numerical results on a hypercubic cell spin glass model the authors study a spin glass hypercubic cell in d dimensions i e hypercubic lattice with l 2 where l is the lattice size for different values of d by means of monte carlo simulation in the limit d to infinity this model converges to the sk model as well as for hypercubic lattices they confirm the usual interpretation of the spin glass phase since this model is more similar to hypercubic lattices than to the infinite range model these results suggest that broken replica symmetry is useful to study finite dimensional hypercubic lattices above a certain critical dimension

reversing the drift of the ehrenfest urn model and three conditionings we consider a drift reversed version of the celebrated ehrenfest urn process with n balls for this dual process the boundaries are assumed to be absorbing and so the killing times at the boundaries play a central role three natural conditionings on the fixation extinction events pertaining to this model are investigated some spectral information on the conditioned markoff chains is obtained allowing us to draw precise new conclusions on their limiting behaviors

multilevel dyson brownian motions via jack polynomials we introduce multilevel versions of dyson brownian motions of arbitrary parameter beta 0 generalizing the interlacing reflected brownian motions of warren for beta 2 such processes unify beta corners processes and dyson brownian motions in a single object our approach is based on the approximation by certain multilevel discrete markov chains of independent interest which are defined by means of jack symmetric polynomials in particular this approach allows to show that the levels in a multilevel dyson brownian motion are intertwined at least for beta ge 1 and to give the corresponding link explicitly

probability laws related to the jacobi theta and riemann zeta functions and brownian excursions this paper reviews known results which connect riemann s integral representations of his zeta function involving jacobi s theta function and its derivatives to some particular probability laws governing sums of independent exponential variables these laws are related to one dimensional brownian motion and to higher dimensional bessel processes we present some characterizations of these probability laws and some approximations of riemann s zeta function which are related to these laws

finite size scaling in the p state mean field potts glass a monte carlo investigation the p state mean field potts glass with bimodal bond distribution j is studied by monte carlo simulations both for p 3 and p 6 states for system sizes from n 5 to n 120 spins considering particularly the finite size scaling behavior at the exactly known glass transition temperature t c it is shown that for p 3 the moments q k of the spin glass order parameter satisfy a simple scaling behavior q k alpha n k 3 tilde f k n 1 3 1 t t c text k 1 2 3 tilde f k being the appropriate scaling function and t the temperature also the specific heat maxima have a similar behavior c v max alpha text const n 1 3 while moments of the magnetization scale as m k alpha n k 2 the approach of the positions t max of these specific heat maxima to t c as n is nonmonotonic for p 6 the results are compatible with a first order transition q k q jump k as n but since the order parameter q jump at t c is rather small a behavior q k n k 3 as n also is compatible with the data thus no firm conclusions on the finite size behavior of the order parameter can be drawn the specific heat maxima c v max behave qualitatively in the same way as for p 3 consistent with the prediction that there is no latent heat a speculative phenomenological discussion of finite size scaling for such transitions is given for small n n 15 for p 3 n 12 for p 6 the monte carlo data are compared to exact partition function calculations and excellent agreement is found we also discuss ratios r x equiv langle x rangle t langle x rangle t text av 2 text av langle x rangle t text av 2 for various quantities x to test the possible lack of self averaging at t c

quasi stationary distributions for structured birth and death processes with mutations we study the probabilistic evolution of a birth and death continuous time measure valued process with mutations and ecological interactions the individuals are characterized by phenotypic traits that take values in a compact metric space each individual can die or generate a new individual the birth and death rates may depend on the environment through the action of the whole population the offspring can have the same trait or can mutate to a randomly distributed trait we assume that the population will be extinct almost surely our goal is the study in this infinite dimensional framework of quasi stationary distributions when the process is conditioned on non extinction we firstly show in this general setting the existence of quasi stationary distributions this result is based on an abstract theorem proving the existence of finite eigenmeasures for some positive operators we then consider a population with constant birth and death rates per individual and prove that there exists a unique quasi stationary distribution with maximal exponential decay rate the proof of uniqueness is based on an absolute continuity property with respect to a reference measure

l2 lower bounds for a special class of random walks we investigate thel 2 speed of convergence to stationarity for a certain class of random walks on a compact connected lie group we give a lower bound on the number of stepsk necessary such that thek fold convolution power of the original step distribution has anl 2 density our method uses work by heckman on the asymptotics of multiplicities along a ray of representations several examples are presented

