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

a general theorem on local controllability we prove a general sufficient condition for local controllability of a nonlinear system at an equilibrium point earlier results of brunovsky hermes jurdjevic crouch and byrnes sussmann and grossmann are shown to be particular cases of this result also a number of new sufficient conditions are obtained all these results follow from one simple general principle namely that local controllability follows whenever brackets with certain symmetries can be neutralized in a suitable way by writing them as linear combinations of brackets of a lower degree both the class of symmetries and the definition of degree can be chosen to suit the problem

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

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

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

extremal laplacian energy of directed trees unicyclic digraphs and bicyclic digraphs abstract let a g and d g be the adjacency matrix of a digraph g with n vertices and the diagonal matrix of vertex outdegrees of g respectively then the laplacian matrix of the digraph g is l g d g a g the laplacian energy of a digraph g is defined as l e g i 1 n i 2 by using second spectral moment where 1 2 n are all the eigenvalues of l g of g in this paper by using arc shifting operation and out star shifting operation we determine the directed trees unicyclic digraphs and bicyclic digraphs which attain maximal and minimal laplacian energy among all digraphs with n vertices respectively

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

the skew energy of a digraph we are interested in the energy of the skew adjacency matrix of a directed graph d which is simply called the skew energy of d in this paper properties of the skew energy of d are studied in particular a sharp upper bound for the skew energy of d is derived in terms of the order of d and the maximum degree of its underlying undirected graph an infinite family of digraphs attaining the maximum skew energy is constructed moreover the skew energy of a directed tree is independent of its orientation and interestingly it is equal to the energy of the underlying undirected tree skew energies of directed cycles under different orientations are also computed some open problems are presented

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

controllability along a trajectory a variational approach this paper unifies and improves most sufficient conditions of local controllability both along a trajectory and at a point this is accomplished by defining high order variations that can be continuously summed the peculiar property of these variations is that they may be generated by thin conditions as relations in the lie algebra associated with a control system this property leads to a variational interpretation of sussmann type sufficient conditions of local controllability neutralization of obstructions and it allows the use of different weights for neutralization

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

5 regular oriented graphs with optimum skew energy abstract let g be a simple undirected graph and g be the corresponding oriented graph of g with the orientation the skew energy of g denoted by e s g is defined as the sum of the singular values of the skew adjacency matrix s g in 2010 adiga et al certified that s g n where is the maximum degree of g of order n it has been shown that every 5 regular oriented graph with optimum skew energy has even neighborhood property that is each pair of neighborhoods of a graph have even number of common vertices in this paper we characterize all connected 5 regular graphs of order n with this property moreover we determine all connected 5 regular oriented graphs of order n with maximum skew energy

lie brackets and local controllability a sufficient condition for scalar input systems we develop a general formalism based on exponential lie series which can be used to study control variations we apply the formalism to prove a result conjectured by h hermes that gives a sufficient condition for small time local controllability for single input systems

small time local controllability of spacecraft attitude using control moment gyros nonlinear controllability theory is utilized to investigate the small time local controllability stlc of the combined dynamics of a spacecraft with control moment gyros cmgs specifically speaking with three or more independent type cmgs the dynamics are stlc from and hence can be stabilized via time invariant piecewise continuous feedback laws to the equilibrium as long as the cmgs never reach certain definite singularities and semidefinite singularities at the equilibrium these results are then applied to a spacecraft carrying four pyramid type cmgs and further extended to a spacecraft using roof type cmgs in addition to the special case of two cmgs

second order conditions for local controllability abstract this paper presents second order depending on the 2 jet sufficient conditions for small time local controllability of affine smooth control systems based on a very simple observation that can also lead to higher order conditions a second order obstruction for a certain class of systems is also discussed as is the gap between the obstruction and the sufficient condition the new results are used for deciding local controllability of systems for which a direct application of the usual criteria is not conclusive

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

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

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

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

controllability of linear systems on generalized heisenberg groups this paper is devoted to the study of global controllability of linear systems on higher dimensional heisenberg groups hn necessary conditions and sufficient ones are provided they are compared to the known necessary and sufficient conditions on h1 stated in dath and jouan 2016 controllability of linear systems on low dimensional nilpotent and solvable lie groups journal of dynamical and control systems 22 2 207 225 the notion of decoupled systems is introduced in order to get examples and counter examples and more precise controllability criteria are stated for them

