violation of the fluctuation dissipation theorem in glassy systems basic notions and the numerical evidence this review reports on the research done during past years on violations of the fluctuation dissipation theorem fdt in glassy systems it is focused on the existence of a quasi fluctuation dissipation theorem qfdt in glassy systems and the current supporting knowledge gained from numerical simulation studies it covers a broad range of non stationary aging and stationary driven systems such as structural glasses spin glasses coarsening systems ferromagnetic models at criticality trap models models with entropy barriers kinetically constrained models sheared systems and granular media the review is divided into four main parts 1 an introductory section explaining basic notions related to the existence of the fdt in equilibrium and its possible extension to the glassy regime qfdt 2 a description of the basic analytical tools and results derived in the framework of some exactly solvable models 3 a detailed report of the current evidence in favour of the qfdt and 4 a brief digression on the experimental evidence in its favour this review is intended for inexpert readers who want to learn about the basic notions and concepts related to the existence of the qfdt as well as for the more expert readers who may be interested in more specific results

facilitated oriented spin models some non equilibrium results we perform the rigorous analysis of the relaxation to equilibrium for some facilitated or kinetically constrained spin models kcsm when the initial distribution is different from the reversible one this setting has been intensively studied in the physics literature to analyze the slow dynamics which follows a sudden quench from the liquid to the glass phase we concentrate on two basic oriented kcsm the east model on for which the constraint requires that the east neighbor of the to be update vertex is vacant and the ad model on the binary tree introduced in aldous and diaconis j stat phys 107 5 6 945 975 2002 for which the constraint requires the two children to be vacant it is important to observe that while the former model is ergodic at any p 1 the latter displays an ergodicity breaking transition at p c 1 2 for the east we prove exponential convergence to equilibrium with rate depending on the spectral gap if is concentrated on any configuration which does not contain a forever blocked site or if is a bernoulli p product measure for any p 1 for the model on the binary tree we prove similar results in the regime p p p c and p p c convergence to equilibrium cannot occur for all local functions finally in a short appendix we present a very simple argument different from the one given in aldous and diaconis j stat phys 107 5 6 945 975 2002 based on a combination of some combinatorial results together with energy barrier considerations which yields the sharp upper bound for the spectral gap of east when p 1

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

integration of thiele continued fractions and the method of fundamental solutions for solving unconfined seepage problems unconfined seepage through an earth dam or a levee is recognized as a challenging problem this complexity is mainly due to the fact that determination of the phreatic line through the dam levee body is not straightforward for simulation purposes mesh generation as well as accurate and smooth alteration of the phreatic line at the junction with the downstream slope referred to as exit point for an unconfined seepage problem with complex geometry generally makes cumbersome numerical solutions this study presents an innovative boundary type meshfree method to determine the phreatic line location in unconfined seepage problems the current study explicitly addresses the problem that alternative methods commonly face to deal with the exit point the method is developed based upon integrating the method of fundamental solutions mfs particle swarm optimization pso algorithm and thiele continued fractions tcf to accurately estimate the phreatic line location the proposed framework uses mfs to solve the flow continuity equation tcf to generate the phreatic line and pso to optimize the phreatic line location generated by tcf as a boundary method mfs only deals with the boundaries of the domain and consequently it only takes the exact position of phreatic line as a variable boundary the proposed approach employs tcf to guarantee that the phreatic line is tangent to the downstream slope at the exit point a characteristics which is important especially for the cases where abrupt changes occur in the phreatic line near the exit point for comparison and validation purposes the phreatic lines determined by the proposed approach for two unconfined seepage problems are compared and verified against those obtained from alternative analytical and numerical methods as well as a set of experimental results an excellent agreement is demonstrated upon comparison of the proposed method to the results attained from the analytical solutions and experimental tests

isogeometric analysis in solution of unconfined seepage problems abstract this article investigates the numerical solution of saturated unconfined seepage problems in which a fluid with a free surface passes through a porous media in such cases the geometry of the wet region is unknown in advance and must be obtained through an iterative process such problems are known as variable domain problems since the use of traditional mesh based numerical methods is not straight forward due to remeshing difficulties the isogeometric analysis is proposed here which eliminates the meshing process and accordingly makes this method an attractive tool to deal with such problems on the other hand the use of isogeometric analysis accounts for other difficulties which this paper aims to address in the isogeometric analysis the boundary control points are not located on the domain boundary therefore in this article a new boundary updating formula is proposed to reconstruct the free surface for modifying the control net the transfinite interpolation is used in order to update the control net when the geometry undergoes any change the domain is considered inhomogeneous with spatially varied permeability three benchmark examples are solved to demonstrate the applicability of the proposed method finally the results are compared with those available in the literature

culling avalanches in bootstrap percolation we study the culling avalanches which occur after the death of a single randomly chosen site in a network where sites are unstable and are culled if they have coordination less than an integer parameter m avalanche distributions are presented for triangular and cubic lattices for values of m where the associated bootstrap transitions are either first or second order in second order cases the culling avalanche distribution is found to be exponential while in first order cases it follows a power law we present an exact relation between culling avalanches and conventional bootstrap percolation and show that a relation proposed by manna physica a 261 351 1998 can be a good approximation for strongly first order bootstrap transitions but not for continuous bootstrap transitions