schur dynamics of the schur processes abstract we construct discrete time markov chains that preserve the class of schur processes on partitions and signatures one application is a simple exact sampling algorithm for q volume distributed skew plane partitions with an arbitrary back wall another application is a construction of markov chains on infinite gelfand tsetlin schemes that represent deterministic flows on the space of extreme characters of the infinite dimensional unitary group

synchronization between interacting ensembles of globally coupled chaotic maps when two groups of globally coupled identical chaotic maps interact full synchronization coherence and different kinds of partial synchronization clustering can arise in this paper we identify the regions in parameter space that lead to coherence and clustering the region of stability for the coherent state is shown to be independent of the number of maps in the ensembles the borders of stability for the coherence region are obtained in terms of the transverse lyapunov exponents and the nonlinearity parameter of the individual map it is shown that for values of the nonlinearity parameter that correspond to periodic windows the coherent state is stable in the unit square of the coupling parameter plane for the simplest non trivial case of four maps we analyse the stability properties of the two cluster states in detail and show cluster stability diagrams for particular values of the nonlinearity parameter of the logistic map in the general case of 2n maps we determine the invariant cluster manifolds in phase space and identify the corresponding cluster states for each cluster state we study the character of the in cluster dynamics which is found to be regular or chaotic

shuffling algorithm for boxed plane partitions we introduce discrete time markov chains that preserve uniform measures on boxed plane partitions elementary markov steps change the size of the box from a b c to a 1 b 1 c or a 1 b 1 c algorithmic realization of each step involves o a b c operations one application is an efficient perfect random sampling algorithm for uniformly distributed boxed plane partitions trajectories of our markov chains can be viewed as random point configurations in the three dimensional lattice we compute the bulk limits of the correlation functions of the resulting random point process on suitable two dimensional sections the limiting correlation functions define a two dimensional determinantal point processes with certain gibbs properties

rates of convergence of random walk on distance regular graphs when run on any non bipartite q distance regular graph from a family containing graphs of arbitrarily large diameter d we show that d steps are necessary and sufficient to drive simple random walk to the uniform distribution in total variation distance and that a sharp cutoff phenomenon occurs for most examples we determine the set on which the variation distance is achieved and the precise rate at which it decays

interlacing eigenvalues in time reversible markov chains for irreducible reversible finite state chains we observe that two sets of eigenvalues related to the transition rate matrix q 0 m and 1 m are interlaced so that 0 1 1 m m many quantities associated with l ta the distribution of the first passage time to a starting in steady state can be simply expressed in terms of these eigenvalues and the interlacing property can be exploited to obtain approximations there is a close connection between interlacing eigenvalues and completely monotone distributions as well as a representation for finite mixtures of exponential distributions which follow from this observation

interacting particle systems at the edge of multilevel dyson brownian motions abstract we study the joint asymptotic behavior of spacings between particles at the edge of multilevel dyson brownian motions when the number of levels tends to infinity despite the global interactions between particles in multilevel dyson brownian motions we observe a decoupling phenomenon in the limit the global interactions become negligible and only the local interactions remain the resulting limiting objects are interacting particle systems which can be described as brownian versions of certain totally asymmetric exclusion processes this is the first appearance of a particle system with local interactions in the context of general random matrix models

improving monte carlo efficiency by increasing variance this paper compares the performances of two well known monte carlo procedures for estimating an unknown quantity as the size of the problem grows one method based on the standard monte carlo approach generates k i i d data points the other derives its data from a single k step sample path generated by a markov chain the paper gives necessary and sufficient conditions for the markov chain approach to perform more efficiently than the standard monte carlo approach does moreover it identifies circumstances under which this better efficiency grows with increasing problem size this improved efficiency comes from reduced sample generating and function evaluating costs in the markov chain approach that more than compensate for the increased variance of the estimator that the markov chain sampling approach induces when compared to the standard monte carlo approach several examples illustrate how the benefits arise one also demonstrates a case in which the standard monte carlo approach becomes increasingly preferred as the problem size grows

commutation relations and markov chains it is shown that the combinatorics of commutation relations is well suited for analyzing the convergence rate of certain markov chains examples studied include random walk on irreducible representations a local random walk on partitions whose stationary distribution is the ewens distribution and some birth death chains

