exact solution of a 1d asymmetric exclusion model using a matrix formulation several recent works have shown that the one dimensional fully asymmetric exclusion model which describes a system of particles hopping in a preferred direction with hard core interactions can be solved exactly in the case of open boundaries here the authors present a new approach based on representing the weights of each configuration in the steady state as a product of noncommuting matrices with this approach the whole solution of the problem is reduced to finding two matrices and two vectors which satisfy very simple algebraic rules they obtain several explicit forms for these non commuting matrices which are in the general case infinite dimensional their approach allows exact expressions to be derived for the current and density profiles finally they discuss briefly two possible generalizations of their results the problem of partially asymmetric exclusion and the case of a mixture of two kinds of particles

invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

mixing and reaction efficiency in closed domains we present a numerical study of mixing and reaction efficiency in closed domains in particular we focus our attention on laminar flows in the case of inert transport the mixing properties of the flows strongly depend on the details of the lagrangian transport we also study the reaction efficiency starting with a little spot of product we compute the time needed to complete the reaction in the container we find that the reaction efficiency is not strictly related to the mixing properties of the flow in particular reaction acts as a dynamical regulator

characterizing the metastable balance between chaos and diffusion abstract we examine some new diagnostics for the behavior of a field evolving in an advective diffusive system one of these diagnostics is approximately the fourier second moment denoted as 2 and the other is the linear entropy or field intensity s the latter being significantly easier to compute or measure we establish that as a result of chaos the increasing structure in is accompanied by increasing exponentially rapidly in time at a rate given by dependent lyapunov exponents i and dominated by the largest one max noise or diffusive coarse graining of causes to decrease as 2 1 4 dt where d is a measure of the diffusion when both effects are present the competition between the processes leads to metastability for followed by a final diffusive tail the initial stages may be chaotic or diffusive depending upon the value of 1max2d 2 0 but the metastable value of 2 is given by 2 i i 2d irrespective since s 2d 2 similar analysis applies to s and in particular there exists a metastable decay rate for s given by s i i these arguments are verified for a simple case the arnol d cat map with added diffusive noise

computing monodromy via continuation methods on random riemann surfaces we consider a riemann surface x defined by a polynomial f x y of degree d whose coefficients are chosen randomly hence we can suppose that x is smooth that the discriminant x of f has d d 1 simple roots and that 0 0 i e the corresponding fiber has d distinct points y1 yd when we lift a loop 0 c by a continuation method we get d paths in x connecting y1 yd hence defining a permutation of that set this is called monodromy here we present experimentations in maple to get statistics on the distribution of transpositions corresponding to loops around each point of multiplying families of neighbor transpositions we construct permutations and the subgroups of the symmetric group they generate this allows us to establish and study experimentally two conjectures on the distribution of these transpositions and on transitivity of the generated subgroups assuming that these two conjectures are true we develop tools allowing fast probabilistic algorithms for absolute multivariate polynomial factorization under the hypothesis that the factors behave like random polynomials whose coefficients follow uniform distributions

logarithmic sobolev inequality for some models of random walks we determine the logarithmic sobolev constant for the bernoulli laplace model and the time to stationarity for the symmetric simple exclusion model up to the leading order our method for proving the logarithmic sobolev inequality is based on a martingale approach and is applied to the random transposition model as well the proof for the time to stationarity is based on a general observation relating the time to stationarity to the hydrodynamical limit

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

cutoff for the ising model on the lattice introduced in 1963 glauber dynamics is one of the most practiced and extensively studied methods for sampling the ising model on lattices it is well known that at high temperatures the time it takes this chain to mix in l 1 on a system of size n is o logn whether in this regime there is cutoff i e a sharp transition in the l 1 convergence to equilibrium is a fundamental open problem if so as conjectured by peres it would imply that mixing occurs abruptly at c o 1 logn for some fixed c 0 thus providing a rigorous stopping rule for this mcmc sampler however obtaining the precise asymptotics of the mixing and proving cutoff can be extremely challenging even for fairly simple markov chains already for the one dimensional ising model showing cutoff is a longstanding open problem

