on the essential features of metastability tunnelling time and critical configurations we consider metropolis markov chains with finite state space and transition probabilities of the form p eta eta q eta eta e beta h eta h eta for given energy function h and symmetric markov kernel q we propose a simple approach to determine the asymptotic behavior for large of the first hitting time to the ground state starting from a particular class of local minima for h called metastable states we separate the asymptotic behavior of the transition time from the determination of the tube of typical paths realizing the transition this approach turns out to be useful when the determination of the tube of typical paths is too difficult as for instance in the case of conservative dynamics we analyze the structure of the saddles introducing the notion of essentiality and describing essential saddles in terms of gates as an example we discuss the case of the 2d ising model in the degenerate case of integer tfrac 2j h

metastable behavior of stochastic dynamics a pathwise approach in this paper a new approach to metastability for stochastic dynamics is proposed the basic idea is to study the statistics of each path performing time averages along the evolution metastability would be characterized by the fact that the process of these time averages converges under a suitable rescaling to a measure valued markov jump process here this convergence is shown for the curie weiss mean field dynamics and also for a model with spatial structure harris contact process

the shortest possible return time of beta mixing processes we consider a stochastic process and a given n string we study the shortest possible return time or shortest return path of the string over all the realizations of process starting from this string for a beta mixing process having complete grammar and for each size n of the strings we approximate the distribution of this short return properly re scaled by a non degenerated distribution under mild conditions on the beta coefficients we prove the existence of the limit of this distribution to a non degenerated distribution we also prove that ergodicity is not enough to guaranty this convergence finally we present a connection between the shortest return and the shannon entropy showing that maximum of the re scaled variables grow as the matching function of wyner and ziv

pointwise estimates and exponential laws in metastable systems via coupling methods we show how coupling techniques can be used in some metastable systems to prove that mean metastable exit times are almost constant as functions of the starting microscopic configuration within a meta stable set in the example of the random field curie weiss model we show that these ideas can also be used to prove asymptotic exponentiallity of normalized metastable escape times

a scaling limit of the glauber dynamics for a class of neural network models the time evolution of d mean field variables is considered for networks of n neurons whose connection matrices jn have d distinct rows certain assumptions are made about the large n behavior of jn which guarantee the convergence of a free energy density function these assumptions are known to be satisfied e g in the hopfield model with p stored patterns for d 2p it is proved that in a scaling limit where n tends to infinity and d stays fixed the time evolution approaches that of a diffusion process in rd this process describes in detail and for times up to o n3 2 iterations the dynamics of the mean field fluctuations near a local minimum of the free energy density

a martingale approach to metastability we presented in beltran and landim j stat phys 140 1065 1114 2010 beltran and landim j stat phys 149 598 618 2012 an approach to derive the metastable behavior of continuous time markov chains we assumed in these articles that the markov chains visit points in the time scale in which it jumps among the metastable sets we replace this condition here by assumptions on the mixing times and on the relaxation times of the chains reflected at the boundary of the metastable sets

hitting time asymptotics for hard core interactions on grids we consider the hard core model with metropolis transition probabilities on finite grid graphs and investigate the asymptotic behavior of the first hitting time between its two maximum occupancy configurations in the low temperature regime in particular we show how the order of magnitude of this first hitting time depends on the grid sizes and on the boundary conditions by means of a novel combinatorial method our analysis also proves the asymptotic exponentiality of the scaled hitting time and yields the mixing time of the process in the low temperature limit as side result in order to derive these results we extended the model independent framework in manzo et al j stat phys 115 1 2 591 642 2004 for first hitting times to allow for a more general initial state and target subset

droplet growth for three dimensional kawasaki dynamics the goal of this paper is to describe metastability and nucleation for a local version of the three dimensional lattice gas with kawasaki dynamics at low temperature and low density

critical droplets in metastable states of probabilistic cellular automata we consider the problem of metastability in a probabilistic cellular automaton pca with a parallel updating rule which is reversible with respect to a gibbs measure the dynamical rules contain two parameters beta and h which resemble but are not identical to the inverse temperature and external magnetic field in a ferromagnetic ising model in particular the phase diagram of the system has two stable phases when beta is large enough and h is zero and a unique phase when h is nonzero when the system evolves at small positive values of h from an initial state with all spins down the pca dynamics give rise to a transition from a metastable to a stable phase when a droplet of the favored phase inside the metastable phase reaches a critical size we give heuristic arguments to estimate the critical size in the limit of zero temperature beta to infty as well as estimates of the time required for the formation of such a droplet in a finite system monte carlo simulations give results in good agreement with the theoretical predictions

