frozen percolation in two dimensions aldous math proc camb philos soc 128 465 477 2000 introduced a modification of the bond percolation process on the binary tree where clusters stop growing freeze as soon as they become infinite we investigate the site version of this process on the triangular lattice where clusters freeze as soon as they reach l infty diameter at least n for some parameter n we show informally speaking that in the limit n rightarrow infty the clusters only freeze in the critical window of site percolation on the triangular lattice hence the fraction of vertices that eventually i e at time 1 are in a frozen cluster tends to 0 as n goes to infinity we also show that the diameter of the open cluster at time 1 of a given vertex is with high probability smaller than n but of order n this shows that the process on the triangular lattice has a behaviour quite different from aldous process we also indicate which modifications have to be made to adapt the proofs to the case of the n parameter frozen bond percolation process on the square lattice this extends our results to the square lattice and answers the questions posed by van den berg et al random struct algorithms 40 220 226 2012

graphical representations and cluster algorithms i discrete spin systems abstract graphical representations similar to the fk representation are developed for a variety of spin systems in several cases it is established that these representations have fkg monotonicity properties which enables characterization theorems for the uniqueness phase and the low temperature phase of the spin system certain systems with intermediate phases and or first order transitions are also described in terms of the percolation properties of the representations in all cases these representations lead in a natural fashion to swendsen wang type algorithms hence at least in the above mentioned instances these algorithms realize the program described by kandel and domany phys rev b 43 1991 8539 8548 all of the algorithms are shown to satisfy a li sokal bound which at least for systems with a divergent specific heat implies critical slowing down however the representations also give rise to invaded cluster algorithms which may allow for the rapid simulation of some of these systems at their transition points

the covariance matrix of the potts model a random cluster analysis we consider the covariance matrix g mm q2 of thed dimensionalq states potts model rewriting it in the random cluster representation of fortuin and kasteleyn in any of theq ordered phases we identify the eigenvalues of this matrix both in terms of representations of the unbroken symmetry group of the model and in terms of random cluster connectivities and covariances thereby attributing algebraic significance to these stochastic geometric quantities we also show that the correlation length corresponding to the decay rate of one of the eigenvalues is the same as the inverse decay rate of the diameter of finite clusers for dimensiond 2 we show that this correlation length and the correlation length of the two point function with free boundary conditions at the corresponding dual temperature are equal up to a factor of two for systems with first order transitions this relation helps to resolve certain inconsistencies between recent exact and numerical work on correlation lengths at the self dual point o for systems with second order transitions this relation implies the equality of the correlation length exponents from above and below threshold as well as an amplitude ratio of two in the course of proving the above results we establish several properties of independent interest including left continuity of the inverse correlation length with free boundary conditions and upper semicontinuity of the decay rate for finite clusters in all dimensions and left continuity of the two dimensional free boundary condition percolation probability at o we also introduce dlr equations for the random cluster model and use them to establish ergodicity of the free measure in order to prove these results we introduce a new class of events which we call decoupling events and two inequalities for these events the first is similar to the fkg inequality but holds for events which are neither increasing nor decreasing the second is similar to the van den berg kesten inequality in standard percolation both inequalities hold for an arbitrary fkg measure

cluster expansion for abstract polymer models a new direct proof of convergence of cluster expansions for polymer contour models is given in an abstract setting it does not rely on kirkwood salsburg type equations or combinatorics of trees a distinctive feature is that at all steps the considered clusters contain every polymer at most once

on the gibbs states of the noncritical potts model on z 2 we prove that all gibbs states of the q state nearest neighbor potts model on mathbb z 2 below the critical temperature are convex combinations of the q pure phases in particular they are all translation invariant to achieve this goal we consider such models in large finite boxes with arbitrary boundary condition and prove that the center of the box lies deeply inside a pure phase with high probability our estimate of the finite volume error term is of essentially optimal order which stems from the brownian scaling of fluctuating interfaces the results hold at any supercritical value of the inverse temperature beta beta c q log left 1 sqrt q right

percolation in half spaces equality of critical densities and continuity of the percolation probability renormalization arguments are developed and applied to independent nearest neighbor percolation on various subsets of d d 2 yielding equality of the critical densities p c for a half space quarter space etc and ford 2 equality with the limit of slab critical densities continuity of the phase transition for the half space quarter space etc i e vanishing of the percolation probability p atp p c

lens spaces and toroidal dehn fillings we show that if m is a hyperbolic 3 manifold with m a torus such that m r 1 is a lens space and m r 2 is toroidal then r 1 r 2 4

a structure theorem for boolean functions with small total influences we show that on every product probability space boolean functions with small total inuences are essentially the ones that are almost measurable with respect to certain natural sub sigma algebras this theorem in particular describes the structure of monotone set properties that do not exhibit sharp thresholds our result generalizes the core of friedgut s seminal work on properties of random graphs to the setting of arbitrary boolean functions on general product probability spaces and improves the result of bourgain in his appendix to friedgut s paper

mean field critical behaviour for percolation in high dimensions the triangle condition for percolation states that sum limits x y tau 0 x tau 0 y cdot tau y 0 is finite at the critical point where x y is the probability that the sitesx andy are connected we use an expansion related to the lace expansion for a self avoiding walk to prove that the triangle condition is satisfied in two situations i for nearest neighbour independent bond percolation on thed dimensional hypercubic lattice ifd is sufficiently large and ii in more than six dimensions for a class of spread out models of independent bond percolation which are believed to be in the same universality class as the nearest neighbour model the class of models in ii includes the case where the bond occupation probability is constant for bonds of length less than some large number and is zero otherwise in the course of the proof an infrared bound is obtained the triangle condition is known to imply that various critical exponents take their mean field bethe lattice values gamma beta 1 delta delta t 2 t underset raise0 3em hbox smash scriptscriptstyle geqslant 2 and that the percolation density is continuous at the critical point we also prove thatv2 in i and ii wherev2 is the critical exponent for the correlation length