a sharper threshold for bootstrap percolation in two dimensions two dimensional bootstrap percolation is a cellular automaton in which sites become infected by contact with two or more already infected nearest neighbours we consider these dynamics which can be interpreted as a monotone version of the ising model on an n n square with sites initially infected independently with probability p the critical probability pc is the smallest p for which the probability that the entire square is eventually infected exceeds 1 2 holroyd determined the sharp first order approximation pc 2 18 log n as n here we sharpen this result proving that the second term in the expansion is log n 3 2 o 1 and moreover determining it up to a poly log log n factor the exponent 3 2 corrects numerical predictions from the physics literature

analysis of inequality and residual flow procedures and an iterative scheme for free surface seepage an efficient block overrelaxation algorithm is presented for the solution of the steady state dam seepage problem the formulation of alt is used in conjunction with a finite element method on a fixed mesh to obtain a finite dimensional problem which is then solved by the algorithm some similarities of the formulation and the residual flow method of desai are also discussed numerical results are compared with results of other authors

k core percolation in four dimensions the k core percolation on the bethe lattice has been proposed as a simple model of the jamming transition because of its hybrid first order second order nature we investigate numerically k core percolation on the four dimensional regular lattice for k 4 the presence of a discontinuous transition is clearly established but its nature is strictly first order in particular the k core density displays no singular behavior before the jump and its correlation length remains finite for k 3 the transition is continuous

induced representations of crossed products by coactions abstract let a m a c r g be a coaction of a locally compact group g on a c algebra a then for any closed normal amenable subgroup h of g we define a coaction a m a c r g h of g h on a we present dense subalgebras of the crossed products a g and a g h and use these to induce representations of a g h to representations of a g we then formulate an imprimitivity theorem for the induction process that is we classify the induced representations we also investigate the continuity of the induction process and are able to prove rieffel s conjecture that the dual action to a coaction is proper and saturated

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

critical points of two dimensional bootstrap percolation like cellular automata cellular automata in two dimensions that generalize the bootstrap percolation dynamics are considered focusing on the thresholdp c of the initial density for convergence to total occupancy to occur these models are classified according top c being 0 1 or strictly between these extreme values explicit upper and lower bounds are provided in the third case

first passage times for threshold growth dynamics on bf z sp 2 in the threshold growth model on an integer lattice the occupied set grows according to a simple local rule a site becomes occupied iff it sees at least a threshold number of already occupied sites in its prescribed neighborhood in this paper we analyze the behavior of two dimensional threshold growth dynamics started from a sparse bernoulli density of occupied sites we explain how nucleation of rare centers invariant shapes and interaction between growing droplets influence the first passage time in the supercritical case we also briefly address scaling laws for the critical case

irreversible conversion of graphs given a graph g a function f v g z and an initial 0 1 vertex labelling c 1 v g 0 1 we study an iterative 0 1 vertex labelling process on g where in each round every vertex v never changes its label from 1 to 0 and changes its label from 0 to 1 if at least f v neighbours have label 1 such processes model opinion disease spreading or fault propagation and have been studied under names such as irreversible threshold majority processes in a large variety of contexts our contributions concern computational aspects related to the minimum cardinality irr f g of sets of vertices with initial label 1 such that during the process on g all vertices eventually change their label to 1 such sets are known as irreversible conversion sets dynamic irreversible monopolies or catastrophic fault patterns answering a question posed by dreyer and roberts p a dreyer jr f s roberts irreversible k threshold processes graph theoretical threshold models of the spread of disease and of opinion discrete appl math 157 2009 1615 1627 we prove a hardness result for irr f g where f v 2 for every email protected v g furthermore we describe a general reduction principle for irr f g which leads to efficient algorithms for graphs with simply structured blocks such as trees and chordal graphs

heterogeneous k core versus bootstrap percolation on complex networks we introduce the heterogeneous k core which generalizes the k core and contrast it with bootstrap percolation vertices have a threshold r i that may be different at each vertex if a vertex has fewer than r i neighbors it is pruned from the network the heterogeneous k core is the subgraph remaining after no further vertices can be pruned if the thresholds r i are 1 with probability f or k ensuremath geqslant 3 with probability 1 ensuremath f the process can be thought of as a pruning process counterpart to ordinary bootstrap percolation which is an activation process we show that there are two types of transitions in this heterogeneous k core process the giant heterogeneous k core may appear with a continuous transition and there may be a second discontinuous hybrid transition we compare critical phenomena critical clusters and avalanches at the heterogeneous k core and bootstrap percolation transitions we also show that the network structure has a crucial effect on these processes with the giant heterogeneous k core appearing immediately at a finite value for any fg0 when the degree distribution tends to a power law p q q ensuremath ensuremath gamma with ensuremath gamma l3

a sharp threshold for a modified bootstrap percolation with recovery bootstrap percolation is a type of cellular automaton on graphs introduced as a simple model of the dynamics of ferromagnetism vertices in a graph can be in one of two states healthy or infected and from an initial configuration of states healthy vertices become infected by local rules while the usual bootstrap processes are monotone in the sets of infected vertices in this paper a modification is examined in which infected vertices can return to a healthy state vertices are initially infected independently at random and the central question is whether all vertices eventually become infected the model examined here is such a process on a square grid for which healthy vertices with at least two infected neighbours become infected and infected vertices with no infected neighbours become healthy sharp thresholds are given for the critical probability of initial infections for all vertices eventually to become infected

a d dimensional nucleation and growth model we analyze the relaxation time of a ferromagnetic d dimensional growth model on the lattice the model is characterized by d parameters which represent the activation energies of a site depending on the number of occupied nearest neighbours this model is a natural generalisation of the model studied by dehghanpour and schonmann probab theory relat fields 107 1 123 135 1997 where the activation energy of a site with more than two occupied neighbours is zero