free vibration of flexible rotating disks abstract perturbation techniques are employed to estimate the free vibration characteristics i e the natural frequencies and mode shapes of a flexible rotating disk the normalized disk stiffness 8 d hb 4 3 v m o 2 is introduced and treated as a small parameter then singular perturbation solutions to the governing eigenvalue problem are derived that are valid up to and including order for any given non zero hub radius the special case of a zero hub radius is then considered and the corresponding solutions are presented next a regular perturbation formulation valid for the limiting case of a very narrow rotating annulus is developed the natural frequency mode shape predictions from the various perturbation formulations are compared with exact values obtained from power series solutions of the eigenvalue problem it is found that the singular perturbation solutions match well with the exact values for small stiffnesses hub radii and nodal circles while the regular perturbation solution provides excellent accuracy for large hub radii

the metastable behavior of infrequently observed weakly random one dimensional diffusion processes consider the one dimensional diffusion process x t satisfying dx t varepsilon 1 2 dw t g x t dt where w t is the standard wiener process for suitable g s with a double well at m and m shape and g m g m and for an appropriate choice of lambda varepsilon the scaled process x lambda varepsilon t converges as varepsilon to 0 in an appropriate sense to the two state m and m jump process with m an absorbing state and transitions from m to m at unit rate

simultaneous exact approximate boundary controllability of thermo elastic plates with variable thermal coefficient and moment control abstract we study a controllability problem exact in the mechanical variables w w t and simultaneously approximate in the thermal variable of thermo elastic plates by means of boundary controls in the hinged dirichlet bc case when the thermal expansion term is variable in space

coalescing random walks and voter model consensus times on the torus in mathbb z d voter models on z d and on the torus of site n in z d are considered the second ones get trapped with probability 1 as t tends to infinity at all 0 s or all 1 s the asymptotic growth of these trapping times are examined as n tends to infinity this is done with limit thewems for coalescing random walk systems on the torus of side n in z d

hitting times of rare events in markov chains a general condition for a markov process with finite states is given for the convergence of the normalized hitting time of a rare event to a mean one exponential law it is then applied to the boundary driven symmetric simple exclusion process for an open set of density profiles which does not contain the stationary density profile

escape from the unstable equilibrium in a random process with infinitely many interacting particles we consider a one dimensional version of the model introduced in ref 1 at each site of z there is a particle with spin 1 particles move according to the stirring process and spins change according to the glauber dynamics in the hydrodynamical limit with the stirring process suitably speeded up the local magnetic density m r is proven in ref 1 to satisfy the reaction diffusion equation 8tmt r 1 2 rmt r v mt v m 89 4 and fl 0 c and fl being determined by the parameters of the glauber dynamics in the present paper we consider an initial state with zero magnetization mo r o we then prove that at long times before taking the hydrodynamical limit the evolution departs from that predicted by and that the microscopic state becomes a nontrivial mixture of states with different magnetizations

metastability effects in bootstrap percolation bootstrap percolation models or equivalently certain types of cellular automata exhibit interesting finite volume effects these are studied at a rigorous level the authors find that for an initial configuration obtained by placing particles independently with probability p or 2 the density of the bootstrapped final configurations in the sequence of cubes l 2 l 2 d typically undergoes an abrupt transition as l is increased from being close to 0 to the value 1 with l fixed at a large value the mean final density as a function of p changes from 0 to 1 around a value which varies only slowly with l the pertinent parameter being lambda p1 d 1 ln l the driving mechanism is the capture of a critical droplet this behaviour is analogous to the decay of a metastable state near a first order phase transition for which the analysis offers some suggestive ideas

a discrete variational problem related to ising droplets at low temperatures we consider a variational problem on thed dimensional latticez d which has applications in the study of the meatastable behavior of the stochastic ising model the problem an isoperimetric one is to find what is the smallest area a finite subset ofz d can have restricted to three classes of subsets ofz d if is one of these subsets we define its volume as the number of points in it and its area as the number of pairs of points inz d which are neighbors and such that only one of them belongs to

metastability of reversible random walks in potential fields let xi be an open and bounded subset of mathbb r d and let f xi rightarrow mathbb r be a twice continuously differentiable function denote by xi n the discretization of xi xi n xi cap n 1 mathbb z d and denote by x n t the continuous time nearest neighbor random walk on xi n which jumps from varvec x to varvec y at rate e 1 2 n f varvec y f varvec x we examine in this article the metastable behavior of x n t among the wells of the potential f

