percolation on infinite graphs and isoperimetric inequalities we consider the bernoulli bond percolation process with parameter p on infinite graphs and we give a general criterion for bounded degree graphs to exhibit a non trivial percolation threshold based either on a single isoperimetric inequality if the graph has a bi infinite geodesic or two isoperimetric inequalities if the graph has not a bi infinite geodesic this new criterion extends previous criteria and brings together a large class of amenable graphs such as regular lattices and non amenable graphs such trees we also study the finite connectivity in graphs satisfying the new general criterion and show that graphs in this class with a bi infinite geodesic always have finite connectivity functions with exponential decay when p is sufficiently close to one on the other hand we show that there are graphs in the same class with no bi infinite geodesic for which the finite connectivity decays sub exponentially down to polynomially in the highly supercritical phase even for p arbitrarily close to one

noise induced drift in stochastic differential equations with arbitrary friction and diffusion in the smoluchowski kramers limit we consider the dynamics of systems with arbitrary friction and diffusion these include as a special case systems for which friction and diffusion are connected by einstein fluctuation dissipation relation e g brownian motion we study the limit where friction effects dominate the inertia i e where the mass goes to zero smoluchowski kramers limit using the ito stochastic integral convention we show that the limiting effective langevin equations has different drift fields depending on the relation between friction and diffusion alternatively our results can be cast as different interpretations of stochastic integration in the limiting equation which can be parametrized by interestingly in addition to the classical ito 0 stratonovich 0 5 and anti ito 1 integrals we show that position dependent x and even stochastic integrals with 0 1 arise our findings are supported by numerical simulations

weighted average importance sampling and defensive mixture distributions importance sampling uses observations from one distribution to estimate for another distribution by weighting the observations including the target distribution as one component of a mixture distribution bounds the weights and makes importance sampling more reliable the usual importance sampling estimate is a weighted average with weights that do not sum to 1 we discuss simple normalization and other more efficient normalization methods these innovations make importance sampling useful in a wider variety of problems we demonstrate with a case study of oil inventory reliability at a large utility

tree graph inequalities and critical behavior in percolation models various inequalities are derived and used for the study of the critical behavior in independent percolation models in particular we consider the critical exponent associated with the expected cluster sizex and the structure of then site connection probabilities n x1 xn it is shown that quite generally 1 the upper critical dimension above which attains the bethe lattice value 1 is characterized both in terms of the geometry of incipient clusters and a diagramatic convergence condition for homogeneousd dimensional lattices with x y o x y d 2 atp p c our criterion shows that 1 if 6 d 3 the connectivity functions n are generally bounded by tree diagrams which involve the two point function we conjecture that above the critical dimension the asymptotic behavior of n in the critical regime is actually given by such tree diagrams modified by a nonsingular vertex factor other results deal with the exponential decay of the cluster size distribution and the function 2 x y

infinite graphs with a nontrivial bond percolation threshold some sufficient conditions we consider bernoulli bond percolation on infinite graphs and we identify a class of graphs for which the critical percolation probability is strictly less than 1 the graphs in this class have to fulfill conditions stated in terms of a minimal cut set property and a logarithmic isoperimetric inequality for the particular case of planar graphs the condition on minimal cut sets can be be replaced by the assumption that the dual of the graph is bounded degree

generalization of the peierls griffiths theorem for the ising model on graphs we present a sufficient condition for the presence of spontaneous magnetization for the ising model on a general graph related to its long range topology applying this condition we are able to prove the existence of a phase transition at temperature t 0 on a wide class of general networks the possibility of further extensions of our results is discussed

a contour method on cayley trees we consider finite range lattice models on cayley trees with two basic properties the existence of only a finite number of ground states and with a peierls type condition we define the notion of a contour for the model on the cayley tree by a contour argument we show the existence of s different where s is the number of ground states gibbs measures

structure of stochastic dynamics near fixed points we analyze the structure of stochastic dynamics near either a stable or unstable fixed point where the force can be approximated by linearization we find that a cost function that determines a boltzmann like stationary distribution can always be defined near it such a stationary distribution does not need to satisfy the usual detailed balance condition but might have instead a divergence free probability current in the linear case the force can be split into two parts one of which gives detailed balance with the diffusive motion whereas the other induces cyclic motion on surfaces of constant cost function by using the jordan transformation for the force matrix we find an explicit construction of the cost function we discuss singularities of the transformation and their consequences for the stationary distribution this boltzmann like distribution may be not unique and nonlinear effects and boundary conditions may change the distribution and induce additional currents even in the neighborhood of a fixed point