spin dynamics confinement in q2r for low concentrations the occupied sites in q2r cellular automata on the square lattice are known to be confined to the smallest rectangles surrounding them however they do not necessarily fill out the whole rectangle allowed for them the effective confinement transition is found numerically to agree well with that of bootstrap percolation

dynamical blume capel model competing metastable states at infinite volume this paper concerns the microscopic dynamical description of competing metastable states we study at infinite volume and very low temperature metastability and nucleation for kinetic blume capel model a ferromagnetic lattice model with spins taking three possible values 1 0 1 in a previous paper mo we considered a simplified irreversible nucleation growth model in the present paper we analyze the full blume capel model we choose a region u of the thermodynamic parameters such that everywhere in u b 1 all minuses corresponds to the highest in energy metastable state b 0 all zeroes corresponds to an intermediate metastable state and b 1 all pluses corresponds to the stable state we start from b 1 and look at a local observable like in mo we find that when crossing a special line in u there is a change in the mechanism of transition towards the stable state b 1 we pass from a situation 1 where the intermediate phase b 0 is really observable before the final transition with a permanence in b 0 typically much longer than the first hitting time to b 0 to the situation 2 where b 0 is not observable since the typical permanence in b 0 is much shorter than the first hitting time to b 0 and moreover large growing 0 droplets are almost full of 1 in their interior so that there are only relatively thin layers of zeroes between 1 and 1

bootstrap percolation in a polluted environment let a low densityp of sites on the lattice z2 be occupied remove a proportionq of them and call the remaining sites empty then update this configuration in discrete time by iteration of the following synchronous rule an empty site becomes occupied by contact with at least two occupied nearest neighbors while occupied and removed sites nerver change their states ifq p2 is large most sites remain unoccupied forever while ifq p2 is small this dynamics eventually makes most sites occupied this demonstrates how sensitive the usual bootstrap percolation rule theq 0 case is to the pollution of space

metastable behavior for bootstrap percolation on regular trees we examine bootstrap percolation on a regular b 1 ary tree with initial law given by bernoulli p the sites are updated according to the usual rule a vacant site becomes occupied if it has at least occupied neighbors occupied sites remain occupied forever it is known that when b 2 the limiting density q q p of occupied sites exhibits a jump at some p t p t b 0 1 from q t q p t p t we investigate the metastable behavior associated with this transition explicitly we pick p p t h with h 0 and show that as h 0 the system lingers around the critical state for time order h 1 2 and then passes to fully occupied state in time o 1 the law of the entire configuration observed when the occupation density is q q t 1 converges as h 0 to a well defined measure

sharp thresholds in bootstrap percolation in the standard bootstrap percolation on the d dimensional grid gnd in the initial position each of the nd sites is occupied with probability p and empty with probability 1 p independently of the state of every other site once a site is occupied it remains occupied for ever while an empty site becomes occupied if at least two of its neighbours are occupied if at the end of the process every site is occupied we say that the initial configuration percolates by making use of a theorem of friedgut and kalai proc amer math soc 124 1996 2993 we shall show that the threshold function of the percolation is sharp we shall prove similar results for three other models of bootstrap percolation as well

metastability in the dilute ising model consider glauber dynamics for the ising model on the hypercubic lattice with a positive magnetic field starting from the minus configuration the system initially settles into a metastable state with negative magnetization slowly the system relaxes to a stable state with positive magnetization schonmann and shlosman showed that in the two dimensional case the relaxation time is a simple function of the energy required to create a critical wulff droplet the dilute ising model is obtained from the regular ising model by deleting a fraction of the edges of the underlying graph in this paper we show that even an arbitrarily small dilution can dramatically reduce the relaxation time this is because of a catalytic effect rare regions of high dilution speed up the transition from minus phase to plus phase

bootstrap automata the dynamics of destruction abstract simulations for bootstrap cellular automata on an oriented square lattice are shown to be consistent with exact results on the critical threshold p c 1 and a finite size scaling bound regimes above below and close to p c for finite samples are detailed and characterized by a wide choice of updating and coding algorithms

rates for the probability of large cubes being non internally spanned in modified bootstrap percolation we find the exact rate of decay for the probability that a large cube is not internally spanned for the modified bootstrap percolation it is proven that for cubes of large side the event that the cube is not internally spanned is essentially the same as the event that the cube possesses a completely vacant line

simulation of three dimensional bootstrap percolation on a simple cubic lattie which is initially occupied randomly with concentration p a site is emptied if it does not have at least four of its six neighbors occupied the threshold concentration below which all sites are emptied after sufficiently many iterations is found to vary logarithmically with l for l l l lattices and l between 32 and 704 it is extrapolated to about 0 936 for infinite lattices two other cases are roughly compatible with the theoretical prediction of unity as a threshold concentration

integrals partitions and cellular automata we prove that 1 0 log f x x dx 2 3ab where f x is the decreasing function that satisfies f a f b x a x b for 0 a b when a is an integer and b a 1 we deduce several combinatorial results these include an asymptotic formula for the number of integer partitions not having a consecutive parts and a formula for the metastability thresholds of a class of threshold growth cellular automaton models related to bootstrap percolation