comparison techniques for random walk on finite groups we develop techniques for bounding the rate of convergence of a symmetric random walk on a finite group to the uniform distribution the techniques gives bounds on the second largest and other eigenvalues in terms of the eigenvalues of a comparison chain with known eigenvalues the techniques yield sharp rates for a host of previously intractable problems on the symmetric group

wave packet pseudomodes of twisted toeplitz matrices the pseudospectra of banded nonsymmetric toeplitz or circulant matrices with varying coefficients are considered such matrices are characterized by a symbol that depends on both position x and wave number k it is shown that when a certain winding number or twist condition is satisfied related to hormander s commutator condition for partial differential equations e pseudoeigenvectors of such matrices for exponentially small values of e exist in the form of localized wave packets the symbol need not be smooth with respect to x just differentiable at a point or less c 2004 wiley periodicals inc

random walks on the groups of upper triangular matrices this paper gives sharp bounds on the eigenvalues of a natural random walk on the group of upper triangular n times n matrices over the field of characteristic p an odd prime with 1 s on the diagonal in particular this includes the finite heisenberg groups as a special case as a consequence we get bounds on the time required to achieve randomness for these walks some of the steps are done using the geometric bounds on the eigenvalues of diaconis and stroock however the crucial step is done using more subtle and idiosyncratic techniques we bound the eigenvalues inductively over a sequence of subspaces

derivation of the leroux system as the hydrodynamic limit of a two component lattice gas the long time behavior of a couple of interacting asymmetric exclusion pro cesses of opposite velocities is investigated in one space dimension we do not allow two particles at the same site and a collision effect exchange takes place when particles of opposite velocities meet at neighboring sites there are two conserved quantities and the model admits hyperbolic euler scaling the hydrodynamic limit results in the classical leroux system of conservation laws even beyond the appearance of shocks actually we prove convergence to the set of entropy solutions the question of unique ness is left open to control rapid oscillations of lax entropies via logarithmic sobolev inequality estimates the symmetric part of the process is speeded up in a suitable way thus a slowly vanishing viscosity is obtained at the macroscopic level following 4 5 the stochastic version of tartar murat theory of compensated compactness is extended to two component stochastic models

the l 2 cutoff for reversible markov processes abstract we consider the problem of proving the existence of an l 2 cutoff for families of ergodic markov processes started from given initial distributions and associated with reversible more generally normal markov semigroups this includes classical examples such as families of finite reversible markov chains and brownian motion on compact riemannian manifolds we give conditions that are equivalent to the existence of an l 2 cutoff and describe the l 2 cutoff time in terms of the spectral decomposition this is illustrated by several examples including the ehrenfest process and the biased p q random walk on the non negative integers both started from an arbitrary point

mixing of the upper triangular matrix walk we study a natural random walk over the upper triangular matrices with entries in the field mathbb z 2 generated by steps which add row i 1 to row i we show that the mixing time of the lazy random walk is o n2 which is optimal up to constants our proof makes key use of the linear structure of the group and extends to walks on the upper triangular matrices over the fields mathbb z q for q prime

on strong anomalous diffusion superdiffusive behavior i e x2 t t2 with 1 2 is in general not completely characterized by a unique exponent we study some systems exhibiting strong anomalous diffusion i e x t q tq q where 2 1 2 and q q is not a linear function of q this feature is different from the weak superdiffusive regime i e q const 1 2 occurring in random shear flows strong anomalous diffusion can be generated by nontrivial chaotic dynamics e g lagrangian motion in 2d time dependent incompressible velocity fields 2d symplectic maps and 1d intermittent maps typically the function q q is piecewise linear this is due to two mechanisms a weak anomalous diffusion for the typical events and a ballistic transport for the rare excursions in order to have strong anomalous diffusion one needs a violation of the hypothesis of the central limit theorem this happens only in a very narrow region of the control parameters space in the presence of strong anomalous diffusion one does not have a unique exponent and therefore one has the failure of the usual scaling p x t t f x t of the probability density this implies that the effective equation at large scale and long time for p x t obeys neither the usual fick equation nor other linear equations involving temporal and or spatial fractional derivatives