curl flux induced drift in stochastic differential equations in the zero mass limit abstract we consider the nonlinear stochastic dynamics of dissipative hamiltonian systems with state dependent friction and diffusion connected by the fluctuation dissipation relation in high dimensions the system under study has a close connection to ao s framework in constructing a dynamical potential for non equilibrium processes without detailed balance we study the limiting case where the mass approaches zero and give a new and complete derivation of effective stochastic differential equations using the ito stochastic integral convention we show that the limiting effective langevin equations have a new drift term this extra term happens to be identical to the corresponding anti ito or isothermal integral requiring constant temperature in one dimension we however cannot obtain this additional drift term using conventional stochastic integrals in high dimension it is interesting to note that in a high dimensional system a curl flux induced drift may appear even if the diffusion matrix is constant our findings are supported by numerical simulations we further analyze and discuss the role of this new drift term in calculating the classic escape time for the first time to our knowledge the relation between the escape rate and the anti ito integral is presented we also demonstrate that the derived diffusion equations give a new sampling algorithm which can increase convergence speed in a simple two dimensional example

optimal non reversible linear drift for the convergence to equilibrium of a diffusion we consider non reversible perturbations of reversible diffusions that do not alter the invariant distribution and we ask whether there exists an optimal perturbation such that the rate of convergence to equilibrium is maximized we solve this problem for the case of linear drift by proving the existence of such optimal perturbations and by providing an easily implementable algorithm for constructing them we discuss in particular the role of the prefactor in the exponential convergence estimate our rigorous results are illustrated by numerical experiments

percolation and local isoperimetric inequalities in this paper we establish some relations between percolation on a given graph g and its geometry our main result shows that if g has polynomial growth and satisfies what we call the local isoperimetric inequality of dimension d 1 then p c g 1 this gives a partial answer to a question of benjamini and schramm electron commun probab 1 8 71 82 1996 as a consequence of this result and haggstrom adv appl probab 32 1 39 66 2000 we derive that these graphs also undergo a non trivial phase transition for the ising model the widom rowlinson model and the beach model our techniques are also applied to dependent percolation processes with long range correlations we provide results on the uniqueness of the infinite percolation cluster and quantitative estimates on the size of finite components finally we leave some remarks and questions that arise naturally from this work

anchored expansion percolation and speed benjamini lyons and schramm random walks and discrete potential theory 1999 56 84 considered properties of an infinite graph g and the simple random walk on it that are preserved by random perturbations in this paper we solve several problems raised by those authors the anchored expansion constant is a variant of the cheeger constant its positivity implies positive lower speed for the simple random walk as shown by virag geom funct anal 10 2000 1588 1605 we prove that if g has a positive anchored expansion constant then so does every infinite cluster of independent percolation with parameter p sufficiently close to 1 a better estimate for the parameters p where this holds is in the appendix we also show that positivity of the anchored expansion constant is preserved under a random stretch if and only if the stretching law has an exponential tail we then study a simple random walk in the infinite percolation cluster in cayley graphs of certain amenable groups known as lamplighter groups we prove that zero speed for a random walk on a lamplighter group implies zero speed for random walk on an infinite cluster for any supercritical percolation parameter p for p large enough we also establish the converse

ito versus stratonovich white noise limits for systems with inertia and colored multiplicative noise we consider the dynamics of systems in the presence of inertia and colored multiplicative noise we study the limit where the particle relaxation time and the correlation time of the noise both tend to zero we show that the limiting equation for the particle position depends on the magnitude of the particle relaxation time relative to the noise correlation time in particular the limiting equation should be interpreted either in the ito or stratonovich sense with a crossover occurring when the two fast time scales are of comparable magnitude at the crossover the limiting stochastic differential equation is neither of ito nor of stratonovich type this means that after adiabatic elimination the governing equations have different drift fields leading to different physical behavior depending on the relative magnitude of the two fast time scales our findings are supported by numerical simulations