partially inner c dynamical systems a c dynamical system a g is called h inner if there is a strictly continuous unitary representation v from the closed subgroup h of g into the invariant multipliers of a such that t ad vt for every t in h assuming that g is a locally compact abelian group we show that in this case the crossed product g a is isomorphic to indh g g ha here g h a is the restricted or twisted crossed product studied by n dang ngoc and p green which naturally carries a dual action of the subgroup h of g and indh g refers to the induced c algebra obtained from this action

on the study of jamming percolation we investigate kinetically constrained models of glassy transitions and determine which model characteristics are crucial in allowing a rigorous proof that such models have discontinuous transitions with faster than power law diverging length and time scales the models we investigate have constraints similar to that of the knights model introduced by toninelli biroli and fisher tbf but differing neighbor relations we find that such knights like models otherwise known as models of jamming percolation need a no parallel crossing rule for the tbf proof of a glassy transition to be valid furthermore most knights like models fail a no perpendicular crossing requirement and thus need modification to be made rigorous we also show how the no parallel crossing requirement can be used to evaluate the provable glassiness of other correlated percolation models by looking at models with more stable directions than the knights model finally we show that the tbf proof does not generalize in any straightforward fashion for three dimensional versions of the knights like models

the time of bootstrap percolation in two dimensions we study the distribution of the percolation time t of 2 neighbour bootstrap percolation on n 2 with initial set a sim text bin n 2 p we determine t up to a constant factor with high probability for all p above the critical probability for percolation and to within a 1 o 1 factor for a large range of p

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

force balance percolation we study models of correlated percolation where there are constraints on the occupation of sites that mimic force balance i e for a site to be stable requires occupied neighboring sites in all four compass directions in two dimensions we prove rigorously that p c 1 for the two dimensional models studied numerical data indicate that the force balance percolation transition is discontinuous with a growing crossover length with perhaps the same form as the jamming percolation models suggesting that all models belong to the same universality class with the same underlying mechanism driving the transition in both cases we find a lower bound for the correlation length in the connected phase and that the correlation function does not appear to be a power law at the transition finally we study the dynamics of the culling procedure invoked to obtain the force balance configurations and find a dynamical exponent similar to that found in sandpile models

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

induction in stages for crossed products of c algebras by maximal coactions let be a maximal coaction of a locally compact group g on a c algebra b and let n and h be closed normal subgroups of g with n h we show that the process indg hg which uses mansfield s bimodule to induce representations of b g from those of b g h is equivalent to the two stage induction process indg ng indg hg n the proof involves a calculus of symmetric imprimitivity bimodules which relates the bimodule tensor product to the fibred product of the underlying spaces

inelastic analysis of 2d solids using a weak form rpim based on deformation theory abstract a radial point interpolation method rpim is presented for inelastic analysis of 2d problems in this method the problem domain is represented by a set of scattered nodes and the field variable is interpolated using the field values of local supporting nodes radial basis function rbf augmented with polynomials is used to construct the shape functions these shape functions possess delta function property that makes the numerical procedure very efficient and many techniques used in the finite element method can be adopted easily galerkin weak form formulation is applied to derive the discrete governing equations and integration is performed using gauss quadrature and stabilized nodal integration the pseudo elastic method proposed by h jahed et al is employed for the determination of stress field and hencky s total deformation theory is used to define the effective material parameters treated as field variables these parameters are functions of the final state of stress fields that can be obtained in an iterative manner based on pseudo linear elastic analysis from one dimensional uniaxial material curve a very long thick walled cylinder and a v notched tension specimen are analyzed and their stress distributions match well with those obtained by finite element commercial software ansys or the available literature

bootstrap percolation on homogeneous trees has 2 phase transitions we study the threshold bootstrap percolation model on the homogeneous tree with degree b 1 2 b and initial density p it is known that there exists a nontrivial critical value for p which we call pf such that a for p pf the final bootstrapped configuration is fully occupied for almost every initial configuration and b if p pc then for almost every initial configuration there are infinite clusters of occupied vertices in the final bootstrapped configuration moreover we show that 3 for p pc the distribution of the occupied cluster size in the final bootstrapped configuration has an exponential tail 4 at p pc the expected occupied cluster size in the final bootstrapped configuration is infinite 5 the probability of percolation of occupied vertices in the final bootstrapped configuration is continuous on 0 pf and analytic on pc pf admitting an analytic continuation from the right at pc and only in the case b also from the left at pf

on the behavior of some cellular automata related to bootstrap percolation convergence to total occupancy and irc the threshold for this convergence to occur exponentially fast we locate these critical points for all the bootstrap percolation models showing that they are both 0 when 1 d for certain rules in which the orientation is important we show that 0 pc ir 1 by relating these systems to oriented site percolation finally these oriented models are used to obtain an estimate for a critical exponent of these models 1 introduction the fields of interacting particle systems mathematical statistical mechanics and percolation have benefited very much from their interrelations here we study a family of models which has arisen in the interface among these areas cellular automata such as those studied in this article may be considered as interacting particle systems see liggett 1985 for a survey of this field the relations between the models that we consider and percolation will become clear in many of the proofs given but can already be guessed from the fact that some of these systems are known as bootstrap percolation finally relations with statistical mechanics while not so explicit in this article were clearly present for instance in the article by chalupa leath and reich 1979 where bootstrap percolation was introduced in connection to disordered magnetic systems also in aizenman and lebowitz 1988 the motivation for studying these systems came from the nonequilibrium statistical mechanics problem of metastability our main concern in this article will be with the critical behavior of our models that is how their behavior changes qualitatively as some parameters cross certain values critical points as usual one of the main tools in the analysis of such phenomena will be a sort of renormalization procedure by