stochastic higher spin six vertex model and q taseps abstract we present two new connections between the inhomogeneous stochastic higher spin six vertex model in a quadrant and integrable stochastic systems from the macdonald processes hierarchy first we show how macdonald q difference operators with t 0 an algebraic tool crucial for studying the corresponding macdonald processes can be utilized to get q moments of the height function h in the higher spin six vertex model first computed in 21 using bethe ansatz this result in particular implies that for the vertex model with the step bernoulli boundary condition the value of h at an arbitrary point n 1 t z 2 z 1 has the same distribution as the last component n of a random partition under a specific t 0 macdonald measure on the other hand it is known that x n n n can be identified with the location of the nth particle in a certain discrete time q tasep started from the step initial configuration the second construction we present is a coupling of this q tasep and the higher spin six vertex model with the step bernoulli boundary condition along time like paths providing an independent probabilistic explanation of the equality of h n 1 t and x n n in distribution as an illustration of our main results we obtain gue tracy widom asymptotics of a certain discrete time q tasep with the step initial configuration and special jump parameters by means of schur measures which are t q macdonald measures this analysis combines our results with the identification of averages of observables between the stochastic higher spin six vertex model and schur measures obtained recently in 8

future independent times and markov chains a random timet is a future independent time for a markov chain x n 0 ift is independent of x t n n 0 and if x t n n 0 is a markov chain with initial distribution and the same transition probabilities as x n 0 this concept is used with the conditional stationary measure to give a new and short proof of the basic limit theorem of markov chains improving somewhat the result in the null recurrent case

fluctuation theory for moran s genetics model abstract a genetics model of moran which includes the bernoulli laplace urn as a special case is examined to give a picture of the model s fluctuation theory and asymptotics as the number of states grows large with particular attention to the time of passage between extreme states these passage times are shown to have means which grow exponentially with the number of states and when normalised converge weakly to an exponential distribution

markov processes on the path space of the gelfand tsetlin graph and on its boundary we construct a four parameter family of markov processes on infinite gelfand tsetlin schemes that preserve the class of central gibbs measures any process in the family induces a feller markov process on the infinite dimensional boundary of the gelfand tsetlin graph or equivalently the space of extreme characters of the infinite dimensional unitary group u the process has a unique invariant distribution which arises as the decomposing measure in a natural problem of harmonic analysis on u posed in olshanski 2003 44 as was shown in borodin and olshanski 2005 11 this measure can also be described as a determinantal point process with a correlation kernel expressed through the gauss hypergeometric function

the mathematics of mixing things up how long should a markov chain monte carlo algorithm be run using examples from statistical physics ehrenfest urn ising model hard discs as well as card shuffling this tutorial paper gives an overview of a body of mathematical results that can give useful answers to practitioners viz seven shuffles suffice for practical purposes it points to new techniques path coupling geometric inequalities and harris recurrence the discovery of phase transitions in mixing times the cutoff phenomenon is emphasized

synchronization of networks of chaotic units with time delayed couplings a network of chaotic units is investigated where the units are coupled by signals with a transmission delay any arbitrary finite network is considered where the chaotic trajectories of the uncoupled units are a solution of the dynamic equations of the network it is shown that chaotic trajectories cannot be synchronized if the transmission delay is larger than the time scales of the individual units for several models the master stability function is calculated which determines the maximal delay time for which synchronization is possible

three dimensional gaussian fluctuations of non commutative random surfaces along time like paths abstract we construct a continuous time non commutative markov operator on u gl n with corresponding morphisms u gl n end l 2 u n at any fixed times 0 t 1 t 2 this is an analog of a continuous time non commutative markov operator on the group von neumann algebra v n u n constructed in 16 and is a variant of discrete time non commutative random walks on u gl n 2 9 it is also shown that when restricting to the gelfand tsetlin subalgebra of u gl n the non commutative markov operator matches a 2 1 dimensional random surface model introduced in 7 as an application it is then proved that the moments converge to an explicit gaussian field along time like paths combining with 7 which showed convergence to the gaussian free field along space like paths this computes the entire three dimensional gaussian field in particular it matches a gaussian field from eigenvalues of random matrices 5