attaining the optimal gaussian diffusion acceleration sampling from probability distributions in high dimensional spaces is generally impractical diffusion processes with invariant equilibrium distributions can be used as a means to generate approximations an important task in such an endeavor is to design an equilibrium preserving drift to accelerate the convergence starting from a reversible diffusion it is desirable to depart for non reversible dynamics via a perturbed drift so that the convergence rate is maximized with the common equilibrium in the gaussian diffusion acceleration this problem can be cast as perturbing the inverse of a given covariance matrix by skew symmetric matrices so that all resulting eigenvalues have identical real part this paper describes two approaches to obtain the optimal rate of gaussian diffusion the asymptotical approach works universally for arbitrary ornstein uhlenbeck processes whereas the direct approach can be implemented as a fast divide and conquer algorithm a comparison with recently proposed lelievre nier pavliotis algorithm is made

adiabatic elimination for systems of brownian particles with nonconstant damping coefficients we discuss the problem of eliminating the momentum variable in the phase space langevin equations for a system of brownian particles in two related situations i position dependent damping and ii existence of hydrodynamic interactions we discuss the problems associated with the conventional elimination and we develop an alternative elimination procedure in the lagevin framework which leads to the correct smoluchowski equation we give a heuristic argument on the basis of stochastic differential equations for the smoluchowski limit and establish rigorously the limit for the general case of position dependent friction and diffusion coefficents

critical points and intermediate phases on wedges of zd the authors examine the phase structure of self avoiding walks the ising magnet and bond percolation defined on subsets of zd d or 2 which have the geometry of wedges they prove that if the cross sectional area of the wedge diverges with its width then the high temperature critical point of any of these models defined on the wedge coincides with that of the corresponding model on the full lattice for the ising magnet and bond percolation they show that there is a non trivial low temperature critical point if the cross sectional area of the wedge diverges logarithmically with its width moreover for bond percolation they show that the low temperature critical point of a logarithmic wedge may be made arbitrarily close to that of the full lattice by taking the coefficient of the logarithm large enough corollaries to these theorems include the existence of an intermediate phase for the ising magnet and bond percolation on logarithmic wedges and the existence of a first order transition for percolation models on a subclass of these wedges

truncated connectivities in a highly supercritical anisotropic percolation model we consider an anisotropic bond percolation model on mathbb z 2 with p p h p v 0 1 2 p v p h and declare each horizontal respectively vertical edge of mathbb z 2 to be open with probability p h respectively p v and otherwise closed independently of all other edges let x x 1 x 2 in mathbb z 2 with 0 mathbb p mathbf p 0 leftrightarrow x in this note we give an affirmative answer in the highly supercritical regime

bernoulli percolation above threshold an invasion percolation analysis on utilise le processus de percolation d invasion pour demontrer des resultats relatifs a la percolation de bernoulli sur z d

exploring a noisy van der pol type oscillator with a stochastic approach based on conventional ito or stratonovich interpretation zero mean multiplicative noise can induce shifts of attractors or even changes of topology to a deterministic dynamics such phenomena usually introduce additional complications in analysis of these systems we employ in this paper a new stochastic interpretation leading to a straightforward consequence the steady state distribution is boltzmann gibbs type with a potential function severing as a lyapunov function for the deterministic dynamics it implies that an attractor corresponds to the local extremum of the distribution function and the probability is equally distributed right on an attractor we consider a prototype of nonequilibrium processes noisy limit cycle dynamics exact results are obtained for a class of limit cycles including a van der pol type oscillator these results provide a new angle for understanding processes without detailed balance and can be verified by experiments

some remarks on the smoluchowski kramers approximation according to the smoluchowski kramers approximation solution qt of the equation mu ddot q t mu b q t mu dot q t mu sigma q t mu dot w t q 0 q dot q p where dot w t is the white noise converges to the solution of equation dot q t b q t sigma q t dot w t q 0 q as 0 many asymptotic problems for the last equation were studied in recent years we consider relations between asymptotics for the first order equation and the original second order equation homogenization large deviations and stochastic resonance approximation of brownian motion wt by a smooth stochastic process stationary distributions are considered

on the poisson equation and diffusion approximation 3 three different results are established which turn out to be closely connected so that the first one implies the second one which in turn implies the third one the first one states the smoothness of an invariant diffusion density with respect to a parameter the second establishes a similar smoothness of the solution of the poisson equation in rd the third one states a diffusion approximation result or in other words an averaging of singularly perturbed diffusion for fully coupled sde systems or sde systems with complete dependence