on the long term behavior of some finite particle systems we consider the problem of comparing large finite and infinite systems with locally interacting components and present a general comparison scheme for the case when the infinite system is nonergodic we show that this scheme holds for some specific models one of these is critical branching random walk onz d let t denote this system and let denote a finite version of t defined on the torus n n d z d ford 3 we prove that for stationary shift ergodic initial measures with density that ift n andt n 2n 1 d s 0 asn then v 0 is the set of extremal invariant measures for the infinite system t andq s is the transition function of feller s branching diffusion we prove several extensions and refinements of this result the other systems we consider are the voter model and the contact process

ising models on power law random graphs we study a ferromagnetic ising model on random graphs with a power law degree distribution and compute the thermodynamic limit of the pressure when the mean degree is finite degree exponent 2 for which the random graph has a tree like structure for this we closely follow the analysis by dembo and montanari ann appl probab 20 2 565 592 2010 which assumes finite variance degrees 3 adapting it when necessary and also simplifying it when possible our results also apply in cases where the degree distribution does not obey a power law

transverse vibration and stability of spinning circular plates of constant thickness and different boundary conditions the lower approximate natural frequencies of a spinning circular plate are determined for various boundary conditions the plate is rotating with constant speed of spin about the axis of symmetry perpendicular to its plane the boundary conditions treated here are clamped simply supported free and guided it is assumed that the normal displacement of the plate is small and that it does not affect during the oscillation the angular and radial variable stress distribution due to the rotating of the plate this divides the problem at hand into two parts of which the first part determines the radial and angular variable planar stresses in the plate while the second part requires the solution of the partial differential equation with variable coefficients the approximate lower natural frequencies are determined as functions of the speed of spin and exhibit in the plates with clamped guided and simply supported boundary conditions instabilities

exact solution of the asymmetric mindlin s plate equations applied to a disk abstract an exact solution is presented for the static and dynamic asymmetric response of a disk governed by mindlin s plate equations forced by a pressure that varies radially as r m the static solution agrees with a modal solution adopting the dynamic mindlin s plate equations in the limit when excitation frequency vanishes this solution is useful in sizing magnitude and shape of surface asymmetries on a disk from pressure loading with slight eccentricity and circumferential non uniformity

behavior of droplets for a class of glauber dynamics at very low temperature we consider a class of glauber dynamics for the two dimensional nearest neighbor ferromagnetic ising model in which the rate with which each spin flips depends only on the increment in energy caused by its flip in a monotonic non increasing fashion

backward uniqueness of the s c semigroup arising in parabolic hyperbolic fluid structure interaction a 2 d or 3 d fluid structure interaction model in its linear form is considered for which semigroup well posedness with explicit generator was recently established in g avalos r triggiani the coupled pde system arising in fluid structure interaction part i explicit semigroup generator and its spectral properties in fluids and waves in contemp math vol 440 amer math soc 2007 pp 15 55 g avalos r triggiani the coupled pde system arising in fluid structure interaction part ii uniform stabilization with boundary dissipation at the interface discrete contin dyn syst in press this is a system which couples at the interface the linear version of the navier stokes equations with the equations of linear elasticity wave like in this paper we establish a backward uniqueness theorem for such a parabolic hyperbolic coupled pde system if eat t 0 is the contraction s c semigroup describing its evolution on the finite energy space h then eaty0 0 for some t 0 and y0 h implies y0 0 this property has implications in establishing unique continuation and controllability properties as in the case of thermoelastic equations m eller i lasiecka r triggiani simultaneous exact approximate boundary controllability of thermoelastic plates with variable coefficient in marcel dekker lect notes pure appl math vol 216 february 2001 pp 109 230 invited paper for the special volume entitled shape optimization and optimal designs j cagnol j p zolesio eds preliminary version is in invited paper in a v balakrishnan ed semigroup of operators and applications birkhauser 2000 pp 335 351 m eller i lasiecka r triggiani simultaneous exact approximate boundary controllability of thermoelastic plates with variable thermal coefficient and moment control j math anal appl 251 2000 452 478 m eller i lasiecka r triggiani simultaneous exact approximate boundary controllability of thermoelastic plates with variable thermal coefficient and clamped controls discrete contin dyn syst 7 2 2001 283 301