almost optimal local graph clustering using evolving sets spectral partitioning is a simple nearly linear time algorithm to find sparse cuts and the cheeger inequalities provide a worst case guarantee for the quality of the approximation found by the algorithm a local graph partitioning algorithm finds a set of vertices with small conductance i e a sparse cut by adaptively exploring part of a large graph g starting from a specified vertex for the algorithm to be local its complexity must be bounded in terms of the size of the set that it outputs with at most a weak dependence on the number n of vertices in g previous local partitioning algorithms find sparse cuts using random walks and personalized pagerank spielman and teng 2013 andersen et al 2006 in this article we introduce a simple randomized local partitioning algorithm that finds a sparse cut by simulating the volume biased evolving set process which is a markov chain on sets of vertices we prove that for any 0 and any set of vertices a that has conductance at most for at least half of the starting vertices in a our algorithm will output with constant probability a set of conductance o we prove that for a given run of the algorithm the expected ratio between its computational complexity and the volume of the set that it outputs is vol a 1 2polylog n where vol a v ad v is the volume of the set a this gives an algorithm with the same guarantee up to a constant factor as the cheeger s inequality that runs in time slightly superlinear in the size of the output this is the first sublinear in the size of the input time algorithm with almost the same guarantee as the cheeger s inequality in comparison the best previous local partitioning algorithm by andersen et al 2006 has a worse approximation guarantee of o log n and a larger ratio of 1 polylog n between the complexity and output volume as a by product of our results we prove a bicriteria approximation algorithm for the expansion profile of any graph for 0 0 finds a set s of volume vol s o k1 and expansion s o k as a new technical tool we show that for any set s of vertices of a graph a lazy t step random walk started from a randomly chosen vertex of s will remain entirely inside s with probability at least 1 s 2 t this itself provides a new lower bound to the uniform mixing time of any finite state reversible markov chain

finite size scaling in the p state mean field potts glass exact statistical mechanics for small samples the mean field potts glass with bimodal bond distribution is studied for p 3 and p 6 potts states via exact summation of the partition function of systems of up to n 15 sites averages over the disorder realizations are also done exactly up to n 8 while random bond configurations are generated for the larger systems our enumeration method is described in some detail and the applicability of finite size scaling fss for our very small systems is discussed for p 3 second order transition the data agree reasonably with the fss relations given by parisi et al for the p 2 ising case for p 6 no clear signs of the predicted first order behaviour could be observed this case is even more affected by finite size effects since the necessary antiferromagnetic bias of the bond distribution introduces an additional correction in the free energy for both p the data are compatible with a vanishing ground state entropy

random forests and networks analysis wilson proceedings of the twenty eight annual acm symposium on the theory of computing pp 296 303 1996 in the 1990s described a simple and efficient algorithm based on loop erased random walks to sample uniform spanning trees and more generally weighted trees or forests spanning a given graph this algorithm provides a powerful tool in analyzing structures on networks and along this line of thinking in recent works avena and gaudilliere in a proof of the transfer current theorem in absence of reversibility in stat probab lett 142 17 22 2018 avena and gaudilliere in j theor probab 2017 https doi org 10 1007 s10959 017 0771 3 avena et al in approximate and exact solutions of intertwining equations though random spanning forests 2017 arxiv 1702 05992v1 avena et al in intertwining wavelets or multiresolution analysis on graphs through random forests 2017 arxiv 1707 04616 to appear in acha 2018 we focused on applications of spanning rooted forests on finite graphs the resulting main conclusions are reviewed in this paper by collecting related theorems algorithms heuristics and numerical experiments a first foundational part on determinantal structures and efficient sampling procedures is followed by four main applications 1 a random walk based notion of well distributed points in a graph 2 a framework to describe metastable like dynamics in finite settings by means of markov intertwining dualities 3 coarse graining schemes for networks and associated processes 4 wavelets like pyramidal algorithms for graph signals