super character theory and comparison arguments for a random walk on the upper triangular matrices abstract consider the random walk on the n n upper triangular matrices with ones on the diagonal and elements over f p where we pick a row at random and either add it or subtract it from the row directly above it the main result of this paper is to prove that the dependency of the mixing time on p is p 2 this is proven by combining super character theory and comparison theory arguments

quantum invariant measures we derive an explicit expression for the haar integral on the quantized algebra of regular functions q k on the compact real form k of an arbitrary simply connected complex simple algebraic group g this is done in terms of the irreducible representations of the hopf algebra q k quantum analogs of the measures on the symplectic leaves of the standard poisson structure on k which are almost invariant under the dressing action of the dual poisson algebraic group k are also obtained they are related to the notion of quantum traces for representations of hopf algebras as an application we define and compute explicitly quantum analogs of harish chandra c functions associated to the elements of the weyl group of g

harmonicity and invariance on slices of the boolean cube in a recent work with kindler and wimmer we proved an invariance principle for the slice for low influence low degree harmonic multilinear polynomials a polynomial in x 1 ldots x n is harmonic if it is annihilated by sum i 1 n frac partial partial x i here we provide an alternative proof for general low degree harmonic multilinear polynomials with no constraints on the influences we show that any real valued harmonic multilinear polynomial on the slice whose degree is o sqrt n has approximately the same distribution under the slice and cube measures our proof is based on ideas and results from the representation theory of s n along with a novel decomposition of random increasing paths in the cube in terms of martingales and reverse martingales while such decompositions have been used in the past for stationary reversible markov chains our decomposition is applied in a non stationary non reversible setup we also provide simple proofs for some known and some new properties of harmonic functions which are crucial for the proof finally we provide independent simple proofs for the known facts that 1 one cannot distinguish between the slice and the cube based on functions of o n coordinates and 2 boolean symmetric functions on the cube cannot be approximated under the uniform measure by functions whose sum of influences is o sqrt n

cutting and shuffling with diffusion evidence for cut offs in interval exchange maps low dimensional dynamical systems are fruitful models for mixing in fluid and granular flows we study a one dimensional discontinuous dynamical system termed cutting and shuffling of a line segment and we present a comprehensive computational study of its finite time mixing properties including the effect of diffusion to explore a large parameter space we introduce fit functions for two mixing metrics of choice the number of cutting interfaces a standard quantity in dynamical systems theory of interval exchange transformations and a mixing norm a more physical measure of mixing we compute averages of the mixing metrics across different permutations shuffling protocols showing that the latter averages are a robust descriptor of mixing for any permutation choice if the decay of the normalized mixing norm is plotted against the number of map iterations rescaled by the characteristic e folding time then universality emerges mixing norm decay curves across all cutting and shuffling protocols collapse onto a single stretched exponential profile next we predict this critical number of iterations using the average length of unmixed subsegments of continuous color during cutting and shuffling and a batchelor scale type diffusion argument this prediction called a stopping time for finite markov chains compares well with the e folding time of the stretched exponential fit finally we quantify the effect of diffusion on cutting and shuffling through a peclet number a dimensionless inverse diffusivity showing that the system transitions more sharply from an unmixed initial state to a mixed final state as the peclet number becomes large our numerical investigation of cutting and shuffling of a line segment in the presence of diffusion thus presents evidence for the latter phenomenon known as a cut off for finite markov chains in interval exchange maps

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

eigenvalues of the natural random walk on the burnside group b 3 n in this paper we give sharp bounds on the eigenvalues of the natural random walk on the burnside group b 3 n most of the argument uses established geometric techniques for eigenvalue bounds however the most interesting bound the upper bound on the second largest eigenvalue cannot be done by existing techniques to give a bound we use a novel method for bounding the eigenvalues of a random walk on a group g or equivalently its cayley graph this method works by choosing eigenvectors which fall into representations of an abelian normal subgroup of g one is then left with a large number one for each representation of easier problems to analyze

entropy dissipation estimates in a zero range dynamics we study the exponential decay of relative entropy functionals for zero range processes on the complete graph for the standard model with rates increasing at infinity we prove entropy dissipation estimates uniformly over the number of particles and the number of vertices