ising critical exponents on random trees and graphs we study the critical behavior of the ferromagnetic ising model on random trees as well as so called locally tree like random graphs we pay special attention to trees and graphs with a power law offspring or degree distribution whose tail behavior is characterized by its power law exponent 2 we show that the critical inverse temperature of the ising model equals the hyperbolic arctangent of the reciprocal of the mean offspring or mean forward degree distribution in particular the critical inverse temperature equals zero when tau in 2 3 where this mean equals infinity

critical behaviour and intermittency in sinai s billiard abstract we study numerically the transition that occurs in the dynamics of sinai s billard when the circles come into contact singular behaviours are predicted by a local analysis of the region of phase space where the motion is nearly integrable the link with intermittency is discussed we use a renormalization transformation and deduce the scaling laws we recover transposed in the language of critical phenomena certain predictions of the ergodic theory

slow motion and metastability for a nonlocal evolution equation in this paper we consider a nonlocal evolution equation in one dimension which describes the dynamics of a ferromagnetic system in the mean field approximation in the presence of a small magnetic field it admits two stationary and homogeneous solutions representing the stable and metastable phases of the physical system we prove the existence of an invariant one dimensional manifold connecting the stable and metastable phases this is the unstable manifold of a distinguished spatially nonhomogeneous stationary solution called the critical droplet 4 10 we show that the points on the manifold are droplets longer or shorter than the critical one and that their motion is very slow in agreement with the theory of metastable patterns we also obtain a new proof of the existence of the critical droplet which is supplied with a local uniqueness result

potential well spectrum and hitting time in renewal processes the potential well of a state can be interpreted physically as the energy that a stationary process needs to leave the state we prove that for discrete time renewal processes the potential well is the right scaling for the hitting and return time distributions of the state we further detail the potential well spectrum of these processes by giving a complete classification of the states according to their potential well

condensation of non reversible zero range processes in this article we investigate the condensation phenomena for a class of non reversible zero range processes on a fixed finite set by establishing a novel inequality bounding the capacity between two sets and by developing a robust framework to perform quantitative analysis on the metastability of non reversible processes we prove that the condensed site of the corresponding zero range processes approximately behaves as a markov chain on the underlying graph whose jump rate is proportional to the capacity with respect to the underlying random walk the results presented in the current paper complete the generalization of the work of beltran and landim probab theory relat fields 152 781 807 2012 on reversible zero range processes and that of landim commun math phys 330 1 32 2014 on totally asymmetric zero range processes on a one dimensional discrete torus

couplings and asymptotic exponentiality of exit times the goal of this note is simply to call attention to the resulting simplification in the proof of asymptotic exponentiality of exit times in the freidlin wentzell regime as proved by f martinelli et al by using the coupling proposed by t lindvall and c rogers

sharp asymptotics for stochastic dynamics with parallel updating rule in this paper we study the metastability problem for a stochastic dynamics with a parallel updating rule in particular we consider a finite volume probabilistic cellular automaton pca in a small external field at low temperature regime we are interested in the nucleation of the system i e the typical excursion from the metastable phase the configuration with all minuses to the stable phase the configuration with all pluses triggered by the formation of a critical droplet the main result of the paper is the sharp estimate of the nucleation time we show that the nucleation time divided by its average converges to an exponential random variable and that the rate of the exponential random variable is an exponential function of the inverse temperature times a prefactor that does not scale with our approach combines geometric and potential theoretic arguments

small random perturbations of finite and infinite dimensional dynamical systems unpredictability of exit times we apply previous results on the pathwise exponential loss of memory of the initial condition for stochastic differential equations with small diffusion to the problem of the asymptotic distribution of the first exit times from an attracted domain we show under general hypotheses that the suitably rescaled exit time converges in the zero noise limit to an exponential random variable then we extend the results to an infinite dimensional case obtained by adding a small random perturbation to a nonlinear heat equation

abrupt convergence and escape behavior for birth and death chains we link two phenomena concerning the asymptotical behavior of stochastic processes i abrupt convergence or cut off phenomenon and ii the escape behavior usually associated to exit from metastability the former is characterized by convergence at asymptotically deterministic times while the convergence times for the latter are exponentially distributed we compare and study both phenomena for discrete time birth and death chains on with drift towards zero in particular this includes energy driven evolutions with energy functions in the form of a single well under suitable drift hypotheses we show that there is both an abrupt convergence towards zero and escape behavior in the other direction furthermore as the evolutions are reversible the law of the final escape trajectory coincides with the time reverse of the law of cut off paths thus for evolutions defined by one dimensional energy wells with sufficiently steep walls cut off and escape behavior are related by time inversion

