central limit theorems for infinite series of queues and applications to simple exclusion we prove that a tagged particle in asymmetric simple exclusion satisfies a central limit theorem when properly rescaled to obtain this result we derive several results of positive or negative correlation for occupation times of a server in a series of queues which imply various central limit theorems

variational formula for the time constant of first passage percolation consider first passage percolation with positive stationary ergodic weights on the square lattice in d dimensions let t x be the first passage time from the origin to x in z d the convergence of t nx n to the time constant as n tends to infinity is a consequence of the subadditive ergodic theorem this convergence can be viewed as a problem of homogenization for a discrete hamilton jacobi bellman hjb equation by borrowing several tools from the continuum theory of stochastic homogenization for hjb equations we derive an exact variational formula duality principle for the timeconstant under a symmetry assumption we will use the variational formula to construct an explicit iteration that produces the limit shape

tropical combinatorics and whittaker functions we establish a fundamental connection between the geometric robinson schensted knuth rsk correspondence and gl n r whittaker functions analogous to the well known relationship between the rsk correspondence and schur functions this gives rise to a natural family of measures associated with gl n r whittaker functions which are the analogues in this setting of the schur measures on integer partitions the corresponding analogue of the cauchy littlewood identity can be seen as a generalization of an integral identity for gl n r whittaker functions due to bump and stade as an application we obtain an explicit integral formula for the laplace transform of the law of the partition function associated with a 1 dimensional directed polymer model with log gamma weights recently introduced by one of the authors

dynamics of a tagged particle in the asymmetric exclusion process with the step initial condition the one dimensional totally asymmetric simple exclusion process tasep is considered we study the time evolution property of a tagged particle in the tasep with the step initial condition calculated is the multi time joint distribution function of its position using the relation of the dynamics of the tasep to the schur process we show that the function is represented as the fredholm determinant we also study the scaling limit the universality of the largest eigenvalue in the random matrix theory is realized in the limit when the hopping rates of all particles are the same it is found that the joint distribution function converges to that of the airy process after the time at which the particle begins to move on the other hand when there are several particles with small hopping rate in front of a tagged particle the limiting process changes at a certain time from the airy process to the process of the largest eigenvalue in the hermitian multi matrix model with external sources

large deviations for increasing sequences on the plane we prove a large deviation principle with explicit rate functions for the length of the longest increasing sequence among poisson points on the plane the rate function for lower tail deviations is derived from a 1977 result of logan and shepp about young diagrams of random permutations for the upper tail we use a coupling with hammersley s particle process and convex analytic techniques along the way we obtain the rate function for the lower tail of a tagged particle in a totally asymmetric hammersley s process

a lower bound for the balanced truncation error for mimo systems we show that for a class of systems which includes state space symmetric systems the balanced truncation error is bounded from below by twice the sum of the tail of the hankel singular values including multiplicities divided by the dimension of the input space

tutte polynomial of scale free networks the tutte polynomial of a graph or equivalently the q state potts model partition function is a two variable polynomial graph invariant of considerable importance in both statistical physics and combinatorics the computation of this invariant for a graph is np hard in general in this paper we focus on two iteratively growing scale free networks which are ubiquitous in real life systems based on their self similar structures we mainly obtain recursive formulas for the tutte polynomials of two scale free networks lattices one is fractal and large world while the other is non fractal but possess the small world property furthermore we give some exact analytical expressions of the tutte polynomial for several special points at x y plane such as the number of spanning trees the number of acyclic orientations etc

model reduction by balanced truncation for systems with nuclear hankel operators we prove the h infinity error bounds for lyapunov balanced truncation and for optimal hankel norm approximation under the assumption that the hankel operator is nuclear this is an improvement of the result from glover curtain and partington siam j control optim 26 1998 pp 863 898 where additional assumptions were made the proof is based on convergence of the schmidt pairs of the hankel operator in a sobolev space we also give an application of this convergence theory to a numerical algorithm for model reduction by balanced truncation