fundamental solutions method for elliptic boundary value problems we consider a procedure for solving boundary value problems for elliptic homogeneous equations known as the fundamental solutions method we prove its applicability for some second order operators as well as for fourth order ones the boundary conditions of an elliptic problem are approximated by using fundamental solutions of the corresponding operator with singularities located outside the domain of interest in a specific case of the laplacian an estimate shows that the method discussed possesses convergence properties as good as those of any method using harmonic polynomials as trial functions numerical examples are given among which are simple signorini s problems for harmonic and lame operators

improved bounds on metastability thresholds and probabilities for generalized bootstrap percolation we generalize and improve results of andrews gravner holroyd liggett and romik on metastability thresholds for generalized two dimensional bootstrap percolation models and answer several of their open problems and conjectures specifically we prove slow convergence and localization bounds for holroyd liggett and romik s k percolation models and in the process provide a unified and improved treatment of existing results for bootstrap modified bootstrap and frobose percolation furthermore we prove improved asymptotic bounds for the generating functions of partitions without k gaps which are also related to certain infinite probability processes relevant to these percolation models one of our key technical probability results is also of independent interest we prove new upper and lower bounds for the probability that a sequence of independent events with monotonically increasing probabilities contains no k gap patterns which interpolates the general markov chain solution that arises in the case that all of the probabilities are equal

minimal contagious sets in random regular graphs the bootstrap percolation or threshold model is a dynamic process modelling the propagation of an epidemic on a graph where inactive vertices become active if their number of active neighbours reach some threshold we study an optimization problem related to it namely the determination of the minimal number of active sites in an initial configuration that leads to the activation of the whole graph under this dynamics with and without a constraint on the time needed for the complete activation this problem encompasses in special cases many extremal characteristics of graphs like their independence decycling or domination number and can also be seen as a packing problem of repulsive particles we use the cavity method including the effects of replica symmetry breaking an heuristic technique of statistical mechanics many predictions of which have been confirmed rigorously in the recent years we have obtained in this way several quantitative conjectures on the size of minimal contagious sets in large random regular graphs the most striking being that 5 regular random graph with a threshold of activation of 3 resp 6 regular with threshold 4 have contagious sets containing a fraction 1 6 resp 1 4 of the total number of vertices equivalently these numbers are the minimal fraction of vertices that have to be removed from a 5 regular resp 6 regular random graph to destroy its 3 core we also investigated survey propagation like algorithmic procedures for solving this optimization problem on single instances of random regular graphs

the sharp threshold for bootstrap percolation in all dimensions in r neighbour bootstrap percolation on a graph g a typically random set a of initially infected vertices spreads by infecting at each time step vertices with at least r already infected neighbours this process may be viewed as a monotone version of the glauber dynamics of the ising model and has been extensively studied on the d dimensional grid n d the elements of the set a are usually chosen independently with some density p and the main question is to determine p c n d r the density at which percolation infection of the entire vertex set becomes likely in this paper we prove for every pair d ge r ge 2 that there is a constant l d r such that p c n d r l d r o 1 log r 1 n d r 1 as n o infty where log r denotes an r times iterated logarithm we thus prove the existence of a sharp threshold for percolation in any fixed number of dimensions moreover we determine l d r for every pair d r

proof of straley s argument for bootstrap percolation we prove thatpc 1 for bootstrap percolation with large void instabilities in particular ifm 3 on the square lattice

narrow band oscillations in probabilistic cellular automata dynamical properties of neural populations are studied using probabilistic cellular automata previous work demonstrated the emergence of critical behavior as the function of system noise and density of long range axonal connections finite size scaling theory identified critical properties which were consistent with properties of a weak ising universality class the present work extends the studies to neural populations with excitatory and inhibitory interactions it is shown that the populations can exhibit narrow band oscillations when confined to a range of inhibition levels with clear boundaries marking the parameter region of prominent oscillations phase diagrams have been constructed to characterize unimodal bimodal and quadromodal oscillatory states the significance of these findings is discussed in the context of large scale narrow band oscillations in neural tissues as observed in electroencephalographic and magnetoencephalographic measurements

higher order corrections for anisotropic bootstrap percolation we study the critical probability for the metastable phase transition of the two dimensional anisotropic bootstrap percolation model with 1 2 neighbourhood and threshold r 3 the first order asymptotics for the critical probability were recently determined by the first and second authors here we determine the following sharp second and third order asymptotics begin aligned p c big l 2 mathcal n scriptscriptstyle 1 2 3 big frac log log l 2 12 log l frac log log l log log log l 3 log l frac left log frac 9 2 1 pm o 1 right log log l 6 log l end aligned we note that the second and third order terms are so large that the first order asymptotics fail to approximate p c even for lattices of size well beyond 10 10 1000

one dimensional caricature of phase transition in the limit as the volume grows and the temperature vanishes it is shown that the one dimensional nearest neighbor ferromagnetic ising model presents a sharp transition between two different regimes fluctuations are studied in one of these regimes and also in the critical case