the large n limit of the threshold values in mandelbrot s fractal percolation process the authors consider mandelbrot s fractal percolation process characterised by a density parameter p in 0 1 and an integer subdivision index n 1 for each n the process is known to have a percolation transition at density pc n in 0 1 they prove that lim pc n pc where pc is the threshold value of the ordinary square lattice site percolation model

slab percolation for the ising model for the fk representation of the ising model we prove that the slab percolation threshold coincides with the critical temperature in any dimension d 3

crabgrass measles and gypsy moths an introduction to modern probability on etudie cinq modeles qui permettent d expliquer des processus lies aux systemes de particules en interaction ainsi que d autres processus

a lower bound for the critical probability of the square lattice site percolation we prove by elementary combinatorial considerations that the critical probability of the square lattice site percolation is larger than 0 503478

renormalisation group approach to delaunay percolation networks with topological disorder a renormalisation group approach is developed for delaunay percolating systems in two and three dimensions using a scaling transformation for a finite lattice in real space considering various renormalisation transformations for two and three dimensional delaunay lattices the authors determine the behaviour of the probabilities under a scale transformation and calculate the fixed point and connectedness length exponent the fixed points for the two dimensional bond lattice and the three dimensional site lattice are 0 3229 and 0 1443 respectively which are in excellent agreement with results of monte carlo simulations the fixed point for the two dimensional site lattice gives the value 1 2 for the critical percolation probability which is equal to the known result for a fully triangulated lattice

an upper bound on the critical percolation probability for the three dimensional cubic lattice we prove that the critical probability for site percolation on the three dimensional cubic lattice satisfies the inequality p 3 c 1 2 an application to the three dimensional ising model is given

strict inequalities for some critical exponents in two dimensional percolation for 2d percolation we slightly improve a result of chayes and chayes to the effect that the critical exponent for the percolation probability isstrictly less than 1 the same argument is applied to prove that ifl x y x r cos y r sin for some r 0 or and limp p c log p p c 1 log pcr ito is connected to by an occupied path inl then is strictly decreasing in on 0 2 similarly limn logn 1 logp cr ito is connected by an occupied path inl to the exterior of n n n n is strictly decreasing in on 0 2

uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation for independent translation invariant irreducible percolation models it is proved that the infinite cluster when it exists must be unique the proof is based on the convexity or almost convexity and differentiability of the mean number of clusters per site which is the percolation analogue of the free energy the analysis applies to both site and bond models in arbitrary dimension including long range bond percolation in particular uniqueness is valid at the critical point of one dimensional 1 x y 2 models in spite of the discontinuity of the percolation density there corollaries of uniqueness and its proof are continuity of the connectivity functions and except possibly at the critical point of the percolation density related to differentiability of the free energy are inequalities which bound the specific heat critical exponent in terms of the mean cluster size exponent and the critical cluster size distribution exponent e g 1 2 1 1

optimal distributed control of nonlocal steady diffusion problems a control problem constrained by a nonlocal steady diffusion equation that arises in several applications is studied the control is the right hand side forcing function and the objective of control is a standard matching functional a recently developed nonlocal vector calculus is exploited to define a weak formulation of the state system when sufficient conditions on certain kernel functions and the volume constraints hold the existence and uniqueness of the optimal state and control is demonstrated and an optimality system is derived we demonstrate the convergence as the nonlocal interactions vanish of the optimal nonlocal state to the optimal state of a local pde constrained control problem we also define continuous and discontinuous galerkin finite element discretizations of the optimality system for which we derive a priori error estimates numerical examples are provided illustrating these convergence results and also illustrating the differences between optimal controls and states obtained f

broadcasting on random directed acyclic graphs we study the following generalization of the well known model of broadcasting on trees consider an infinite directed acyclic graph dag with a unique source vertex x let the collection of vertices at distance k from x be called the k th layer and suppose every non source vertex has indegree d geq 2 at layer 0 the source vertex is given a random bit at layer k geq 1 each vertex receives d bits from its parents in the k 1 th layer which are transmitted along edges that are independent binary symmetric channels bscs with crossover probability delta in left 0 frac 1 2 right each vertex combines its d noisy inputs using a deterministic d ary boolean processing function that generates the value at the vertex the goal is to be able to reconstruct the original bit x with probability of error bounded away from frac 1 2 using the values of all vertices at an arbitrarily deep layer k this question is closely related to models of reliable computation and storage and information flow in biological networks in this paper we treat the case of randomly constructed dags for which we show that broadcasting is only possible if the bsc noise level delta is below a certain degree and function dependent critical threshold for d geq 3 and random dags with layers of size omega log k and majority processing functions we identify the critical threshold for d 2 we establish a similar result for the nand processing function we also prove a partial converse result for odd d geq 3 illustrating that the identified thresholds are impossible to improve by selecting different processing functions if the decoder is restricted to using a single vertex s value finally for any bsc noise level delta we construct explicit dags using regular bipartite lossless expander graphs with bounded degree and layers of size theta log k admitting reconstruction in particular we show that the first r layers of such dags can be generated in either deterministic quasi polynomial time or randomized polylogarithmic time in r these results portray a doubly exponential advantage for storing a bit in bounded degree dags compared to trees where d 1 but layer sizes need to grow exponentially with depth in order for broadcasting to be possible