approximation of subadditive functions and convergence rates in limiting shape results for a nonnegative subadditive function h on mathbb z d with limiting approximation g x lim n h nx n it is of interest to obtain bounds on the discrepancy between g x and h x typically of order x nu with nu 1 for certain subadditive h x particularly those which are expectations associated with optimal random paths from 0 to x in a somewhat standardized way a more natural and seemingly weaker property can be established every x is in a bounded multiple of the convex hull of the set of sites satisfying a similar bound we show that this convex hull property implies the desired bound for all x applications include rates of convergence in limiting shape results for first passage percolation standard and oriented and longest common subsequences and bounds on the error in the exponential decay approximation to the off axis connectivity function for subcritical bernoulli bond percolation on the integer lattice

potts model on sierpinski carpets and their relation to hypercubic lattices of non integer dimensionality the authors study the critical frontiers and the high and low temperature asymptotic behavior of the potts model two site correlation function for several sierpinski carpets with fractal dimension lying between one and two they simulate the sierpinski carpets by means of appropriate hierarchical lattices obtaining from their basic cells the two graphs among others that arise from the anisotropic bond moving schemes they correspond to the asymptotic high and low temperature dominant terms for the hierarchical lattice two site correlation function exact result by taking the infinite limit size of the basic cell these asymptotic behaviors tends to those of the sierpinski carpets it is then possible to compare them with similar results obtained from analytical extension to d 2 of the hypercubic lattice their results indicate that these asymptotic behaviors do not always coincide suggesting that possibly they may not identify the sierpinski carpets exhibited by gefen et al 1984 as a correct implementation of the analytical extension of hypercubic lattices

limiting shape for directed percolation models we consider directed first passage and last passage percolation on the nonnegative lattice z d d 2 with i i d weights at the vertices under certain moment conditions on the common distribution of the weights the limits g x lim n n 1 t nx exist and are constant a s for x r d where t z is the passage time from the origin to the vertex z z d we show that this shape function g is continuous on r d in particular at the boundaries in two dimensions we give more precise asymptotics for the behavior of g near the boundaries these asymptotics depend on the common weight distribution only through its mean and variance in addition we discuss growth models which are naturally associated to the percolation processes giving a shape theorem and illustrating various possible types of behavior with output from simulations

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

on hopf s formulas for solutions of hamilton jacobi equations soit h r n r on considere l equation de hamilton jacobi u t h du 0 dans r n 1 r n 0 on etudie les proprietes des formules de hopf pour les solutions

a growth model in a random environment we consider a model of interface growth in two dimensions given by a height function on the sites of the one dimensional integer lattice according to the discrete time update rule the height above the site x increases to the height above x 1 if the latter height is larger otherwise the height above x increases by 1 with probability p x we assume that p x are chosen independently at random with a common distribution f and that the initial state is such that the origin is far above the other sites we explicitly identify the asymptotic shape and prove that in the pure regime the fluctuations about that shape normalized by the square root of time are asymptotically normal this contrasts with the quenched version conditioned on the environment and normalized by the cube root of time the fluctuations almost surely approach a distribution known from random matrix theory

limit theorems for height fluctuations in a class of discrete space and time growth models we introduce a class of one dimensional discrete space discrete time stochastic growth models described by a height function ht x with corner initialization we prove with one exception that the limiting distribution function of ht x suitably centered and normalized equals a fredholm determinant previously encountered in random matrix theory in particular in the universal regime of large x and large t the limiting distribution is the fredholm determinant with airy kernel in the exceptional case called the critical regime the limiting distribution seems not to have previously occurred the proofs use the dual rsk algorithm gessel s theorem the borodin okounkov identity and a novel rigorous saddle point analysis in the fixed x large t regime we find a brownian motion representation this model is equilvalent to the seppalainen johansson model hence some of our results are not new but the proofs are