siegmund duality with applications to the neutral moran model conditioned on never being absorbed we first consider the classical neutral moran model with two alleles whose fate is either to become extinct or to reach fixation we study an ergodic version of the moran model obtained by conditioning it to never hit the boundaries making use of a doob transform we call it the recurrent moran model we show that the siegmund dual of the recurrent moran process exists and is a substochastic birth and death chain conditioning this process to exit in its natural absorbing state we construct a process with a unique absorbing state which is intertwined to the original recurrent moran process the time needed for the intertwined process to first hit its absorbing state is related to the time needed to reach stationarity for the recurrent moran process using spectral information on the intertwined chain we extract limiting information on this first hitting time that shows that there is no abrupt relaxation to equilibrium for the recurrent moran chain this makes use of the relation between duality and intertwining and strong stationary times other related transition times of the recurrent moran chain are also briefly investigated namely the first return time to the ground state and the expected time needed to move from one end to the other end of the state space

random surface growth with a wall and plancherel measures for o we consider a markov evolution of lozenge tilings of a quarter plane and study its asymptotics at large times one of the boundary rays serves as a reflecting wall we observe frozen and liquid regions prove convergence of the local correlations to translation invariant gibbs measures in the liquid region and obtain new discrete jacobi and symmetric pearcey determinantal point processes near the wall the model can be viewed as the one parameter family of plancherel measures for the infinite dimensional orthogonal group and we use this interpretation to derive the determinantal formula for the correlation functions at any finite time moment

loss of coherence in a system of globally coupled maps we study the formation of symmetric i e equally sized or nearly symmetric clusters in an ensemble of globally coupled identical chaotic maps it is shown that the loss of synchronization for the coherent state and the emergence of subgroups of oscillators with synchronized behavior are two distinct processes and that the type of behavior that arises after the loss of total synchronization depends sensitively on the dynamics of the individual map for our system of globally coupled logistic maps symmetric two cluster formation is found to proceed through a periodic state associated with the stabilization either of an asynchronous period 2 cycle or of an asynchronous period 4 cycle with further reduction of the coupling strength each of the principal clustering states undergoes additional bifurcations leading to cycles of higher periodicity or to quasiperiodic and chaotic dynamics if desynchronization of the coherent chaotic state occurs before the formation of stable clusters becomes possible high dimensional chaotic motion is observed in the intermediate parameter interval

the cutoff phenomenon in finite markov chains abstract natural mixing processes modeled by markov chains often show a sharp cutoff in their convergence to long time behavior this paper presents problems where the cutoff can be proved card shuffling the ehrenfests urn it shows that chains with polynomial growth drunkard s walk do not show cutoffs the best general understanding of such cutoffs high multiplicity of second eigenvalues due to symmetry is explored examples are given where the symmetry is broken but the cutoff phenomenon persists

evolving sets mixing and heat kernel bounds we show that a new probabilistic technique recently introduced by the first author yields the sharpest bounds obtained to date on mixing times of markov chains in terms of isoperimetric properties of the state space also known as conductance bounds or cheeger inequalities we prove that the bounds for mixing time in total variation obtained by lovasz and kannan can be refined to apply to the maximum relative deviation p n x y y 1 of the distribution at time n from the stationary distribution we then extend our results to markov chains on infinite state spaces and to continuous time chains our approach yields a direct link between isoperimetric inequalities and heat kernel bounds previously this link rested on analytic estimates known as nash inequalities

ehrenfest urn revisited playing the game on a realistic fluid model the ehrenfest urn process also known as the dogs and fleas model is realistically simulated by molecular dynamics of the lennard jones fluid the key variable is delta z i e the absolute value of the difference between the number of particles in one half of the simulation box and in the other half this is a pure jump stochastic process induced under coarse graining by the deterministic time evolution of the atomic coordinates we discuss the markov hypothesis by analyzing the statistical properties of the jumps and of the waiting times between jumps in the limit of a vanishing integration time step the distribution of waiting times becomes closer to an exponential and therefore the continuous time jump stochastic process is markovian the random variable delta z behaves as a markov chain and in the gas phase the observed transition probabilities follow the predictions of the ehrenfest theory

on the tail of the overlap probability distribution in the sherrington kirkpatrick model we investigate the large deviation behaviour of the overlap probability density in the sherrington kirkpatrick sk model using the coupled replica scheme and we compare with the results of a large scale numerical simulation in the spin glass phase we show that generically for any model with continuous replica symmetry breaking rsb 1 n log pn q q qea 3 and we compute the first correction to the expansion of in powers of tc t for the sk model we also study the paramagnetic phase where results are obtained in the replica symmetric scheme that do not involve an expansion in powers of q qea or tc t finally we give precise semi analytical estimates of p q 1 the overall agreement between the various points of view is very satisfactory