quenched survival of bernoulli percolation on galton watson trees we explore the survival function for percolation on galton watson trees letting g t p represent the probability a tree t survives bernoulli percolation with parameter p we establish several results about the behavior of the random function g mathbf t cdot where mathbf t is drawn from the galton watson distribution these include almost sure smoothness in the supercritical region an expression for the k text th order taylor expansion of g mathbf t cdot at criticality in terms of limits of martingales defined from mathbf t this requires a moment condition depending on k and a proof that the k text th order derivative extends continuously to the critical value each of these results is shown to hold for almost every galton watson tree

on the critical percolation probabilities we prove that the critical probabilities of site percolation on the square lattice satisfy the relation p c p 1 furthermore we prove the continuity of the function percolation probability

infinite ab percolation clusters exist on the triangular lattice infinite ab percolation clusters are shown to exist almost surely on the triangular lattice for an interval of parameter values it was not previously known that infinite ab percolation occurred with positive probability on any planar lattice

sos approximants for potts crystal shapes abstract an approximation made when replacing the orientation dependent potts model surface tension by the sos one is rigorously estimated as a result an sos model may be used for an explicit and precise construction of crystal and meniscus shapes for potts models at low temperatures

critical behavior and the limit distribution for long range oriented percolation i we consider oriented percolation on mathbb z d times mathbb z whose bond occupation probability is pd where p is the percolation parameter and d is a probability distribution on mathbb z d suppose that d x decays as x d for some 0 we prove that the two point function obeys an infrared bound which implies that various critical exponents take on their respective mean field values above the upper critical dimension d c 2 alpha wedge2 we also show that for every k the fourier transform of the normalized two point function at time n with a proper spatial scaling has a convergent subsequence to e c k alpha wedge2 for some c 0

a sharp transition for the two dimensional ising percolation we show that the percolation transition for the two dimensional ising model is sharp namely we show that for every reciprocal temperature 0 there exists a critical valueh c of external magnetic fieldh such that the following two statements hold we also shows that the percolation probability is continuous in h except on the half line 0 c

rigorous analysis of discontinuous phase transitions via mean field bounds we consider a variety of nearest neighbor spin models defined on the d dimensional hypercubic lattice z d our essential assumption is that these models satisfy the condition of reflection positivity we prove that whenever the associated mean field theory predicts a discontinuous transition the actual model also undergoes a discontinuous transition which occurs near the mean field transition temperature provided the dimension is sufficiently large or the first order transition in the mean field model is sufficiently strong as an application of our general theory we show that for d sufficiently large the 3 state potts ferromagnet on z d undergoes a first order phase transition as the temperature varies similar results are established for all q state potts models with q 3 the r component cubic models with r 4 and the o n nematic liquid crystal models with n 3

first order phase transitions in lattice and continuous systems extension of pirogov sinai theory we generalize the notion of ground states in the pirogov sinai theory of first order phase transitions at low temperatures applicable to lattice systems with a finite number of periodic ground states to that of restricted ensembles with equal free energies a restricted ensemble is a gibbs ensemble i e equilibrium probability measure on a restricted set of configurations in the phase space of the system when a restricted ensemble contains only one configuration it coincides with a ground state in the more general case the entropy is also important

mixing times of critical two dimensional potts models we study dynamical aspects of the q state potts model on an n n box at its critical c q heat bath glauber dynamics and cluster dynamics such as swendsen wang that circumvent low temperature bottlenecks are all expected to undergo critical slowdowns in the presence of periodic boundary conditions the inverse spectral gap which in the subcritical regime is o 1 should at criticality be polynomial in n for 1 4 in accordance with the predicted discontinuous phase transition this was confirmed for q 2 the ising model by the second author and sly and for sufficiently large q by borgs et al here we show that the following holds for the critical potts model on the torus for q 3 the inverse gap of glauber dynamics is no 1 for q 4 it is at most no log n and for every q 4 in the phase coexistence regime the inverse gaps of both glauber dynamics and swendsen wang dynamics are exponential in n for free or monochromatic boundary conditions and large q we show that the dynamics at criticality is faster than on the torus unlike the ising model where free periodic boundary conditions induce similar dynamical behavior at all temperatures the inverse gap of swendsen wang dynamics is exp no 1 2017 wiley periodicals inc

subsequential scaling limits for liouville graph distance for 0 gamma 2 and delta 0 we consider the liouville graph distance which is the minimal number of euclidean balls of gamma liouville quantum gravity measure at most delta whose union contains a continuous path between two endpoints in this paper we show that the renormalized distance is tight and thus has subsequential scaling limits at delta rightarrow 0 in particular we show that for all delta 0 the diameter with respect to the liouville graph distance has the same order as the typical distance between two endpoints

on the 2d ising wulff crystal near criticality we study the behavior of the two dimensional ising model in a finite box at temperatures that are below but very close to the critical temperature in a regime where the temperature approaches the critical point and simultaneously the size of the box grows fast enough we establish a large deviation principle that proves the appearance of a round wulff crystal

nonlocal optimal design a new perspective about the approximation of solutions in optimal design abstract it is well known from the recent literature that nonlocal integral models are suitable to approximate integral functionals or partial differential equations in the present work a nonlocal optimal design model has been considered as approximation of the corresponding classical or local optimal control problem the new model is driven by a nonlocal elliptic equation and the cost functional belongs to a broad class of nonlocal functional integrals the purpose of this paper is to prove existence of optimal design for the new model this work is complemented by showing that the limit of the nonlocal problem is the local one when the cost to minimize is the compliance functional see 14