from hammersley s lines to hammersley s trees we construct a stationary random tree embedded in the upper half plane with prescribed offspring distribution and whose vertices are the atoms of a unit poisson point process this process which we call hammersley s tree process extends the usual hammersley s line process just as hammersley s process is related to the problem of the longest increasing subsequence this model also has a combinatorial interpretation it counts the number of heaps i e increasing trees required to store a random permutation this problem was initially considered by byers et al 2011 and istrate and bonchis 2015 in the case of regular trees we show in particular that the number of heaps grows logarithmically with the size of the permutation

ferromagnetic phase transitions of inhomogeneous systems modelled by square ising models with diamond type bond decorations abstract the two dimensional ising model defined on square lattices with diamond type bond decorations is employed to study the nature of the ferromagnetic phase transitions of inhomogeneous systems the model is studied analytically under the bond renormalization scheme for an n level decorated lattice the long range ordering occurs at the critical temperature given by the fitting function k b t c j n 1 6410 0 6281 exp 0 5857 n and the local ordering inside n level decorated bonds occurs at the temperature given by the fitting function k b t m j n 1 6410 0 8063 exp 0 7144 n the critical amplitude a sin g n of the logarithmic singularity in specific heat characterizes the width of the critical region and it varies with the decoration level n as a sin g n 0 2473 exp 0 3018 n obtained by fitting the numerical results the cross over from a finite decorated system to an infinite decorated system is not a smooth continuation for the case of infinite decorations the critical specific heat becomes a cusp with the height c n 0 639852 the results are compared with those obtained in the cell decorated ising model

critical properties of the s4 model on diamond type hierarchial lattices we study the critical properties of the s 4 model on diamond type hierarchical lattices in the presence of an external magnetic field it is assumed that for this type of inhomogenous fractal lattice the gaussian distribution constant the four spins interaction parameter and the external magnetic field in the s4 model depend on the coordination number of the site on the fractal lattices by combining the real space renormalization group scheme with the cumulative expansion method we obtain the critical points and further calculate critical exponents according to the scaling theory the results show that on diamond type hierarchical lattices with branches m 4 of 2 bonds the critical point of the s4 model is just the gaussian fixed point and therefore critical exponents are in full agreement with those of the gaussian model and that on those with m 4 the wilson fisher fixed point as well as the gaussian fixed point is obtained the wilson fisher fixed point has a decisive influence on the critical behavior of the system and critical exponents are related to the fractal dimensionality we found that the s4 model on the fractal lattices and that on the translation symmetric lattices show similar behaviors in the dependence of the critical properties on the dimensionality c 2005 elsevier b v all rights reserved

asymptotic shapes for stationary first passage percolation this paper deals with first passage percolation where the usual i i d condition is weakened to stationarity and ergodicity the well known asymptotic shape result is known to extend to this case it is easy to give necessary conditions for a compact set b subset mathbb r d to arise as the asymptotic shape for some stationary measure on the passage times our main result says that these conditions are also sufficient

directed last passage percolation and random matrices this paper is a survey of various techniques recent results and applications of directed last passage percolation model and its relation to random matrices the connection can be explained mainly via the universality of famous tracy widom distribution and airy process some unsolved problems are also surveyed

tutte polynomials of two self similar network models the tutte polynomial t g x y of a graph g or equivalently the q state potts model partition function is a two variable polynomial graph invariant of considerable importance in combinatorics and statistical physics graph operations have been extensively applied to model complex networks recently in this paper we study the tutte polynomials of the diamond hierarchical lattices and a class of self similar fractal models which can be constructed through graph operations firstly we find out the behavior of the tutte polynomial under k inflation and k subdivision which are two graph operations secondly we compute and gain the tutte polynomials of this two self similar fractal models by using their structure characteristic moreover as an application of the obtained results some evaluations of their tutte polynomials are derived such as the number of spanning trees and the number of spanning forests