exact solution for a class of random walk on the hypercube a class of families of markov chains defined on the vertices of the n dimensional hypercube n 0 1 n is studied the single step transition probabilities pn ij with i j n are given by p n ij frac 1 alpha d ij 2 alpha n where 0 1 and dij is the hamming distance between i and j this corresponds to flip independently each component of the vertex with probability frac 1 alpha 2 alpha the m step transition matrix p n ij m is explicitly computed in a close form the class is proved to exhibit cutoff a model independent result about the vanishing of the first m terms of the expansion in of p n ij m is also proved

spectral gap and logarithmic sobolev inequality for kawasaki and glauber dynamics we prove that the spectral gap of the kawasaki dynamics shrink at the rate of 1 l2 for cubes of sizel provided that some mixing conditions are satisfied we also prove that the logarithmic sobolev inequality for the glauber dynamics in standard cubes holds uniformly in the size of the cube if the dobrushin shlosman mixing condition holds for standard cubes

mixing of the symmetric exclusion processes in terms of the corresponding single particle random walk we prove an upper bound for the e mixing time of the symmetric exclusion process on any graph g with any feasible number of particles our estimate is proportional to trw g ln v e where v is the number of vertices in g and trw g is the 1 4 mixing time of the corresponding single particle random walk this bound implies new results for symmetric exclusion on expanders percolation clusters the giant component of the erdos renyi random graph and poisson point processes in rd our technical tools include a variant of morris s chameleon process

regression adjustment in completely randomized experiments with a diverging number of covariates extending r a fisher and d a freedman s results on the analysis of covariance lin 2013 proposed an ordinary least squares adjusted estimator of the average treatment effect in completely randomized experiments we further study its statistical properties under the potential outcomes model in the asymptotic regimes allowing for a diverging number of covariates we show that lin 2013 s estimator is consistent when kappa log p rightarrow 0 and asymptotically normal when kappa p rightarrow 0 under mild moment conditions where kappa is the maximum leverage score of the covariate matrix in the favorable case where leverage scores are all close together his estimator is consistent when p o n log n and is asymptotically normal when p o n 1 2 in addition we propose a bias corrected estimator that is consistent when kappa log p rightarrow 0 and is asymptotically normal with the same variance in the fixed p regime when kappa 2 p log p rightarrow 0 in the favorable case the latter condition reduces to p o n 2 3 log n 1 3 similar to lin 2013 our results hold for non random potential outcomes and covariates without any model specification our analysis requires novel analytic tools for sampling without replacement which complement and potentially enrich the theory in other areas such as survey sampling matrix sketching and transductive learning

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

the logarithmic sobolev constant of the lamplighter we give estimates on the logarithmic sobolev constant of some finite lamplighter graphs in terms of the spectral gap of the underlying base also we give examples of application

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

statistics of advective stretching in three dimensional incompressible flows we present a method to quantify kinematic stretching in incompressible unsteady isoviscous three dimensional flows we extend the method of kellogg and turcotte j geophys res 95 421 432 1990 to compute the axial stretching thinning experienced by infinitesimal ellipsoidal strain markers in arbitrary three dimensional incompressible flows and discuss the differences between our method and the computation of finite time lyapunov exponent ftle we use the cellular flow model developed in solomon and mezic nature 425 376 380 2003 to study the statistics of stretching in a three dimensional unsteady cellular flow we find that the probability density function of the logarithm of normalised cumulative stretching log s for a globally chaotic flow with spatially heterogeneous stretching behavior is not gaussian and that the coefficient of variation of the gaussian distribution does not decrease with time as t frac 1 2 however it is observed that stretching becomes exponential log s t and the probability density function of log s becomes gaussian when the time dependence of the flow and its three dimensionality are increased to make the stretching behaviour of the flow more spatially uniform we term these behaviors weak and strong chaotic mixing respectively we find that for strongly chaotic mixing the coefficient of variation of the gaussian distribution decreases with time as t frac 1 2 this behavior is consistent with a random multiplicative stretching process

i factorial quantum torsors and heisenberg algebras of quantized universal enveloping type abstract we introduce a notion of i factorial quantum torsor which consists of an integrable ergodic action of a locally compact quantum group on a type i factor such that also the crossed product is a type i factor we show that any such i factorial quantum torsor is at the same time a i factorial quantum torsor for the dual locally compact quantum group in such a way that the construction is involutive as a motivating example we show that quantized compact semisimple lie groups when amplified via a crossed product construction with the function algebra on the associated weight lattice admit i factorial quantum torsors and give an explicit realization of the dual quantum torsor in terms of a deformed heisenberg algebra for the borel part of a quantized universal enveloping algebra