metastability and low lying spectra in reversible markov chains we study a large class of reversible markov chains with discrete state space and transition matrix pn we define the notion of a set of metastable points as a subset of the state space n such that i this set is reached from any point x n without return to x with probability at least bn while ii for any two points x y in the metastable set the probability t 1x y to reach y from x without return to x is smaller than an 1 bn under some additional non degeneracy assumption we show that in such a situation

slow droplet driven relaxation of stochastic ising models in the vicinity of the phase coexistence region we consider the stochastic ising models glauber dynamics corresponding to the infinite volume basic ising model in arbitrary dimensiond 2 with nearest neighbor interaction and under a positive external magnetic fieldh under minimal assumptions on the rates of flip so that all the common choices are included we obtain results which state that when the system is at low temperaturet the relaxation time when the evolution is started with all the spins down blows up whenh 0 as exp t hd 1 the precise results are lower and upper bounds of this form moreover after a time which does not scale withh and before a time which also grows as an exponential of a multiple of 1 hd 1 ash 0 the law of the state of the process stays whenh is small close to the minus phase of the same ising model without an external field these results may be considered as a partial vindication of a conjecture raised by aizenman and lebowitz in connection to the metastable behavior of these stochastic ising models

extinction time for the contact process on general graphs we consider the contact process on finite and connected graphs and study the behavior of the extinction time that is the amount of time that it takes for the infection to disappear in the process started from full occupancy we prove without any restriction on the graph g that if the infection rate lambda is larger than the critical rate of the one dimensional process then the extinction time grows faster than exp g log g kappa for any constant kappa 1 where g denotes the number of vertices of g also for general graphs we show that the extinction time divided by its expectation converges in distribution as the number of vertices tends to infinity to the exponential distribution with parameter 1 these results complement earlier work of mountford mourrat valesin and yao in which only graphs of bounded degrees were considered and the extinction time was shown to grow exponentially in n here we also provide a simpler proof of this fact

small random perturbations of dynamical systems exponential loss of memory of the initial condition we consider markov processes arising from small random perturbations of non chaotic dynamical systems under rather general conditions we prove that with large probability the distance between two arbitrary paths starting close to a same attractor of the unperturbed system decreases exponentially fast in time the case of paths starting in different basins of attraction is also considered as well as some applications to the analysis of the invariant measure and to elliptic problems with small parameter in front to the second derivatives the proof is based on a multiscale analysis of the typical trajectories of the markov process this analysis is done using techniques involved in the proof of anderson localization for disordered quantum systems

wulff droplets and the metastable relaxation of kinetic ising models we consider the kinetic ising models glauber dynamics corresponding to the infinite volume ising model in dimension 2 with nearest neighbor ferromagnetic interaction and under a positive external magnetic field h minimal conditions on the flip rates are assumed so that all the common choices are being considered we study the relaxation towards equilibrium when the system is at an arbitrary subcritical temperature t and the evolution is started from a distribution which is stochastically lower than the phase we show that as h 0 the relaxation time blows up as exp c t h with lgr c t w t 2 12 t m t here m t is the spontaneous magnetization and w t is the integrated surface tension of the wulff body of unit volume moreover for 0 c the state of the process at time exp h is shown to be close when h is small to the phase the difference between this state and the phase can be described in terms of an asymptotic expansion in powers of the external field this expansion can be interpreted as describing a set of continuations in h of the family of gibbs distributions with the negative magnetic fields into the region of positive fields

the consensus times of the majority vote process on a torus we study the majority vote process on a two dimensional torus in which every voter adopts the minority of opinion with small probability we identify the exponent that the mean of consensus time is asymptotically 1 with that exponent as goes to 0 the proof is by a formula for mean exit time and by the metastable theory of markov chains developed in the study of the stochastic ising model

a limit theorem for a class of inhomogeneous markov processes pour un processus de markov xh irreductible et aperiodique avec un espace d etats fini et avec un taux de transition q ij t x t devient homogene dans le cas q ij t q ij et admet une loi limite qui est independante de x o le but de cet article est de demontrer qu un resultat similaire peut etre obtenu dans le cas de processus de markov non homogenes avec un certain type de taux de transition

