j s 0 1 2 provided that i 2s j in this letter we derive the analogues of this formula for charlier and meixner polynomials

unpredictable paths and percolation we construct a nearest neighbor process s n on z that is less predictable than simple random walk in the sense that given the process until time n the conditional probability that s n k x is uniformly bounded by ck for some 1 2 from this process we obtain a probability measure on oriented paths in z 3 such that the number of intersections of two paths chosen independently according to has an exponential tail for d 4 the uniform measure on oriented paths from the origin in z d has this property we show that on any graph where such a measure on paths exists oriented percolation clusters are transient if the retention parameter p is close enough to 1 this yields an extension of a theorem of grimmett kesten and zhang who proved that supercritical percolation clusters in z d are transient for all d 3

surface order large deviations for high density percolation we derive surface order large deviation estimates for the volume of the largest cluster and for the volume of the largest region surrounded by a cluster of a bernoulli percolation process restricted to a big finite box with sufficiently large parameter we also establish a useful version of the isoperimetric inequality which is the main tool of our proofs

existence of the harmonic measure for random walks on graphs and in random environments we give a sufficient condition for the existence of the harmonic measure from infinity of transient random walks on weighted graphs in particular this condition is verified by the random conductance model on d d 3 when the conductances are i i d and the bonds with positive conductance percolate the harmonic measure from infinity also exists for random walks on supercritical clusters of 2 this is proved using results of barlow ann probab 32 3024 3084 2004 and barlow and hambly electron j probab 14 1 1 27 2009

the speed of biased random walk on percolation clusters we consider biased random walk on supercritical percolation clusters in 2 we show that the random walk is transient and that there are two speed regimes if the bias is large enough the random walk has speed zero while if the bias is small enough the speed of the random walk is positive

quenched invariance principle for simple random walk on percolation clusters we consider the simple random walk on the unique infinite cluster of super critical bond percolation in d with d 2 we prove that for almost every percolation configuration the path distribution of the walk converges weakly to that of non degenerate isotropic brownian motion our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square integrable martingale the size of the deformation expressed by the so called corrector is estimated by means of ergodicity arguments

the supercritical phase of percolation is well behaved we prove a general result concerning the critical probabilities of subsets of a lattice l it is a consequence of this result that the critical probability of a percolation process on l equals the limit of the critical probability of a slice of l as the thickness of the slice tends to infinity this verification of one of the standard hypotheses of the subject settles many questions concerning supercritical percolation

on the chemical distance for supercritical bernoulli percolation we prove large deviation estimates at the correct order for the graph distance of two sites lying in the same cluster of an independent percolation process we improve earlier results of gartner and molchanov and grimmett and marstrand and answer affirmatively a conjecture of kozlov

darcy equation for random porous media many attempts have been made recently to deduce darcy equations for flows through porous media much has been done for models having a periodic microstructure this paper presents a mathematical analysis based on the more general and more realistic assumption that the microstructure of porous media is random and stochastically homogeneous for this type of random domain we consider homogenization of the poisson and stokes equations supplemented by homogeneous dirichlet boundary conditions on the random boundary with this approach we get the darcy equation in general as well as present details for the particular case of a checkerboard model of porous media 1996 john wiley sons inc

parabolic problems for the anderson model i intermittency and related topics the objective of this paper is a mathematically rigorous investigation of intermittency and related questions intensively studied in different areas of physics in particular in hydrodynamics on a qualitative level intermittent random fields are distinguished by the appearance of sparsely distributed sharp peaks which give the main contribution to the formation of the statistical moments the paper deals with the cauchy problem t u t x hu t x u 0 x t 0 x 0 t x d for the anderson hamiltonianh x x d where is a generally unbounded spatially homogeneous random potential this first part is devoted to some basic problems using percolation arguments a complete answer to the question of existence and uniqueness for the cauchy problem in the class of all nonnegative solutions is given in the case of i i d random variables necessary and sufficient conditions for intermittency of the fieldsu t ast are found in spectral terms ofh rough asymptotic formulas for the statistical moments and the almost sure behavior ofu t x ast are also derived

rate of convergence of the mean for sub additive ergodic sequences abstract for sub additive ergodic processes x m n with weak dependence we analyze the rate of convergence of e x 0 n n to its limit g we define an exponent given roughly by e x 0 n n g n and assuming existence of a fluctuation exponent that gives var x 0 n n 2 we provide a lower bound for of the form the main requirement is that 1 2 in the case 1 2 and under the assumption var x 0 n o n log n for some 0 we prove c for a dependent constant c these results show in particular that non diffusive fluctuations are associated to non trivial various models including first passage percolation directed polymers the minimum of a branching random walk and bin packing fall into our general framework and the results apply assuming exists in the case of first passage percolation in z d we provide a version of 1 2 without assuming existence of