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

strange eigenmodes and decay of variance in the mixing of diffusive tracers we prove the existence of asymptotic spatial patterns for diffusive tracers advected by unsteady velocity fields the asymptotic patterns arise from convergence to a time dependent inertial manifold in the underlying advection diffusion equation for time periodic velocity fields we find that the inertial manifold is spanned by a finite number of floquet solutions the strange eigenmodes observed first numerically by pierrehumbert these strange eigenmodes only admit a regular asymptotic expansion in the diffusivity if the velocity field is completely integrable

entropy driven cutoff phenomena in this paper we present in the context of diaconis paradigm a general method to detect the cutoff phenomenon we use this method to prove cutoff in a variety of models some already known and others not yet appeared in literature including a non reversible random walk on a cylindrical lattice all the given examples clearly indicate that a drift towards the opportune quantiles of the stationary measure could be held responsible for this phenomenon in the case of birth and death chains this mechanism is fairly well understood our work is an effort to generalize this picture to more general systems such as systems having stationary measure spread over the whole state space or systems in which the study of the cutoff may not be reduced to a one dimensional problem in those situations the drift may be looked for by means of a suitable partitioning of the state space into classes using a statistical mechanics language it is then possible to set up a kind of energy entropy competition between the weight and the size of the classes under the lens of this partitioning one can focus the mentioned drift and prove cutoff with relative ease

chaos and lyapunov exponents in classical and quantal distribution dynamics we analytically establish the role of a spectrum of lyapunov exponents in the evolution of phase space distributions rho p q of particular interest is lambda 2 an exponent which quantifies the rate at which chaotically evolving distributions acquire structure at increasingly smaller scales and which is generally larger than the maximal lyapunov exponent lambda for trajectories the approach is trajectory independent and is therefore applicable to both classical and quantum mechanics in the latter case we show that the hbar to 0 limit yields the classical fully chaotic result for the quantum cat map

kinetics of ring formation we study reversible polymerization of rings in this stochastic process two monomers bond and as a consequence two disjoint rings may merge into a compound ring or a single ring may split into two fragment rings this aggregation fragmentation process exhibits a percolation transition with a finite ring phase in which all rings have microscopic length and a giant ring phase where macroscopic rings account for a finite fraction of the entire mass interestingly while the total mass of the giant rings is a deterministic quantity their total number and their sizes are stochastic quantities the size distribution of the macroscopic rings is universal although the span of this distribution increases with time moreover the average number of giant rings scales logarithmically with system size we introduce a card shuffling algorithm for efficient simulation of the ring formation process and we present numerical verification of the theoretical predictions

random walk on upper triangular matrices mixes rapidly we present an upper bound o n 2 for the mixing time of a simple random walk on upper triangular matrices we show that this bound is sharp up to a constant and find tight bounds on the eigenvalue gap we conclude by applying our results to indicate that the asymmetric exclusion process on a circle indeed mixes more rapidly than the corresponding symmetric process

eigenvalues of random walks on groups in this paper we discuss and apply a novel method for bounding the eigenvalues of a random walk on a group g or equivalently on its cayley graph this method works by looking at the action of an abelian normal subgroup h of g on g we may then choose eigenvectors which fall into representations of h one is then left with a large number one for each representation of h of easier problems to analyze this analysis is carried out by new geometric methods this method allows us to give bounds on the second largest eigenvalue of random walks on nilpotent groups with low class number the method also lets us treat certain very easy solvable groups and to give better bounds for certain nice nilpotent groups with large class number for example we will give sharp bounds for two natural random walks on groups of upper triangular matrices