universality of cutoff for the ising model on any locally finite geometry the stochastic ising model is known to be contractive when the inverse temperature beta is small enough via classical results of dobrushin and of holley in the 1970s by a general principle proposed by peres the dynamics is then expected to exhibit cutoff however so far cutoff for the ising model has been confirmed mainly for lattices heavily relying on amenability and log sobolev inequalities without these cutoff was unknown at any fixed beta 0 no matter how small even in basic examples such as the ising model on a binary tree or a random regular graph we use the new framework of information percolation to show that in any geometry there is cutoff for the ising model at high enough temperatures precisely on any sequence of graphs with maximum degree d the ising model has cutoff provided that beta kappa d for some absolute constant kappa a result which up to the value of kappa is best possible moreover the cutoff location is established as the time at which the sum of squared magnetizations drops to 1 and the cutoff window is o 1 just as when beta 0 finally the mixing time from almost every initial state is not more than a factor of 1 varepsilon beta faster then the worst one with varepsilon beta to0 as beta to0 whereas the uniform starting state is at least 2 varepsilon beta times faster

metastability for reversible probabilistic cellular automata with self interaction the problem of metastability for a stochastic dynamics with a parallel updating rule is addressed in the freidlin wentzel regime namely finite volume small magnetic field and small temperature the model is characterized by the existence of many fixed points and cyclic pairs of the zero temperature dynamics in which the system can be trapped in its way to the stable phase our strategy is based on recent powerful approaches not needing a complete description of the fixed points of the dynamics but relying on few model dependent results we compute the exit time in the sense of logarithmic equivalence and characterize the critical droplet that is necessarily visited by the system during its excursion from the metastable to the stable state we need to supply two model dependent inputs 1 the communication energy that is the minimal energy barrier that the system must overcome to reach the stable state starting from the metastable one 2 a recurrence property stating that for any configuration different from the metastable state there exists a path starting from such a configuration and reaching a lower energy state such that its maximal energy is lower than the communication energy

metastability in a condensing zero range process in the thermodynamic limit zero range processes with decreasing jump rates are known to exhibit condensation where a finite fraction of all particles concentrates on a single lattice site when the total density exceeds a critical value we study such a process on a one dimensional lattice with periodic boundary conditions in the thermodynamic limit with fixed super critical particle density we show that the process exhibits metastability with respect to the condensate location i e the suitably accelerated process of the rescaled location converges to a limiting markov process on the unit torus this process has stationary independent increments and the rates are characterized by the scaling limit of capacities of a single random walker on the lattice our result extends previous work for fixed lattices and diverging density in beltran and landim probab theory relat fields 152 3 4 781 807 2012 and we follow the martingale approach developed there and in subsequent publications besides additional technical difficulties in estimating error bounds for transition rates the thermodynamic limit requires new estimates for equilibration towards a suitably defined distribution in metastable wells corresponding to a typical set of configurations with a particular condensate location the total exit rates from individual wells turn out to diverge in the limit which requires an intermediate regularization step using the symmetries of the process and the regularity of the limit generator another important novel contribution is a coupling construction to provide a uniform bound on the exit rates from metastable wells which is of a general nature and can be adapted to other models

shapes of growing droplets a model of escape from a metastable phase nucleation from a metastable state is studied for an ising ferromagnet with nearest and next nearest neighbor interaction and at very low temperatures the typical escape path is shown to follow a sequence of configurations with a growing droplet of stable phase whose shape is determined by dynamical considerations and differs significantly from the equilibrium shape corresponding to the instantaneous volume

diffusions conditioned on occupation measures a markov process fluctuating away from its typical behavior can be represented in the long time limit by another markov process called the effective or driven process having the same stationary states as the original process conditioned on the fluctuation observed we construct here this driven process for diffusions spending an atypical fraction of their evolution in some region of state space corresponding mathematically to stochastic differential equations conditioned on occupation measures as an illustration we consider the langevin equation conditioned on staying for a fraction of time in different intervals of the real line including the positive half line which leads to a generalization of the brownian meander problem other applications related to quasi stationary distributions metastable states noisy chemical reactions queues and random walks are discussed

detecting nonergodicity in continuous time spin systems we develop an empirical test aimed at detecting nonergodicity from a single sample of a spin system we show that the test is asymptotically correct and we give explicit asymptotics for the error probability the key tool consists in some new large deviation estimates