anisotropic 2 1 d growth and gaussian limits of q whittaker processes we consider a discrete model for anisotropic 2 1 dimensional growth of an interface height function owing to a connection with q whittaker functions this system enjoys many explicit integral formulas by considering certain gaussian stochastic differential equation limits of the model we are able to prove a space time limit of covariances to those of the 2 1 dimensional additive stochastic heat equation or edwards wilkinson equation along characteristic directions in particular the bulk height function converges to the gaussian free field which evolves according to this stochastic pde

extremality of invariant measures and ergodicity of stochastic systems in this paper we first present some conditions on the extremality of invariant measures of a markov process these conditions involve average convergence and invariant functions of the process we then combine these with coupling and duality to study some aspects of convergence to invariant measures of markov processes in particular we suggest a new way of studying complete convergence applications to interacting particle systems are given

quantum operators in classical probability theory ii the concept of duality in interacting particle systems duality has proved to be a powerful tool in the theory of interacting particle systems the approach in this paper is algebraic rather than via harris diagrams a form of duality is found which includes coalescing and annihilating duality as special cases this enables new results for the branching annihilating random walk and the biased annihilating branching process to be derived

on eigenfunctions of markov processes on trees we begin by studying the eigenvectors associated to irreducible finite birth and death processes showing that the i nontrivial eigenvector i admits a succession of i decreasing or increasing stages each of them crossing zero imbedding naturally the finite state space into a continuous segment one can unequivocally define the zeros of i which are interlaced with those of i 1 these kind of results are deduced from a general investigation of minimax multi sets dirichlet eigenproblems which leads to a direct construction of the eigenvectors associated to birth and death processes this approach can be generically extended to eigenvectors of markov processes living on trees this enables to reinterpret the eigenvalues and the eigenvectors in terms of the previous dirichlet eigenproblems and a more general conjecture is presented about related higher order cheeger inequalities finally we carefully study the geometric structure of the eigenspace associated to the spectral gap on trees

nearest neighbor markov dynamics on macdonald processes abstract macdonald processes are certain probability measures on two dimensional arrays of interlacing particles introduced by borodin and corwin in 7 they are defined in terms of nonnegative specializations of the macdonald symmetric functions and depend on two parameters q t 0 1 our main result is a classification of continuous time nearest neighbor markov dynamics on the space of interlacing arrays that act nicely on macdonald processes the classification unites known examples of such dynamics and also yields many new ones when t 0 one dynamics leads to a new integrable interacting particle system on the one dimensional lattice which is a q deformation of the pushtasep long range tasep when q t the macdonald processes become the schur processes of okounkov and reshetikhin 41 in this degeneration we discover new robinson schensted type correspondences between words and pairs of young tableaux that govern some of our dynamics

numerical study of the two replica overlap of the 3d edwards anderson ising spin glass abstract we present results of recent high statistics monte carlo simulations of the edwards anderson ising spin glass model in three dimensions the study is based on a non boltzmann sampling technique the multi self overlap algorithm which is specifically tailored for sampling rare event states we thus concentrate on those properties which are difficult to obtain with standard canonical boltzmann sampling such as the free energy barriers f b q in the probability densities p j q of the parisi overlap parameter q and the behavior of the tails of the disorder averaged density p q p j q av our results for the tails disagree with mean field predictions and support extreme order statistics over many orders of magnitude

patterns of chaos synchronization small networks of chaotic units which are coupled by their time delayed variables are investigated in spite of the time delay the units can synchronize isochronally i e without time shift moreover networks cannot only synchronize completely but can also split into different synchronized sublattices these synchronization patterns are stable attractors of the network dynamics different networks with their associated behaviors and synchronization patterns are presented in particular we investigate sublattice synchronization symmetry breaking spreading chaotic motifs synchronization by restoring symmetry and cooperative pairwise synchronization of a bipartite tree

dynamic exponent of the 3d ising spin glass the nonequilibrium critical dynamics of the 3d ising spin glass following a quench to the critical temperature from t infinity are used to measure both the dynamical exponent z and the static exponent eta the results for z and eta are quite sensitive to the value assumed for tc although the ratio 2 eta z is rather insensitive and close to the value 0 39 obtained by huse 1989 taking tc 1 18 gives z 5 85 or 0 3 and eta 0 29 or 0 07