random walks that find perfect objects and the lovasz local lemma we give an algorithmic local lemma by establishing a sufficient condition for the uniform random walk on a directed graph to reach a sink quickly our work is inspired by moser s entropic method proof of the lovasz local lemma lll for satisfiability and completely bypasses the probabilistic method formulation of the lll in particular our method works when the underlying state space is entirely unstructured similarly to moser s argument the key point is that the inevitability of reaching a sink is established by bounding the entropy of the walk as a function of time

the emergence of typical entanglement in two party random processes we investigate the entanglement within a system undergoing a random local process we find that there is initially a phase of very fast generation and spread of entanglement at the end of this phase the entanglement is typically maximal in oliveira et al 2007 phys rev lett 98 130502 we proved that the maximal entanglement is reached to a fixed arbitrary accuracy within o n3 steps where n is the total number of qubits here we provide a detailed and more pedagogical proof we demonstrate that one can use the so called stabilizer gates to simulate this process efficiently on a classical computer furthermore we discuss three ways of identifying the transition from the phase of rapid spread of entanglement to the stationary phase i the time when saturation of the maximal entanglement is achieved ii the cutoff moment when the entanglement probability distribution is practically stationary and iii the moment block entanglement exhibits volume scaling we furthermore investigate the mixed state and multipartite setting numerically we find that the mutual information appears to behave similarly to the quantum correlations and that there is a well behaved phase space flow of entanglement properties towards an equilibrium we describe how the emergence of typical entanglement can be used to create a much simpler tripartite entanglement description the results form a bridge between certain abstract results concerning typical also known as generic entanglement relative to an unbiased distribution on pure states and the more physical picture of distributions emerging from random local interactions

upper bound on the characters of the symmetric groups for balanced young diagrams and a generalized frobenius formula we study asymptotics of an irreducible representation of the symmetric group sn corresponding to a balanced young diagram a young diagram with at most rows and columns for some fixed constant c in the limit as n tends to infinity we show that there exists a constant d which depends only on c with a property that where p denotes the length of a permutation the minimal number of factors necessary to write p as a product of transpositions our main tool is an analogue of the frobenius character formula which holds true not only for cycles but for arbitrary permutations

the cutoff profile for the simple exclusion process on the circle in this paper we give a very accurate description of the way the simple exclusion process relaxes to equilibrium let ptpt denote the semi group associated the exclusion on the circle with 2n2n sites and nn particles for any initial condition and for any t 4n29 2lognt 4n29 2log n we show that the probability density pt pt is given by an exponential tilt of the equilibrium measure by the main eigenfunction of the particle system as 4n29 2logn4n29 2log n is smaller than the mixing time which is n22 2lognn22 2log n this allows to give a sharp description of the cutoff profile if dn t dn t denote the total variation distance starting from the worse initial condition we have limn dn n22 2logn n2 2s erf 2 e s limn dn n22 2log n n2 2s erf 2 e s where erferf is the gauss error function

random walks and hyperplane arrangements let be the set of chambers of a real hyperplane arrangement we study a random walk on introduced by bidigare hanlon and rockmore this includes various shuffling schemes used in computer science biology and card games it also includes random walks on zonotopes and zonotopal tilings we find the stationary distributions of these markov chains give good bounds on the rate of convergence to stationarity and prove that the transition matrices are diagonalizable the results are extended to oriented matroids

cutoff for general spin systems with arbitrary boundary conditions the cutoff phenomenon describes a sharp transition in the convergence of a markov chain to equilibrium in recent work the authors established cutoff and its location for the stochastic ising model on the d dimensional torus n d for any d 1 the proof used the symmetric structure of the torus and monotonicity in an essential way here we enhance the framework and extend it to general geometries boundary conditions and external fields to derive a cutoff criterion that involves the growth rate of balls and the log sobolev constant of the glauber dynamics in particular we show there is cutoff for the stochastic ising model on any sequence of bounded degree graphs with subexponential growth under arbitrary external fields provided the inverse log sobolev constant is bounded for lattices with homogenous boundary such as all plus we identify the cutoff location explicitly in terms of spectral gaps of infinite volume dynamics on half plane intersections analogous results establishing cutoff are obtained for nonmonotone spin systems at high temperatures including the gas hard core model the potts model the antiferromagnetic potts model and the coloring model 2014 wiley periodicals inc