exact thresholds for ising gibbs samplers on general graphs we establish tight results for rapid mixing of gibbs samplers for the ferromagnetic ising model on general graphs we show that if d 1 tanh 1 then there exists a constant c such that the discrete time mixing time of gibbs samplers for the ferromagnetic ising model on any graph of n vertices and maximal degree d where all interactions are bounded by and arbitrary external fields are bounded by cnlogn moreover the spectral gap is uniformly bounded away from 0 for all such graphs as well as for infinite graphs of maximal degree d we further show that when dtanh 1 with high probability over the erd s renyi random graph g n d n it holds that the mixing time of gibbs samplers is n1 1 loglogn both results are tight as it is known that the mixing time for random regular and erd s renyi random graphs is with high probability exponential in n when d 1 tanh 1 and dtanh 1 respectively to our knowledge our results give the first tight sufficient conditions for rapid mixing of spin systems on general graphs moreover our results are the first rigorous results establishing exact thresholds for dynamics on random graphs in terms of spatial thresholds on trees

interfaces and typical gibbs configurations for one dimensional kac potentials we consider a one dimensional ising spin system with a ferromagnetic kac potential j r j having compact support we study the system in the limit 0 below the lebowitz penrose critical temperature where there are two distinct thermodynamic phases with different magnetizations we prove that the empirical spin average in blocks of size 1 for any positive converges as 0 to one of the two thermodynamic magnetizations uniformly in the intervals of size p for any given positivep 1 we then show that the intervals where the magnetization is approximately constant have lengths of the order of exp c 1 c 0 and that when normalized they converge to independent variables with exponential distribution we show this by proving large deviation estimates and applying the ventsel and friedlin methods to gibbs random fields finally if the temperature is low enough we characterize the interface namely the typical magnetization pattern in the region connecting the two phases

markov chains with exponentially small transition probabilities first exit problem from a general domain i the reversible case we consider general ergodic aperiodic markov chains with finite state space whose transition probabilities between pairs of different communicating states are exponentially small in a large parameter we extend previous results by m i freidlin and a d wentzell fw on the first exit problem from a general domainq in the present paper we analyze the case ofreversible markov chains the general case will be studied in a forthcoming paper we probe in a purely probabilistic way and without using the fw graphical technique some results on the first exit problem from a general domainq containing many attractors in particular we analyze the properties of special domains calledcycles and by using the new concept oftemporal entropy we obtain new results leading to a complete description of the typical tube of trajectories during the first excursion outsideq

the contact process on a finite set ii in this paper we complete the work started in part i we show that if sigma n is the time that the contact process on 1 ldots n first hits the empty set then for lambda lambda c the critical value for the process on z there is a positive constant gamma lambda so that log sigma n n rightarrow gamma lambda in probability as n rightarrow infty we also give a new simple proof that sigma n e sigma n converges to a mean one exponential the keys to the proof of the first result are a planar graph duality for the contact process and an observation of j chayes and l chayes that exponential decay rates for connections in strips approach the decay rates in the plane as the width of the strip goes to infty

large density fluctuations for the one dimensional supercritical contact process we consider the one dimensional supercritical contact process lettv be the first time the process reaches a densityq larger than the equilibrium one in the region 1 n we prove that starting from equilibrium tn e tn converges to an exponential random time of mean one in this way we extend previous results of lebowitz and schonmann

edge fluctuations for the one dimensional supercritical contact process we consider the one dimensional supercritical contact process with initial configurations having infinitely many particles to the left of the origin and only finitely many to its right starting from any such configuration we first prove that in the limit as time goes to infinity the law of the process as seen from the edge converges to the invariant distribution constructed by durrett 12 we then prove a functional central limit theorem for the fluctuations of the edge around its average showing that the corresponding diffusion coefficient is strictly positive we finally characterize the space time structure of the system in particular we prove that its distribution shifted in space by at t denotes the time and a the drift of the edge converges when t goes to infinity to a 2 2 mixture of the two extremal invariant measures for the contact process 1 introduction in this article we study the one dimensional supercritical contact process a stochastic process with infinitely many interacting particles which move on z the set of all the integers we consider initial states with infinitely many particles to the left of the origin and finitely many to its right we first prove that at large times the state of the system approaches some definite structure which as time increases rigidly shifts to the right a steady state propagates from the left to the right with constant speed through the formerly empty region the system is a very schematic and rough microscopic model for the formation and propagation of one dimensional shock waves as we shall see in some more detail in the sequel at large times we can distinguish three different space regions the first two are semiinfinite intervals which extend to oo and oo respectively while the third one is the interval which connects the first two in the first region the state looks like the nonzero density equilibrium state for the supercritical contact process while in the second it is the empty state the third is the interesting region it describes the wave front we find that the state there is close to a superposition of the two extremal equilibrium states the weight of the decomposition ranges from 1 all the weight to the nonzero density state to 0 as one moves from left to right as the time t increases the previous picture shifts to the right and at the same time the size of the front wave increases like tl 2 the picture is very similar indeed to that found by wick 25 in a zero range model our result is a consequence of a very detailed analysis of the motion of the first particle in the system the edge and of a precise knowledge of the structure of the state as seen from the edge such