dehn fillings of knot manifolds containing essential once punctured tori in this paper we study exceptional dehn fillings on hyperbolic knot manifolds which contain an essential once punctured torus let m be such a knot manifold and let be the boundary slope of such an essential once punctured torus we prove that if dehn filling m with slope produces a seifert fibred manifold then 5 furthermore we classify the triples m when 4 more precisely when 5 then m is the unique manifold w h 3 2 obtained by dehn filling one boundary component of the whitehead link exterior with slope 3 2 and is the pair of slopes 5 0 further 4 if and only if m is the triple w h 2n 1 n 4 0 for some integer n with n 1 combining this with known results we classify all hyperbolic knot manifolds m and pairs of slopes on m where is the boundary slope of an essential once punctured torus in m and is an exceptional filling slope of distance 4 or more from refined results in the special case of hyperbolic genus one knot exteriors in s3 are also given

forbidden gap argument for phase transitions proved by means of chessboard estimates chessboard estimates are one of the standard tools for proving phase coexistence in spin systems of physical interest in this note we show that the method not only produces a point in the phase diagram where more than one gibbs states coexist but that it can also be used to rule out the existence of shift ergodic states that differ significantly from those proved to exist for models depending on a parameter say the temperature this shows that the values of the conjugate thermodynamic quantity the energy inside the transitional gap are forbidden in all shift ergodic gibbs states we point out several models where our result provides useful additional information concerning the set of possible thermodynamic equilibria

infinite ab percolation clusters exist the authors provide an example of a lattice on which infinite ab percolation clusters can occur answering a question of halley they identify parameter values for which infinite ab percolation occurs and does not occur on the lattice

connectivity properties of mandelbrot s percolation process in 1974 mandelbrot introduced a process in 0 1 2 which he called canonical curdling and later used in this book s on fractals to generate self similar random sets with hausdorff dimension d 0 2 in this paper we will study the connectivity or percolation properties of these sets proving all of the claims he made in sect 23 of the fractal geometry of nature and a new one that he did not anticipate there is a probability pc 0 1 so that if p pc then the set is duslike i e the largest connected component is a point whereas if p pc notice the opposing sides are connected with positive probability and furthermore if we tile the plane with independent copies of the system then there is with probability one a unique unbounded connected component which intersects a positive fraction of the tiles more succinctly put the system has a first order phase transition

the q state potts model in the standard pirogov sinai theory surface tensions and wilson loops theq state potts model both scalar and gauge versions is rewritten with the help of the duality transformation into a form of the pirogov sinai theory with noninteracting contours that can be controlled by cluster expansions onceq is large enough this is then used in a new proof of the existence of a unique transition inverse temperature t where the mean internal energy is discontinuous moreover we prove for the scalar model again forq large enough that there are discontinuities at t of the magnetization and of the mass gap with the magnetization vanishing below t and the mass gap vanishing above t we also show that the surface tensions between ordered stable phases are strictly positive up to t and the surface tension between an ordered phase and the disordered one is strictly positive at t for the three dimensional gauge model the wilson parameter exhibits a direct transition from an area law decay quark confinement to a perimeter law decay deconfinement

surface tension and phase coexistence for general lattice systems we investigate the surface tension between coexisting phases of general discrete lattice systems in particular the different phases need not be connected by any symmetry we prove the positivity of the surface tension in the low temperature regime where the pirogov sinai theory of first order phase transitions is valid finite range hamiltonian having a finite number of periodic ground states we give a brief description with some extensions of this theory

rigidity of the interface in percolation and random cluster models we study conditioned random cluster measures with edge parameter p and cluster weighting factor q satisfying q 1 the conditioning corresponds to mixed boundary conditions for a spin model interfaces may be defined in the sense of dobrushin and these are proved to be rigid in the thermodynamic limit in three dimensions and for sufficiently large values of p this implies the existence of non translation invariant conditioned random cluster measures in three dimensions the results are valid in the special case q 1 thus indicating a property of three dimensional percolation not previously noted

a study of perfect wetting for potts and blume capel models with correlation inequalities the so called perfect wetting phenomenon is studied for theq state d 2 potts model using a new correlation inequality a general inequality is established for the surface tension between ordered phases a b and the surface tension between an ordered and the disordered phases a f for any even value ofq this result implies in particular sigma beta l a b geqslant sigma beta l a f sigma beta l b f 0 at the transition point t where the previous phases coexist forq large this inequality is connected to perfect wetting at the transition point using thermodynamic considerations the same kinds of results are derived for the blume capel model

on a sharp transition from area law to perimeter law in a system of random surfaces we introduce and study a phase transition which is associated with the spontaneous formation of infinite surface sheets in a bernoulli system of random plaquettes the transition is manifested by a change in the asymptotic behavior of the probability of the formation of a surface spanning a prescribed loop as such this transition offers a generalization of the bond percolation phenomenon at low plaquette densities the probability for large loops is shown to decay exponentially with the loops area whereas for high densities the decay is by a perimeter law furthermore we show that the two phases of the three dimensional plaquette system are in a precise correspondence with the two phases of the dual system of random bonds thus if a natural conjecture about the phase structure of the bond percolation model is true then there is a sharp transition in the asymptotic behavior of the surface events our analysis incorporates block variables in terms of which a non critical system is transformed into one which is close to a trivial high or low density fixed point stochastic geometric effects like those discussed here play an important role in lattice gauge theories