clustering coding switching hierarchical ordering and control in a network of chaotic elements abstract a network of chaotic elements is investigated with the use of globally coupled maps a simple coding of many attractors with clustering is shown through the coding the attractors are organized so that their change exhibits bifurcation like phenomena a precision dependent tree is constructed which leads to the similarity of our attractor with those of spin glasses hierarchical dynamics is constructed on the tree which leads to the dynamical change of trees and the temporal change of effective degrees of freedom by a simple input on a site we can switch among attractors and tune the strength of chaos a threshold on a cluster size is found beyond which a peculiar posi nega switch occurs possible application to biological information processing is discussed with the emphasis on the fuzzy switch chaotic search and hierarchical code categorization

proof of aldous spectral gap conjecture aldous spectral gap conjecture asserts that on any graph the random walk process and the random transposition or interchange process have the same spectral gap we prove the conjecture using a recursive strategy the approach is a natural extension of the method already used to prove the validity of the conjecture on trees the novelty is an idea based on electric network reduction which reduces the problem to the proof of an explicit inequality for a random transposition operator involving both positive and negative rates the proof of the latter inequality uses suitable coset decompositions of the associated matrices with rows and columns indexed by permutations

four dimensional ising spin glass scaling within the spin glass phase we investigate the nature of the spin glass phase in the four dimensional king spin glass we study he probability dishibutions of overlaps of energy overlaps and ullrmetricity for several sizes we discover the existence of finite size scaling in the fails of the first and second of these this allows us to exmct mame exponents within the spin glass phase we also perform studies on quenched thermalization of large samples our results together with previous work favour the mean field picfure of the spin glass phase

on the concept of attractor this note proposes a definition for the concept of attractor based on the probable asymptotic behavior of orbits the definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor

mixing time for random walk on supercritical dynamical percolation we consider dynamical percolation on the d dimensional discrete torus znd of side length n where each edge refreshes its status at rate n 1 2 to be open with probability p we study random walk on the torus where the walker moves at rate 1 2d along each open edge in earlier work of two of the authors with a stauffer it was shown that in the subcritical case p pc zd the mixing time is n2 1 here the implied constants depend only on d and p we prove a quenched and hence annealed version of this conjecture up to a poly logarithmic factor under the assumption p 1 2 when p 0 we prove a version of this conjecture for an alternative notion of mixing time involving randomised stopping times the latter implies sharp up to poly logarithmic factors upper bounds on exit times of large balls throughout the supercritical regime our proofs are based on percolation results e g the grimmett marstrand theorem and an analysis of the volume biased evolving set process the key point is that typically the evolving set has a substantial intersection with the giant percolation cluster at many times this allows us to use precise isoperimetric properties of the cluster due to g pete to infer rapid growth of the evolving set which in turn yields the upper bound on the mixing time

total variation cutoff in birth and death chains the cutoff phenomenon describes a case where a markov chain exhibits a sharp transition in its convergence to stationarity diaconis proc natl acad sci usa 93 4 1659 1664 1996 surveyed this phenomenon and asked how one could recognize its occurrence in families of finite ergodic markov chains peres american institute of mathematics aim research workshop palo alto http www aimath org wwn mixingtimes 2004 noted that a necessary condition for cutoff in a family of reversible chains is that the product of the mixing time and spectral gap tends to infinity and conjectured that in many settings this condition should also be sufficient diaconis and saloff coste ann appl probab 16 4 2098 2122 2006 verified this conjecture for continuous time birth and death chains started at an endpoint with convergence measured in separation it is natural to ask whether the conjecture holds for these chains in the more widely used total variation distance in this work we confirm the above conjecture for all continuous time or lazy discrete time birth and death chains with convergence measured via total variation distance namely if the product of the mixing time and spectral gap tends to infinity the chains exhibit cutoff at the maximal hitting time of the stationary distribution median with a window of at most the geometric mean between the relaxation time and mixing time in addition we show that for any lazy or continuous time birth and death chain with stationary distribution the separation 1 p t x y y is maximized when x y are the endpoints together with the above results this implies that total variation cutoff is equivalent to separation cutoff in any family of such chains