correlated percolation and tricriticality the recent proliferation of correlated percolation models models where the addition of edges and or vertices is no longer independent of other edges and or vertices has been motivated by the quest to find discontinuous percolation transitions the leader in this proliferation is what is known as explosive percolation a recent proof demonstrates that a large class of explosive percolation type models does not in fact exhibit a discontinuous transition riordan and warnke science 333 322 2011 here we discuss two lesser known correlated percolation models the k 3 core model on random graphs and the counter balance model in two dimensions both exhibiting discontinuous transitions to search for tricriticality we construct mixtures of these models with other percolation models exhibiting the more typical continuous transition using a powerful rate equation approach we demonstrate that a mixture of k 2 core and k 3 core vertices on the random graph exhibits a tricritical point however for a mixture of k core and counter balance vertices in two dimensions as the fraction of counter balance vertices is increased numerics and heuristic arguments suggest that there is a line of continuous transitions with the line ending at a discontinuous transition i e when all vertices are counter balanced interestingly these heuristic arguments may help identify the ingredients needed for a discontinuous transition in low dimensions in addition our results may have potential implications for glassy and jamming systems

metastable states for the becker doring cluster equations the becker doring equations in whichc l t can represent the concentration ofl particle clusters or droplets in say a condensing vapour at timet are begin array 20 c dc l t mathord left vphantom dc l t dt right kern nulldelimiterspace dt j l 1 t j l t l 2 3 end array with j l t a l c 1 t c l t b l 1 c l 1 t and eitherc1 const case a or rho sum limits 1 infty lc l const case b the equilibrium solutions arec l q l z l where q l prod limits 2 l a r 1 mathord left vphantom a r 1 b r right kern nulldelimiterspace b r the density of the saturated vapour defined as rho s sum limits 1 infty lq l z s l wherez s is the radius of convergence of the series is assumed finite it is proved here that subject to some further plausible conditions on the kinetic coefficientsa l andb l there is a class of metastable solutions of the equations withc1 z s small and positive which take an exponentially long time to decay to their asymptotic steady states an exponentially long time means one that increases more rapidly than any negative power of the given value ofc1 z s or in caseb s as the latter tends to zero the main ingredients in the proof are i a time independent upper bound on the solution of the kinetic equations this upper bound is a steady state solution of case a of the equations of the type used in the becker doring theory of nucleation and ii an upper bound on the total concentration of particles in clusters greater than a certain critical size which with suitable initial conditions remains exponentially small until the time becomes exponentially large

polluted bootstrap percolation with threshold two in all dimensions in the polluted bootstrap percolation model the vertices of a graph are independently declared initially occupied with probability p or closed with probability q at subsequent steps a vertex becomes occupied if it is not closed and it has at least r occupied neighbors on the cubic lattice mathbb z d of dimension d ge 3 with threshold r 2 we prove that the final density of occupied sites converges to 1 as p and q both approach 0 regardless of their relative scaling our result partially resolves a conjecture of morris and contrasts with the d 2 case where gravner and mcdonald proved that the critical parameter is q p 2

finite size scaling behavior of a biased majority rule cellular automaton abstract we consider the biased majority rule cellular automaton in which the sites take the majority of the states 0 or 1 of their neighbors to update their states with a bias towards 1 in case of ties we observe that in any dimension for any initial density of 1 s the probability of final complete occupancy by 1 s converges to 1 as the lattice grows in two dimensions the value p l of the initial density for which the probability of eventual complete occupancy of a lattice l l is 0 1 is shown to satisfy for fixed the bounds a log l b log l for large l where a and b are positive and finite constants

the transport exponent in percolation models with additional loops abstract several percolation models with additional loops were studied the transport exponents for these models were estimated numerically by means of a transfer matrix approach it was found that the transport exponent has a drastically changed value for some of the models this result supports some previous numerical studies on the vibrational properties of similar models with additional loops

sharp metastability threshold for two dimensional bootstrap percolation in the bootstrap percolation model sites in an l by l square are initially independently declared active with probability p at each time step an inactive site becomes active if at least two of its four neighbours are active we study the behaviour as p 0 and l simultaneously of the probability i l p that the entire square is eventually active we prove that i l p 1 if and i l p 0 if where 2 18 we prove the same behaviour with the same threshold for the probability j l p that a site is active by time l in the process on the infinite lattice the same results hold for the so called modified bootstrap percolation model but with threshold 2 6 the existence of the thresholds settles a conjecture of aizenman and lebowitz 3 while the determination of their values corrects numerical predictions of adler stauffer and aharony 2

exact solution of a jamming transition closed equations for a bootstrap percolation problem jamming or dynamical arrest is a transition at which many particles stop moving in a collective manner in nature it is brought about by for example increasing the packing density changing the interactions between particles or otherwise restricting the local motion of the elements of the system the onset of collectivity occurs because when one particle is blocked it may lead to the blocking of a neighbor that particle may then block one of its neighbors these effects propagating across some typical domain of size named the dynamical correlation length when this length diverges the system becomes immobile even where it is finite but large the dynamics is dramatically slowed such phenomena lead to glasses gels and other very long lived nonequilibrium solids the bootstrap percolation models are the simplest examples describing these spatio temporal correlations we have been able to solve one such model in two dimensions exactly exhibiting the precise evolution of the jamming correlations on approach to arrest we believe that the nature of these correlations and the method we devise to solve the problem are quite general both should be of considerable help in further developing this field

nucleation and growth for the ising model in d dimensions at very low temperatures this work extends to dimension d 3 the main result of dehghanpour and schonmann we consider the stochastic ising model on zd evolving with the metropolis dynamics under a fixed small positive magnetic field h starting from the minus phase when the inverse temperature goes to the relaxation time of the system defined as the time when the plus phase has invaded the origin behaves like exp d the value d is equal to d 1d 1 1 d where i is the energy of the i dimensional critical droplet of the ising model at zero temperature and magnetic field h