large deviations for the contact process and two dimensional percolation summary the following results are proved 1 for the upper invariant measure of the basic one dimensional supercritical contact process the density of l s has the usual large deviation behavior the probability of a large deviation decays exponentially with the number of sites considered 2 for supercritical two dimensional nearest neighbor site or bond percolation the density ya of sites inside a square a which belong to the infinite cluster has the following large deviation properties the probability that ya deviates from its expected value by a positive amount decays exponentially with the area of a while the probability that it deviates from its expected value by a negative amount decays exponentially with the perimeter of a these two problems are treated together in this paper because similar techniques renormalization are used for both

the 2d ising model near criticality a fk percolation analysis we study the 2d ising model defined on finite boxes at temperatures that are below but very close from the critical point when the temperature approaches the critical point and the size of the box grows fast enough we establish large deviations estimates on fk percolation events that concern the phenomenon of phase coexistence

optimal design in conductivity under locally constrained heat flux this paper addresses the relaxation of an optimal design problem in conductivity under a point wise constraint on the heat flux by using a variational approach developed by the authors in a previous work we obtained a complete relaxation involving the explicit computation of constrained quasiconvex envelopes this relaxed formulation turns out to be very simple in the remarkable compliance case for which numerical simulations are performed those simulations show that our approach and results are interesting from a theoretical point of view as well as from a practical perspective the emphasis of our work is placed on the local constraint on the heat flux which seems to be a new ingredient for this kind of optimal design problems

an alternate version of pirogov sinai theory a new approach to the pirogov sinai theory of phase transitions is developed not employing the contour models with a parameter the completeness of the phase diagram is proven

translation invariant equilibrium states of ferromagnetic abelian lattice systems the structure of the set of all translation invariant equilibrium states is determined for all temperatures for which the free energy is differentiable models with several phase transitions are discussed rigorously

translation invariant gibbs states in the q state potts model we describe the set of all translation invariant gibbs states in theq state potts model for the case ofq large enough and the other parameters to be arbitrary

upper bounds on fourier entropy given a function f 0 1 n rightarrow mathbb r its fourier entropy is defined to be sum s widehat f 2 s log widehat f 2 s where hat f denotes the fourier transform of f this quantity arises in a number of applications especially in the study of boolean functions an outstanding open question is a conjecture of friedgut and kalai 1996 called the fourier entropy influence fei conjecture asserting that the fourier entropy of any boolean function f is bounded above up to a constant factor by the total influence average sensitivity of f

geometric aspects of quantum spin states a number of interesting features of the ground states of quantum spin chains are analyzed with the help of a functional integral representation of the system s equilibrium states methods of general applicability are introduced in the context of the su 2s 1 invariant quantum spin s chains with the interaction p o wherep o is the projection onto the singlet state of a pair of nearest neighbor spins the phenomena discussed here include the absence of neel order the possibility of dimerization conditions for the existence of a spectral gap and a dichotomy analogous to one found by affleck and lieb stating that the systems exhibit either slow decay of correlations or translation symmetry breaking our representation elucidates the relation evidence for which was found earlier of the p o spin s systems with the potts and the fortuin kasteleyn random cluster models in one more dimension the method reveals the geometric aspects of the listed phenomena and gives a precise sense to a picture of the ground state in which the spins are grouped into random clusters of zero total spin e g within such structure the dichotomy is implied by a topological argument and the alternatives correspond to whether or not the clusters are of finite mean length

the incipient infinite cluster for high dimensional unoriented percolation we consider bond percolation on mathbb z d at the critical occupation density p c for d 6 in two different models the first is the nearest neighbor model in dimension d 6 the second model is a spread out model having long range parameterized by l in dimension d 6 in the spread out case we show that the cluster of the origin conditioned to contain the site x weakly converges to an infinite cluster as x when d 6 and l is sufficiently large we also give a general criterion for this convergence to hold which is satisfied in the case d 6 in the nearest neighbor model by work of hara 12 we further give a second construction by taking p 6 the probability for invasion percolation to reach a site x is asymptotic to c x d 4 as x

mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions in this paper we study the linear bond based nonlocal peridynamic models with a particular focus on problems associated with nonstandard nonlocal displacement loading conditions both stationary and time dependent problems are considered for a one dimensional scalar equation defined on a finite bar and for a two dimensional system defined on a square the related peridynamic operators and associated functional spaces are analyzed applications to the numerical analysis of the finite dimensional approximations to peridynamic models are also illustrated

influence and sharp threshold theorems for monotonic measures the influence theorem for product measures on the discrete space 0 1 n may be extended to probability measures with the property of monotonicity which is equivalent to strong positive association corresponding results are valid for probability measures on the cube 0 1 n that are absolutely continuous with respect to lebesgue measure these results lead to a sharp threshold theorem for measures of random cluster type and this may be applied to box crossings in the two dimensional random cluster model

dirichlet s principle and wellposedness of solutions for a nonlocal p laplacian system abstract we prove dirichlet s principle for a nonlocal p laplacian system which arises in the nonlocal setting of peridynamics when p 2 this nonlinear model includes boundary conditions imposed on a nonzero volume collar surrounding the domain our analysis uses nonlocal versions of integration by parts techniques that resemble the classical green and gauss identities the nonlocal energy functional associated with this elliptic type system exhibits a general kernel which could be weakly singular the coercivity of the system is shown by employing a nonlocal poincare s inequality we use the direct method in calculus of variations to show existence and uniqueness of minimizers for the nonlocal energy from which we obtain the wellposedness of this steady state diffusion system