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

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

space time large deviations for interacting particle systems the asymptotic behavior of the space time empirical averages of several classes of interacting particle systems is studied here the authors prove that the empirical averages obey a process level large deviation principle whose rate function is given explicitly in the case of spin systems the authors establish some relevant properties of the rate function such as the correspondence between its minima and the invariant measures for the process finally for discrete time non ergodic systems the authors give some results on the critical behavior of the empirical averages

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

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

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

3 regular digraphs with optimum skew energy abstract the skew energy of a digraph d is defined as the sum of the singular values of its skew adjacency matrix s d in this paper we first interpret the entries of the power of the skew adjacency matrix of a digraph in terms of the number of its walks and then focus on the question posed by adiga et al c adiga r balakrishnan wasin so the skew energy of a graph linear algebra appl 432 2010 1825 1835 of determining all 3 regular connected digraphs with optimum skew energy

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

geometric conditions for local controllability abstract small time local controllability of a c1 affine control system with a generic drift vector field and n 1 linearly independent input vector fields on an n dimensional connected c1 manifold m is studied showing that a necessary condition namely the drift vector field belonging to the convex hull of the control vector fields at the given point is also sufficient on an open dense subset h1 of the set of points where it is verified local controllability at the points in the interior of h 1 is also considered when the input vector fields generate an involutive distribution and a criterion based on the type of contract of h with the hyperplane spanned by the input vector fields is proved an example shows a situation where the sufficient condition for small time local controllability of h sussmann siam j control option 25 1987 158 194 cannot be applied but in which the method now presented distinguishes the possible cases

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

the energy of a graph abstract the energy e g of a simple graph g is defined to be the sum of the absolute values of the eigen values of g if g is a k regular graph on n vertices then e g k k n 1 n k b 2 and this bound is sharp it is shown that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k e g b 2 two graphs with the same number of vertices are equienergetic if they have the same energy we show that for any positive integer n 3 there exist two equienergetic graphs of order 4n that are not cospectral

4 regular oriented graphs with optimum skew energy abstract let g be a simple undirected graph and g be an oriented graph of g with the orientation and skew adjacency matrix s g the skew energy of the oriented graph g denoted by e s g is defined as the sum of the absolute values of all the eigenvalues of s g in this paper we characterize the underlying graphs of all 4 regular oriented graphs with optimum skew energy and give orientations of these underlying graphs such that the skew energy of the resultant oriented graphs indeed attain optimum it should be pointed out that there are infinitely many 4 regular connected optimum skew energy oriented graphs while the 3 regular case only has two graphs k 4 the complete graph on 4 vertices and q 3 the hypercube

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

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

local controllability in 3 manifolds small time local controllability of a two input ck affine control system on a 3 dimesional connected ck manifold m is studied showing that in general a necessary condition is also sufficient on an open dense subset of the set of points where it is verified an example shows a situation where the sufficient condition for small time local controllability of sussmann 8 cannot be applied but in which the method now presented distinguishes the possible cases

graded approximations and controllability along a trajectory graded approximations of an affine control system are defined and their properties are investigated in particular it is proved that the local controllability along a reference trajectory of an approximating system implies the local controllability along the corresponding reference trajectory of the original system using graded approximations sufficient conditions of local controllability along a reference trajectory that generalize some known results are given

low order controllability and kinematic reductions for affine connection control systems controllability and kinematic modeling notions are investigated for a class of mechanical control systems first low order controllability results are given for the class of mechanical control systems second a precise connection is made between those mechanical systems which are dynamic i e have forces as inputs and those which are kinematic i e have velocities as inputs interestingly and surprisingly these two subjects are characterized and linked by a certain intrinsic vector valued quadratic form that can be associated to an affine connection control system

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

on the skew energy of orientations of hypercubes abstract let d be a digraph of order n and 1 2 n denote all the eigenvalues of the skew adjacency matrix of d the skew energy e s d of d is defined as e s d i 1 n i in this paper it is proved that for any positive integer k 3 there exists a k regular graph of order n having an orientation d with e s d n k this work positively answers a problem proposed by adiga et al c adiga r balakrishnan wasin so the skew energy of a digraph linear algebra appl 432 2010 1825 1835 in addition a digraph is also constructed such that its skew energy is the same as the energy of its underlying graph

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