random growth models with polygonal shapes we consider discrete time random perturbations of monotone cellular automata ca in two dimensions under general conditions we prove the existence of half space velocities and then establish the validity of the wulff construction for asymptotic shapes arising from finite initial seeds such a shape converges to the polygonal invariant shape of the corresponding deterministic model as the perturbation decreases in many cases exact stability is observed that is for small perturbations the shapes of the deterministic and random processes agree exactly we give a complete characterization of such cases and show that they are prevalent among threshold growth ca with box neighborhood we also design a nontrivial family of ca in which the shape is exactly computable for all values of its probability parameter

exact limiting shape for a simplified model of first passage percolation on the plane we derive the limiting shape for the following model of first passage bond percolation on the two dimensional integer lattice the percolation is directed in the sense that admissible paths are nondecreasing in both coordinate directions the passage times of horizontal bonds are bernoulli distributed while the passage times of vertical bonds are all equal to a deterministic constant to analyze the percolation model we couple it with a one dimensional interacting particle system this particle process has nonlocal dynamics in the sense that the movement of any given particle can be influenced by far away particles we prove a law of large numbers for a tagged particle in this process and the shape result for the percolation is obtained as a corollary

q plane zeros of the potts partition function on diamond hierarchical graphs we report exact results concerning the zeros of the partition function of the potts model in the complex q plane as a function of a temperature like boltzmann variable v for the m th iterate graphs dm of the diamond hierarchical lattice including the limit m in this limit we denote the continuous accumulation locus of zeros in the q planes at fixed v v0 as bq v0 we apply theorems from complex dynamics to establish the properties of bq v0 for v 1 the zero temperature potts antiferromagnet or equivalently chromatic polynomial we prove that bq 1 crosses the real q axis at i a minimal point q 0 ii a maximal point q 3 iii q 32 27 iv a cubic root that we give with the value q q1 1 638 896 9 and v an infinite number of points smaller than q1 converging to 32 27 from above similar results hold for bq v0 for any 1 0 potts ferromagnet we also provide the computer generated plots of bq v0 at various values of v0 in both the antiferromagnetic and ferromagnetic regimes and compare them to the numerically computed zeros of z d4 q v0

a limit theorem for the position of a tagged particle in a simple exclusion process on montre l extremalite d une mesure invariante pour le processus n a partir d une particule etiquetee et on utilise les proprietes d ergodicite du processus initial pour montrer que la position suit une loi des grands nombres

improved upper bounds on the asymptotic growth velocity of eden clusters we consider the asymptotic shape of clusters in the eden model on a d dimensional hypercubical lattice we discuss two improvements for the well known upper bound to the growth velocity in different directions by that of the independent branching process ibp in the ibp each cell gives rise to a daughter cell at a neighboring site at a constant rate in the first improvement we do not allow such births along the bond connecting the cell to its mother cell in the second we iteratively evolve the system by a growth as ibp for a duration delta followed by culling process in which if any cell produced a descendant within this interval who occupies the same site as the cell itself then the descendant is removed we study the improvement on the upper bound on the velocity for different dimensions d the bounds are asymptotically exact in the large d limit but in d 2 the improvement over the ibp approximation is only a few percent

divergence of shape fluctuations in two dimensions we consider stochastic growth models such as standard first passage percolation on z d where to leading order there is a linearly growing deterministic shape under natural hypotheses we prove that for d 2 the shape fluctuations grow at least logarithmically in all directions although this bound is far from the expected power law behavior with exponent 1 3 it does prove divergence with additional hypotheses we obtain inequalities involving and the related exponent which is expected to equal 2 3 for d 2 combining these inequalities with previously known results we obtain for standard first passage percolation the bounds 1 8 for d 2 and 3 4 for all d

first passage percolation the stationary case if the passage time of the edges of the d lattice are stationary ergodic and have finite moment of orderp d then a s the set of vertices that can be reached within timet has an asymptotic shape ast