the time of bootstrap percolation with dense initial sets let r n in r neighbour bootstrap percolation on the vertex set of a graph g vertices are initially infected independently with some probability p at each time step the infected set expands by infecting all uninfected vertices that have at least r infected neighbours when p is close to 1 we study the distribution of the time at which all vertices become infected given t t n o logn loglogn we prove a sharp threshold result for the probability that percolation occurs by time t in d neighbour bootstrap percolation on the d dimensional discrete torus t d n moreover we show that for certain ranges of p p n the time at which percolation occurs is concentrated either on a single value or on two consecutive values we also prove corresponding results for the modified d neighbour rule

finite size scaling in three dimensional bootstrap percolation we consider the problem of bootstrap percolation on a three dimensional lattice and we study its finite size scaling behavior bootstrap percolation is an example of cellular automata defined on the d dimensional lattice 1 2 l d in which each site can be empty or occupied by a single particle in the starting configuration each site is occupied with probability p occupied sites remain occupied forever while empty sites are occupied by a particle if at least among their 2d nearest neighbor sites are occupied when d is fixed the most interesting case is the one d this is a sort of threshold in the sense that the critical probability p c for the dynamics on the infinite lattice z d switches from zero to one when this limit is crossed finite size effects in the three dimensional case are already known in the cases 2 in this paper we discuss the case 3 and we show that the finite size scaling function for this problem is of the form f l const in in l we prove a conjecture proposed by a c d van enter

test of analytical approximations for kinetic ising models with sharp blocking transition abstract mode coupling approximations and an effective medium approximation are applied to two kinetic ising models exhibiting a sharp blocking transition to a non ergodic phase the models are variants of the two spin facilitated kinetic ising model with a directed rather than isotropic kinetic constraint it is shown that the effective medium approximation qualitatively describes the blocking transition while the mode coupling approximations fail completely

kinetically constrained spin models we analyze the density and size dependence of the relaxation time for kinetically constrained spin models kcsm intensively studied in the physics literature as simple models sharing some of the features of the glass transition kcsm are interacting particle systems on mathbb z d with glauber like dynamics reversible w r t a simple product i i d bernoulli p measure the essential feature of a kcsm is that the creation destruction of a particle at a given site can occur only if the current configuration around it satisfies certain constraints which completely define each specific model no other interaction is present in the model from the mathematical point of view the basic issues concerning positivity of the spectral gap inside the ergodicity region and its scaling with the particle density p remained open for most kcsm with the notably exception of the east model in d 1 aldous and diaconis in j stat phys 107 5 6 945 975 2002 here for the first time we i identify the ergodicity region by establishing a connection with an associated bootstrap percolation model ii develop a novel multi scale approach which proves positivity of the spectral gap in the whole ergodic region iii establish sometimes optimal bounds on the behavior of the spectral gap near the boundary of the ergodicity region and iv establish pure exponential decay at equilibrium for the persistence function i e the probability that the occupation variable at the origin does not change before time t our techniques are flexible enough to allow a variety of constraints and our findings disprove certain conjectures which appeared in the physical literature on the basis of numerical simulations

zero temperature glauber dynamics on mathbb z d we study zero temperature glauber dynamics on mathbb z d which is a dynamic version of the ising model of ferromagnetism spins are initially chosen according to a bernoulli distribution with density p and then the states are continuously and randomly updated according to the majority rule this corresponds to the sudden quenching of a ferromagnetic system at high temperature with an external field to one at zero temperature with no external field define p c mathbb z d to be the infimum over p such that the system fixates at with probability 1 it is a folklore conjecture that p c mathbb z d 1 2 for every 2 le d in mathbb n we prove that p c mathbb z d to 1 2 as d

recursive dynamics in an asymmetrically constrained kinetic ising chain abstract for a hierarchically constrained kinetic ising chain it is shown that the process of spin up propagation for small concentration c of facilitating up spins occurs recursively near the bottom of the configuration space which has a self similar structure for chains with a finite number n of spins the spin autocorrelation function is calculated analytically for n 2 and 4 and numerically for n 4 and 8 from the conjectured behaviour for arbitrary n it is concluded that the mean spin relaxation time on the infinite chain for c 0 diverges more strongly than any positive power of 1 c

the second term for two neighbour bootstrap percolation in two dimensions in the r neighbour bootstrap process on a graph g vertices are infected in each time step if they have at least r already infected neighbours motivated by its close connections to models from statistical physics such as the ising model of ferromagnetism and kinetically constrained spin models of the liquid glass transition the most extensively studied case is the two neighbour bootstrap process on the two dimensional grid n 2 around 15 years ago in a major breakthrough holroyd determined the sharp threshold for percolation in this model and his bounds were subsequently sharpened further by gravner and holroyd and by gravner holroyd and morris in this paper we strengthen the lower bound of gravner holroyd and morris by proving that the critical probability p c big n 2 2 big for percolation in the two neighbour model on n 2 satisfies p c big n 2 2 big frac pi 2 18 log n frac theta 1 log n 3 2 the proof of this result requires a very precise understanding of the typical growth of a critical droplet and involves a number of technical innovations we expect these to have other applications for example to the study of more general two dimensional cellular automata and to the r neighbour process in higher dimensions

bootstrap percolation on the hypercube in the bootstrap percolation on the n dimensional hypercube in the initial position each of the 2n sites is occupied with probability p and empty with probability 1 p independently of the state of the other sites every occupied site remains occupied for ever while an empty site becomes occupied if at least two of its neighbours are occupied if at the end of the process every site is occupied we say that the initial position spans the hypercube we shall show that there are constants c1 c2 0 such that for open image in new window the probability of spanning tends to 1 as n while for open image in new window the probability tends to 0 furthermore we shall show that for each n the transition has a sharp threshold function