existence and construction of dynamical potential in nonequilibrium processes without detailed balance the existence of a dynamical potential with both local and global meanings in general nonequilibrium processes has been controversial following an earlier heuristic argument in a letter by one of us in the present paper we show rigorously its existence for a generic class of situations in physical and biological sciences the local dynamical meaning of this potential function is demonstrated via a special stochastic differential equation and its global steady state meaning via a novel and explicit form of the fokker planck equation we also give a procedure to obtain the special stochastic differential equation for any given fokker planck equation no detailed balance condition is required in our demonstration for the first time we obtain here a formula to describe the noise induced shift in drift force compared to the steady state distribution a phenomenon extensively observed in numerical studies

multiplicity of phase transitions and mean field criticality on highly non amenable graphs we consider independent percolation ising and potts models and the contact process on infinite locally finite connected graphs

percolation beyond ℤd the contributions of oded schramm oded schramm 1961 2008 influenced greatly the development of percolation theory beyond the usual d setting in particular the case of nonamenable lattices here we review some of his work in this field

a manifold of pure gibbs states of the ising model on a cayley tree we study the ising model on a cayley tree a wide class of new extreme gibbs states is exhibited

safe and effective importance sampling abstract we present two improvements on the technique of importance sampling first we show that importance sampling from a mixture of densities using those densities as control variates results in a useful upper bound on the asymptotic variance that bound is a small multiple of the asymptotic variance of importance sampling from the best single component density this allows one to benefit from the great variance reductions obtainable by importance sampling while protecting against the equally great variance increases that might take the practitioner by surprise the second improvement is to show how importance sampling from two or more densities can be used to approach a zero sampling variance even for integrands that take both positive and negative values

the correlation length for the high density phase of bernoulli percolation l auteur examine 2 types standards de fonctions de connectivite dans la phase de densite elevee de filtration de bernoulli du plus proche voisin il demontre que les 2 quantites decroissent exponentiellement avec le meme taux constant la reciproque de cette constante definit donc une longueur de correlation pour laquelle on n arrive pas a montrer qu elle est finie au dessus d une densite critique associee a la transition de phases des proprietes de semicontinuite et de convexite sont montrees pour cette longueur de correlation et on obtient des resultats analogues a ceux concernant le modele d ising en deux dimensions

the smoluchowski kramers limit of stochastic differential equations with arbitrary state dependent friction we study a class of systems of stochastic differential equations describing diffusive phenomena the smoluchowski kramers approximation is used to describe their dynamics in the small mass limit our systems have arbitrary state dependent friction and noise coefficients we identify the limiting equation and in particular the additional drift term that appears in the limit is expressed in terms of the solution to a lyapunov matrix equation the proof uses a theory of convergence of stochastic integrals developed by kurtz and protter the result is sufficiently general to include systems driven by both white and ornstein uhlenbeck colored noises we discuss applications of the main theorem to several physical phenomena including the experimental study of brownian motion in a diffusion gradient

dynamical behaviors determined by the lyapunov function in competitive lotka volterra systems dynamical behaviors of the competitive lotka volterra system even for 3 species are not fully understood in this paper we study this problem from the perspective of the lyapunov function we construct explicitly the lyapunov function using three examples of the competitive lotka volterra system for the whole state space 1 the general 2 species case 2 a 3 species model and 3 the model of may leonard the basins of attraction for these examples are demonstrated including cases with bistability and cyclical behavior the first two examples are the generalized gradient system where the energy dissipation may not follow the gradient of the lyapunov function in addition under a new type of stochastic interpretation the lyapunov function also leads to the boltzmann gibbs distribution on the final steady state when multiplicative noise is added

the exit distribution on the stochastic separatrix in kramers exit problem kramers exit problem is concerned with noise activated escape from a potential well in the case when the noise strength varepsilon temperature measured in units of potential barrier height is small this becomes a singular perturbation problem unexpectedly the noise induced escape from the domain of attraction of the stable equilibrium at the bottom of the well does not occur at points on the separatrix where the energy is minimal rather the distribution of exit points on the separatrix in the phase plane is shown to be spread away from the saddle point where the energy is minimal it is shown that actually most of the probability of the exit points on the separatrix is located at a distance o sqrt varepsilon from the saddle point and the probability vanishes altogether at the saddle point in this problem large deviations theory fails to predict the distribution of the exit point for finite noise we construct an asymptotic approximation to the solution of the fokker planck eq