metastability of nonreversible random walks in a potential field and the eyring kramers transition rate formula we consider nonreversible random walks in a bounded domain of d evolving on a potential field we describe the complete metastable behavior of the random walk among the landscape of valleys and we derive the eyring kramers formula for the mean transition time from a metastable set to a stable set 2017 wiley periodicals inc

metastability in stochastic dynamics of disordered mean field models we study a class of markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures our main purpose is to give a precise relation between the metastable time scales in the problem to the properties of the rate functions of the corresponding gibbs measures we derive the analog of the wentzell freidlin theory in this case showing that any transition can be decomposed with probability exponentially close to one into a deterministic sequence of admissible transitions for these admissible transitions we give upper and lower bounds on the expected transition times that differ only by a constant factor the distributions of the rescaled transition times are shown to converge to the exponential distribution we exemplify our results in the context of the random field curie weiss model

backwards uniqueness of the c_0 semigroup associated with a parabolic hyperbolic stokes lame partial differential equation system in this paper the backward uniqueness property is ascertained for a two or three dimensional fluid structure interactive partial differential equation pde system for which an explicit co semigroup formulation was recently given by avalos and triggiani 2007 on the natural finite energy space h see also their contemporary mathematics article of 2007 for a preliminary simplified canonical model this system of coupled pdes comprises the parabolic stokes equations and the hyperbolic lame system of dynamic elasticity each dynamic evolves within its respective domain while being coupled on the boundary interface between fluid and structure in terms of said fluid structure semigroup e at posed on the associated finite energy space h the backward uniqueness property can be stated in this way if for given initial data y 0 h e at y 0 0 for some t 0 then necessarily y 0 0 the proof of this property hinges on establishing necessary pde estimates for a certain static fluid structure equation in order to invoke the abstract backward uniqueness resolvent based criterion by lasiecka renardy and triggiani 2001 the backward uniqueness property for the coupled stokes lame pde is motivated by and has positive implications to the problem of exact controllability in the hyperbolic state variables w w t and simultaneously approximate controllability in the parabolic state variable u of the present coupled pde model under boundary control a similar situation occurred for thermoelastic models as shown in papers by m eller v isakov i lasiecka m renardy and r triggiani

the weak limit of ising models on locally tree like graphs we consider the ising model with inverse temperature and without external field on sequences of graphs g n which converge locally to the k regular tree we show that for such graphs the ising measure locally weakly converges to the symmetric mixture of the ising model with boundary conditions and the boundary conditions on the k regular tree with inverse temperature in the case where the graphs g n are expanders we derive a more detailed understanding by showing convergence of the ising measure conditional on positive magnetization sum of spins to the measure on the tree

precise asymptotics of small eigenvalues of reversible diffusions in the metastable regime we investigate the close connection between metastability of the reversible diffusion process x defined by the stochastic differential equation d xt f xt d t 2e d wt e 0 and the spectrum near zero of its generator l f where f d and w denotes brownian motion on d for generic f to each local minimum of f there corresponds a metastable state we prove that the distribution of its rescaled relaxation time converges to the exponential distribution as 0 with optimal and uniform error estimates each metastable state can be viewed as an eigenstate of l with eigenvalue which converges to zero exponentially fast in 1 modulo errors of exponentially small order in 1 this eigenvalue is given as the inverse of the expected metastable relaxation time the eigenstate is highly concentrated in the basin of attraction of the corresponding trap

metastability of the ising model on random regular graphs at zero temperature we study the metastability of the ferromagnetic ising model on a random r regular graph in the zero temperature limit we prove that in the presence of a small positive external field the time that it takes to go from the all minus state to the all plus state behaves like exp beta r 2 mathcal o sqrt r n when the inverse temperature beta rightarrow infty and the number of vertices n is large enough but fixed the proof is based on the so called pathwise approach and bounds on the isoperimetric number of random regular graphs