finite size effects for some bootstrap percolation models the consequences of schonmann s new proof that the critical threshold is unity for certain bootstrap percolation models are explored it is shown that this proof provides an upper bound for the finite size scaling in these systems comparison with data for one case demonstrates that this scaling appears to give the correct asymptotics we show that the threshold for a finite system of sizel scales aso ln lnl for the isotropic model in three dimensions where sites that fail to have at least four neighbors are culled

sharp metastability threshold for an anisotropic bootstrap percolation model bootstrap percolation models have been extensively studied during the two past decades in this article we study the following anisotropic boot strap percolation model the neighborhood of a point m n is the set m 2 n m 1 n m n 1 m 1 n m 2 n m n 1 at time 0 sites are occupied with probability p at each time step sites that are occupied remain occupied while sites that are not occupied become occupied if and only if three of more sites in their neighborhood are occupied we prove that it exhibits a sharp metastability threshold this is the first mathematical proof of a sharp threshold for an anisotropic bootstrap percolation model

universality for critical kcm finite number of stable directions kinetically constrained models kcm are reversible interacting particle systems on mathbb z d with continuous time constrained glauber dynamics they are a natural non monotone stochastic version of the family of cellular automata with random initial state known as mathscr u bootstrap percolation kcm have an interest in their own right owing to their use for modelling the liquid glass transition in condensed matter physics in two dimensions there are three classes of models with qualitatively different scaling of the infection time of the origin as the density of infected sites vanishes here we study in full generality the class termed critical together with the companion paper by hartarsky et al universality for critical kcm finite number of stable directions arxiv e prints arxiv 1910 06782 2019 we establish the universality classes of critical kcm and determine within each class the critical exponent of the infection time as well as of the spectral gap in this work we prove that for critical models with an infinite number of stable directions this exponent is twice the one of their bootstrap percolation counterpart this is due to the occurrence of energy barriers which determine the dominant behaviour for these kcm but which do not matter for the monotone bootstrap dynamics our result confirms the conjecture of martinelli et al commun math phys 369 2 761 809 https doi org 10 1007 s00220 018 3280 z 2019 who proved a matching upper bound

induced c algebras and landstad duality for twisted coactions suppose n is a closed normal subgroup of a locally compact group g a coaction e a m a c n of n on a c algebra a can be inflated to a coaction s of g on a and the crossed product a x g is then isomorphic to the induced c algebra ind a xe n we prove this and a natural generalization in which a xf n is replaced by a twisted crossed product a x g n g in case g is abelian we recover a theorem of olesen and pedersen we then use this to extend the landstad duality of the first author to twisted crossed products and give several applications in particular we prove that if l n d g n 1 is topologically trivial but not necessarily split as a group extension then every twisted crossed product a x g n g is isomorphic to a crossed product of the form a x n

blocking transitions in lattice spin models with directed kinetic constraints abstract we present a class of kinetic ising models with a directed constraint which exhibit a dynamocal phase transition at a critical up spin concentration at lower up spin concentrations a non ergodic state exists for which the fraction of permanently blocked spins plays the role of an order parameter the spin autocorrelation function has been calculated by monte carlo simulation for two particular models one for the square lattice with the constraint that the neighboring spins in the north and in the east are up and one for a cayley tree with branching ratio 2 where both neighbors above a given spin must be up the transition is of second order and the integrated autocorrelation time diverges on both sides of the transition the models define a distribution of blocking lengths which can be calculated analytically for the cayley tree we conjecture that this length distribution is related to the autocorrelation time by a dynamic critical exponent which is numerically estimated for small lattices up to a length 11 to be about 5 and 2 7 for the two models respectively

duality of restriction and induction for c coactions consider a coaction of a locally compact group g on a c algebra a and a closed normal subgroup n of g we prove following results of echterhoff for abelian g that mansfield s imprimitivity between a g n and a g r n implements equivalences between mansfield induction of representations from a g n to a g and restriction of representations froma g rn to a g and between restriction of representations from a g to a g n and green induction of representations from a g to a g r n this allows us to deduce properties of mansfield induction from the known theory of ordinary crossed products

on some growth models with a small parameter we consider the behavior of the asymptotic speed of growth and the asymptotic shape in some growth models when a certain parameter becomes small the basic example treated is the variant of richardson s growth model on d in which each site which is not yet occupied becomes occupied at rate 1 if it has at least two occupied neighbors at rate 1 if it has exactly 1 occupied neighbor and of course at rate 0 if it has no occupied neighbor occupied sites remain occupied forever starting from a single occupied site this model has asymptotic speeds of growth in each direction as time goes to infinity and these speeds determine an asymptotic shape in the usual sense it is proven that as tends to 0 the asymptotic speeds scale as 1 d and the asymptotic shape when renormalized by dividing it by 1 d converges to a cube other similar models which are partially oriented are also studied

a nucleation and growth model we consider the following simple nucleation and growth model on the lattice d starting with all sites unoccupied a site becomes occupied at rate e if it has no occupied neighbors at rate e if it has 1 occupied neighbor and at rate 1 if it has 2 or more occupied neighbors occupied sites remain occupied forever the parameters are fixed and we are interested in the behavior of the system as we show that the relaxation time of this system scales as e c where c max d 1