criticality of the potts ferromagnet in migdal kadanoff like hierarchical lattices within the real space renormalisation group framework the authors discuss the critical point and exponent nu of the potts ferromagnet in b sized migdal kadanoff like hierarchical lattices both b to infinity and b to 1 limits are exhibited the important discrepancies that might exist between the exact results for d dimensional hierarchical lattices and d dimensional bravais lattices are illustrated

existence of hydrodynamics for the totally asymmetric simple k exclusion process in a totally asymmetric simple k exclusion process particles take nearest neighbor steps to the right on the lattice z under the constraint that each site contain at most k particles we prove that such processes satisfy hydrodynamic limits under euler scaling and that the limit of the empirical particle profile is the entropy solution of a scalar conservation law with a concave flux function our technique requires no knowledge of the invariant measures of the process which is essential because the equilibria of asymmetric k exclusion are unknown but we cannot calculate the flux function precisely the proof proceeds via a coupling with a growth model on the two dimensional lattice in addition to the basic k exclusion with constant exponential jump rates we treat the site disordered case where each site has its own jump rate randomly chosen but frozen for all time the hydrodynamic limit under site disorder is new even for the simple exclusion process the case k 1 our proof makes no use of the markov property so at the end of the paper we indicate how to treat the case with arbitrary waiting times

thermodynamical approach to the longest common subsequence problem we introduce an interacting particle model in a random media and show that this particle process is equivalent to the longest common subsequence lcs problem of two binary sequences we derive a differential equation which links the mean lcs curve to the average speed of the particles given their density and prove that the average speed of the particles and density converges uniformly on every scale which is somewhat larger than sqrt n

euclidean models of first passage percolation we introduce a new family of first passage percolation fpp models in the context of poisson voronoi tesselations of d compared to standard fpp on d these models have some technical complications but also have the advantage of statistical isotropy we prove two almost sure results a shape theorem where isotropy implies an exact euclidean ball for the asymptotic shape and nonexistence of certain doubly infinite geodesics where isotropy yields a stronger result than in standard fpp

order of the variance in the discrete hammersley process with boundaries we discuss the order of the variance on a lattice analogue of the hammersley process with boundaries for which the environment on each site has independent bernoulli distributed values the last passage time is the maximum number of bernoulli points that can be collected on a piecewise linear path where each segment has strictly positive but finite slope we show that along characteristic directions the order of the variance of the last passage time is of order n 2 3 in the model with boundary these characteristic directions are restricted in a cone starting at the origin and along any direction outside the cone the order of the variance changes to o n in the boundary model and to o 1 for the non boundary model this behaviour is the result of the two flat edges of the shape function

the random average process and random walk in a space time random environment in one dimension we study space time fluctuations around a characteristic line for a one dimensional interacting system known as the random average process the state of this system is a real valued function on the integers new values of the function are created by averaging previous values with random weights the fluctuations analyzed occur on the scale n1 4 where n is the ratio of macroscopic and microscopic scales in the system the limits of the fluctuations are described by a family of gaussian processes in cases of known product form invariant distributions this limit is a two parameter process whose time marginals are fractional brownian motions with hurst parameter 1 4 along the way we study the limits of quenched mean processes for a random walk in a space time random environment these limits also happen at scale n1 4 and are described by certain gaussian processes that we identify in particular when we look at a backward quenched mean process the limit process is the solution of a stochastic heat equation

critical behaviour of the bond diluted potts model on sierpinski carpets a four parameter bond moving renormalisation is used to study the critical properties of the bond diluted q state potts model on sierpinski carpets fixed points critical exponents and a phase diagram are obtained in the authors results there is a borderline value q qt for q qt there is crossover to a new diluted fixed point this behaviour is similar to that of the diluted system on a regular lattice with translational invariance they give some values of qt for carpets with different b l where b and l are structure parameters of carpets