nonlocal elasticity and related variational principles abstract the eringen model of nonlocal elasticity is considered and its implications in solid mechanics studied the model is refined by assuming an attenuation function depending on the geodetical distance between material particles such that in the diffusion processes of the nonlocality effects certain obstacles as holes or cracks existing in the domain can be circumvented a suitable thermodynamic framework with nonlocality is also envisaged as a firm basis of the model the nonlocal elasticity boundary value problem for infinitesimal displacements and quasi static loads is addressed and the conditions for the solution uniqueness are established three variational principles nonlocal counterparts of classical ones i e the total potential energy the complementary energy and the mixed hu washizu principles are provided the former is used to derive a nonlocal type fem nl fem in which the symmetric global stiffness matrix reflects the nonlocality features of the problem an alternative standard fem based solution method is also provided which consists in an iterative procedure of the type local prediction nonlocal correction in which the nonlocality is simulated by an imposed like correction strain the potentialities of these analysis methods are pointed out their numerical implementations being the object of an ongoing research work

interfaces in the potts model ii antonov s rule and rigidity of the order disorder interface within the ferromagneticq state potts model we discuss the wetting of the interface between two ordered phasesa andb by the disordered phasef at the transition temperature in two or more dimensions and forq large we establish the validity of the antonov s rule ab af fb where denotes the surface tension between the considered phases we also prove that at this temperature in three or more dimensions the interface between any ordered phase and the disordered one is rigid

discontinuity of the wilson string tension in the 4 dimensional lattice pure gauge potts model we consider the 4 dimensionalq state pure gauge potts model forq large enough we give a new proof of the existence of a unique coupling constant t where a first order phase transition occurs moreover we prove the following new results the string tension is discontinuous at t the wilson parameter exhibits at t a direct transition from an area law decay quark confinement to a perimeter law decay quark deconfinement

crossing probabilities for voronoi percolation we prove that the standard russo seymour welsh theory is valid for voronoi percolation this implies that at criticality the crossing probabilities for rectangles are bounded by constants depending only on their aspect ratio this result has many consequences such as the polynomial decay of the one arm event at criticality

sharp thresholds of graph properties and the sat problem consider g n p to be the probability space of random graphs on n vertices with edge probability p we will be considering subsets of this space defined by monotone graph properties a monotone graph property p is a property of graphs such that a p is invariant under graph automorphisims b if graph h has property p then so does any graph g having h as a subgraph a monotone symmetric family of graphs is a family defined by such a property one of the first observations made about random graphs by erdos and renyi in their seminal work on random graph theory 12 was the existence of threshold phenomena the fact that for many interesting properties p the probability of p appearing in g n p exhibits a sharp increase at a certain critical value of the parameter p bollobas and thomason proved the existence of threshold functions for all monotone set properties 6 and in 14 it is shown that this behavior is quite general and that all monotone graph properties exhibit threshold behavior i e the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval more precise analysis of the size of the threshold interval is done in 7 this threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics one of the main questions that arises in studying phase transitions is how sharp is the transition for example one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random kcnf boolean formula nati linial who introduced me to this problem suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition answering the question posed in 14 as to the relation between the length of the threshold interval and the value of the critical probability in this paper we indeed introduce a simple condition and prove it is sufficient stated roughly in the setting of random graphs the main theorem states that if a property has a coarse threshold then it can be approximated by the property of having certain given graphs as a subgraph this condition can be applied in a more

decay of correlations in random cluster models we prove exponential decay for the tail of the radius r of the cluster at the origin for subcritical random cluster models under an assumption slightly weaker than that here d is the number of dimensions specifically if throughout the subcritical phase then for some 0 this implies the exponential decay of the two point correlation function of subcritical potts models subject to a hypothesis of at least polynomial decay of this function similar results are known already for percolation and ising models and for potts models when the number q of available states is sufficiently large indeed the hypothesis of polynomial decay has been proved rigorously for these cases in two dimensions the hypothesis that is weaker than requiring that the susceptibility be finite i e that the two point function be summable the principal new technique is a form of russo s formula for random cluster models reported by bezuidenhout grimmett and kesten for the current application this leads to an analysis of a first passage problem for random cluster models and a proof that the associated time constant is strictly positive if and only if the tail of r decays exponentially

inhomogeneous percolation problems and incipient infinite clusters the authors consider inhomogeneous percolation models with density pc f x and examine the forms of f x which produce incipient structures taking f x approximately mod x mod lambda and assuming the existence of a correlation length exponent v for the homogeneous percolation model they prove that in d 2 the borderline value of lambda is lambda b 1 v if lambda 1 v then with probability one there is no infinite cluster while if lambda 2 the models studied suggest what sort of incipient objects should be examined in random surface models

another critical exponent inequality for percolation β 2 δ the inequality in the title is derived for standard site percolation in any dimension assuming only that the percolation density vanishes at the critical point the proof based on a lattice animal expansion is fairly simple and is applicable to rather general site or bond short or long range independent percolation models

percolation beyond ℤd the contributions of oded schramm oded schramm 1961 2008 influenced greatly the development of percolation theory beyond the usual d setting in particular the case of nonamenable lattices here we review some of his work in this field

dehn surgery on knots in d dehn considered the following method for constructing 3 manifolds remove a solid torus neighborhood n k of some knot x in the 3 sphere s and sew it back differently in particular he showed that taking x to be the trefoil one could obtain infinitely many non simply connected homology spheres o in this way let mx s n k then the different resewings are parametrized by the isotopy class r of the simple closed curve on the torus omk that bounds a meridional disk in the re attached solid torus we denote the resulting closed oriented 3 manifold by mk and say that it is obtained by r dehn surgery on x more generally one can consider the manifolds ml obtained by r dehn surgery on a fc component link l xi u u kk in s where r r it turns out that every closed oriented 3 manifold can be constructed in this way wal lie thus a good understanding of dehn surgery might lead to progress on general questions about the structure of 3 manifolds starting with the case of knots it is natural to extend the context a little and consider the manifolds m r obtained by attaching a solid torus v to an arbitrary compact oriented irreducible every 2 sphere bounds a 3 ball 3 manifold m with dm an incompressible torus where r is the isotopy class slope on dm of the boundary of a meridional disk of v we say that m r is the result of r dehn filling on m an observed feature of this construction is that