surface order large deviations for ising potts and percolation models we derive uniform surface order large deviation estimates for the block magnetization in finite volume ising or potts models with plus or free or a combination of both boundary conditions in the phase coexistence regime ford 3 the results are valid up to a limit of slab thresholds conjectured to agree with the critical temperature our arguments are based on the renormalization of the random cluster model withq 1 andd 3 and on corresponding large deviation estimates for the occurrence in a box of a largest cluster with density close to the percolation probability the results are new even for the case of independent percolation q 1 as a byproduct of our methods we obtain further results in the fk model concerning semicontinuity inp andq of the percolation probability the second largest cluster in a box and the tail of the finite cluster size distribution

localization for random walks among random obstacles in a single euclidean ball place an obstacle with probability 1 p independently at each vertex of mathbb z d and run a simple random walk before hitting one of the obstacles for d ge 2 and p strictly above the critical threshold for site percolation we condition on the environment where the origin is contained in an infinite connected component free of obstacles and we show that for environments with probability tending to one as n rightarrow infty there exists a unique discrete euclidean ball of volume d log 1 p n asymptotically such that the following holds conditioned on survival up to time n we have that at any time t in epsilon n n n for some epsilon n rightarrow n rightarrow infty 0 with probability tending to one the simple random walk is in this ball this work relies on and substantially improves a previous result of the authors on localization in a region of volume poly logarithmic in n for the same problem

on the scaling of the chemical distance in long range percolation models we consider the unoriented long range percolation on z d in dimensions d 1 where distinct sites x y z d get connected with probability p xy 0 1 assuming p xy x y s o 1 as x y where s 0 and is a norm distance on z d and supposing that the resulting random graph contains an infinite connected component c we let d x y be the graph distance between x and y measured on c our main result is that for s d 2d d x y log x y o 1 x y c x y where 1 is the binary logarithm of 2d s and o 1 is a quantity tending to zero in probability as x y besides its interest for general percolation theory this result sheds some light on a question that has recently surfaced in the context of small world phenomena as part of the proof we also establish tight bounds on the probability that the largest connected component in a finite box contains a positive fraction of all sites in the box

on the anisotropic walk on the supercritical percolation cluster we investigate in this work the asymptotic behavior of an anisotropic random walk on the supercritical cluster for bond percolation on d d 2 in particular we show that for small anisotropy the walk behaves in a ballistic fashion whereas for strong anisotropy the walk is sub diffusive for arbitrary anisotropy we also prove the directional transience of the walk and construct a renewal structure

disorder entropy and harmonic functions we study harmonic functions on random environments with particular emphasis on the case of the infinite cluster of supercritical percolation on z d we prove that the vector space of harmonic functions growing at most linearly is d 1 dimensional almost surely in particular there are no non constant sublinear harmonic functions thus implying the uniqueness of the corrector the main ingredient of the proof is a quantitative annealed version of the kaimanovich vershik entropy argument this also provides bounds on the derivative of the heat kernel simplifying and generalizing existing results the argument applies to many different environments even reversibility is not necessary we also mention several open problems and conjectures on the behavior of harmonic functions on stationary random graphs

nearest neighbor walks with low predictability profile and percolation in 2 epsilon dimensions a few years ago grimmett kesten and zhang proved that for supercritical bond percolation on z 3 simple random walk on the infinite cluster is a s transient we generalize this result to a class of wedges in z 3 including for any e 0 1 the wedge w e x y z z 3 x 0 z x e which can be thought of as representing a 2 e dimensional lattice our proof builds on recent work of benjamini pemantle and peres and involves the construction of finite energy flows using nearest neighbor walks on z with low predictability profile along the way we obtain some new results on attainable decay rates for predictability profiles of nearest neighbor walks

sharpness of the phase transition in percolation models the equality of two critical points the percolation threshold ph and the point p where the cluster size distribution ceases to decay exponentially is proven for all translation invariant independent percolation models on homogeneous d dimensional lattices d 1 the analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter m h which for h q reduces to the percolation density p at the bond density p l e in the single parameter case these are 1 m hdm dh m2 mdm d and 2 dm d j mdm dh inequality 1 is intriguing in that its derivation provides yet another hint of a 3 structure in percolation models moreover through the elimination of one of its derivatives 1 yields a pair of ordinary differential inequalities which provide information on the critical exponents and one of these resembles an ising model inequality of fr hlich and sokal and yields the mean field bound 5 2 and the other implies the result of chayes and chayes that an inequality identical to 2 is known for ising models where it provides the basis for newman s universal relation 5 1 1 and for certain extrapolation principles which are now made applicable also to independent percolation these results apply to both finite and long range models with or without orientation and extend to periodic and weakly inhomogeneous systems

random conductance models with stable like jumps heat kernel estimates and harnack inequalities abstract we establish two sided heat kernel estimates for random conductance models with non uniformly elliptic possibly degenerate stable like jumps on graphs these are long range counterparts of the well known two sided gaussian heat kernel estimates by m t barlow for nearest neighbor short range random walks on the supercritical percolation cluster unlike the cases of nearest neighbor conductance models we cannot use parabolic harnack inequalities since even elliptic harnack inequalities do not hold in the present setting as an application we establish the local limit theorem for the models