distance between toroidal surgeries on hyperbolic knots in the 3 sphere for a hyperbolic knot in the 3 sphere at most finitely many dehn surgeries yield non hyperbolic 3 manifolds as a typical case of such an ex ceptional surgery a toroidal surgery is one that yields a closed 3 manifold containing an incompressible torus the slope corresponding to a toroidal surgery called a toroidal slope is known to be integral or half integral we show that the distance between two integral toroidal slopes for a hyperbolic knot except the figure eight knot is at most four

closed form solution for a nonlocal elastic bar in tension a simple mechanical one dimensional problem in the context of nonlocal integral elasticity is solved analytically the nonlocal elastic material behaviour is described by the eringen model whose nonlocality features all reside in the constitutive relation this relation of integral type contains an attenuation function usually assumed symmetric aimed to capture the diffusion process of the nonlocality effects it also exhibits a convolution format the governing equation is a fredholm integral equation of second kind whose analytical treatment even for the usual choice of a symmetric kernel is not easy to develop in the present paper assuming a specific shape for the attenuation function a closed form solution in terms of strains is alternatively obtained by solving a volterra integral equation of second kind the latter can be easily solved with standard techniques at least for the adopted kernel taking also advantage from the symmetry of the solution such a closed form solution is an essential result to validate the effectiveness of numerical procedures aimed to solve more complex mechanical problems in the context of nonlocal elasticity 2002 elsevier science ltd all rights reserved

mean field theory of the potts gas we consider a gas of classical particles in r d having q distinct colours interacting via a mean field potts potential and subject to an external field a colour independent molecular interaction of mean field type is also admitted in contrast to the usual lattice potts model the potts gas exhibits the specific volume v as a parameter in addition to colour ordering the v dependence of the potts gas is studied in detail and the complete phase diagram is derived it turns out that for q 3 the transition to colour ordering implies a jump of density for q 2 this transition is continuous but may become discontinuous under the influence of a suitable molecular interaction

sharpness of the phase transition for continuum percolation in mathbb r 2 r 2 we study the phase transition of random radii poisson boolean percolation around each point of a planar poisson point process we draw a disc of random radius independently for each point the behavior of this process is well understood when the radii are uniformly bounded from above in this article we investigate this process for unbounded and possibly heavy tailed radii distributions under mild assumptions on the radius distribution we show that both the vacant and occupied sets undergo a phase transition at the same critical parameter lambda c moreover the techniques we develop in this article can be applied to other models such as the poisson voronoi and confetti percolation

interfaces in the potts model i pirogov sinai theory of the fortuin kasteleyn representation we develop a new analysis of the order disorder transition in ferromagnetic potts models for large numberq of spin states we use the pirogov sinai theory which we adapt to the fortuin kasteleyn representation of the models this theory applies in a rather direct way in our approach and leads to a system of non interacting contours with small activities as a consequence simpler and more natural techniques are found allowing us to recover previous results on the bulk properties of the model which then extend to non integer values ofq and to deal with non translation invariant boundary conditions this will be applied in a second part of this work to study the behaviour of the interfaces at the transition point

the self dual point of the two dimensional random cluster model is critical for q 1 we prove a long standing conjecture on random cluster models namely that the critical point for such models with parameter q 1 on the square lattice is equal to the self dual point p sd q sqrt q 1 sqrt q this gives a proof that the critical temperature of the q state potts model is equal to log 1 sqrt q for all q 2 we further prove that the transition is sharp meaning that there is exponential decay of correlations in the sub critical phase the techniques of this paper are rigorous and valid for all q 1 in contrast to earlier methods valid only for certain given q the proof extends to the triangular and the hexagonal lattices as well

analyticity properties and power law estimates of functions in percolation theory we consider percolation on the sites of a graphg e g a regulard dimensional lattice all sites ofg are occupied vacant with probabilityp respectively q 1 p independently of each other w denotes the cluster of occupied sites containing a fixed site which will usually be taken to be the origin andw the cardinality ofw the percolation probability is the probability that w i e p pp w some critical values ofp ph andpt are defined respectively as the smallest value ofp for which p 0 and for which the expectation of w is infinite formally ph inf p p 0 andpt inf p ep w we show for fairly general graphsgthat ifp pt thenpp w n decreases exponentially inn for the special casesg g0 the simple quadratic lattice andg1 the graph which corresponds to bond percolation on 2 we obtain upper and lower bounds for p of the formc p ph and bounds forep w of the formc p ph we also investigate smoothness properties of p ep number of clusters per site ep w 1 w 1 this function was introduced by sykes and essam who assumed that has exactly one singularity namely atp ph for the graphsg0 andg1 i e site or bond percolation on 2 we show that p is analytic atp ph and has two continuous derivatives atp ph the emphasis is on rigorous proofs

the triangle condition for percolation aizenman and newman introduced an unverified condition the triangle condition which has been shown to imply that a number of percolation critical exponents take their mean field values and which is expected to hold above six dimensions for nearest neighbour percolation we prove that the triangle condition is satisfied in sufficiently high dimensions for the nearest neighbour model and above six dimensions for a class of spread out models the proof uses an expansion which is related to the lace expansion for self avoiding walk percolation is a simple probabilistic model for which many interesting problems remain unsolved the basic objects in percolation are random graphs on an infinite lattice typically there is a critical density for the graph edges known as bonds below which there is zero probability that a fixed point in the lattice is part of an infinite connected graph but above which this probability is strictly positive this abrupt change in behaviour has been used in chemistry and statistical physics to model phase transitions in a variety of physical systems such as fluid flow through a porous medium hence the name percolation random resistor networks and the gelation of branched polymers for an introduction see 15 20 to define the models we are interested in we consider the of dimensional integer lattice z whose elements are referred to as sites and the set of pairs b x y of distinct sites the bonds to each bond b is associated an independent bernoulli random variable nb9 with nb 1 with probability p jb and nb 0 with probability p jh the jb are fixed and z invariant and p is a parameter the nearest neighbour model is defined by taking jb 1 for those b x y such that x y i 1 and jb 0 otherwise expectation with respect to the joint distribution of the nb is denoted by if nb 1 we say b is occupied and otherwise b is vacant given a bond configuration nb two sites x and y are said to be connected if there is a path from x to y which consists of occupied bonds the set of sites which are connected to x is called the connected cluster of x and is denoted c x the probability that x is connected to y is written tp x y 1 ip x y i x and y are connected i y e c x p where i denotes the indicator function denoting by c x the number of sites in c x the expected cluster size or susceptibility is given by

the incipient infinite cluster in two dimensional percolation letpp be the probability measure on the configurations of occupied and vacant vertices of a two dimensional graphg under which all vertices are independently occupied respectively vacant with probabilityp respectively 1 p letph be the critical probability for this system andw the occupied cluster of some fixed vertexw0 we show that for many graphsg such as mathbb z 2 or its covering graph which corresponds to bond percolation on mathbb z 2 the following two conditional probability measures converge and have the same limit v say

percolation of poisson sticks on the plane we consider a percolation model on the plane which consists of 1 dimensional sticks placed at points of a poisson process onr 2 each stick having a random but bounded length and a random direction the critical probabilities are defined with respect to the occupied clusters and vacant clusters and they are shown to be equal the equality is shown through a pivotal cell argument using a version of the russo seymour welsh theorem which we obtain for this model

the stochastic random cluster process and the uniqueness of random cluster measures the random cluster model is a generalization of percolation and ferromagnetic potts models due to fortuin and kasteleyn not only is the random cluster model a worthwhile topic for study in its own right but also it provides much information about phase transitions in the associated physical models this paper serves two functions first we introduce and survey random cluster measures from the probabilist s point of view giving clear statements of some of the many open problems second we present new results for such measures as follows we discuss the relationship between weak limits of random cluster measures and measures satisfying a suitable dlr condition using an argument based on the convexity of pressure we prove the uniqueness of random cluster measures for all but at most countably many values of the parameter p related results concerning phase transition in two or more dimensions are included together with various stimulating conjectures the uniqueness of the infinite cluster is employed in an intrinsic way in part of these arguments in the second part of this paper is constructed a markov process whose level sets are reversible markov processes with random cluster measures as unique equilibrium measures this construction enables a coupling of random cluster measures for all values of p furthermore it leads to a proof of the semicontinuity of the percolation probability and provides a heuristic probabilistic justification for the widely held belief that there is a first order phase transition if and only if the cluster weighting factor q is sufficiently large

phases coexistence and surface tensions for the potts model theq states potts model exhibits a first order phase transition at some inverse temperature t between ordered and disordered phases forq large as proved in 1 in space dimension 2 we use theduality transformation as aninternal symmetry of the partition function at t to derive an estimate on the probability of a contour this enables us to prove the preceding result and the following new results i the discontinuity of the mass gap at t ii the existence of astrictly positive surface tension between two ordered phases up to t iii the existence of a non zero surface tension between an ordered and the disordered phase at t

sharpness of the phase transition in percolation models the equality of two critical points the percolation threshold ph and the point p where the cluster size distribution ceases to decay exponentially is proven for all translation invariant independent percolation models on homogeneous d dimensional lattices d 1 the analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter m h which for h q reduces to the percolation density p at the bond density p l e in the single parameter case these are 1 m hdm dh m2 mdm d and 2 dm d j mdm dh inequality 1 is intriguing in that its derivation provides yet another hint of a 3 structure in percolation models moreover through the elimination of one of its derivatives 1 yields a pair of ordinary differential inequalities which provide information on the critical exponents and one of these resembles an ising model inequality of fr hlich and sokal and yields the mean field bound 5 2 and the other implies the result of chayes and chayes that an inequality identical to 2 is known for ising models where it provides the basis for newman s universal relation 5 1 1 and for certain extrapolation principles which are now made applicable also to independent percolation these results apply to both finite and long range models with or without orientation and extend to periodic and weakly inhomogeneous systems

approximate zero one laws and sharpness of the percolation transition in a class of models including two dimensional ising percolation one of the most well known classical results for site percolation on the square lattice is the equation p c p c 1 in words this equation means that for all values p c of the parameter p the following holds either a s there is an infinite open cluster or a s there is an infinite closed star cluster this result is closely related to the percolation transition being sharp below p c the size of the open cluster of a given vertex is not only a s finite but has a distribution with an exponential tail the analog of this result has been proven by higuchi in 1993 for two dimensional ising percolation at fixed inverse temperature c with external field h the parameter of the model using sharp threshold results approximate zero one laws and a modification of an rsw like result by bollobas and riordan we show that these results hold for a large class of percolation models where the vertex values can be nicely represented in a sense which will be defined precisely by i i d random variables we point out that the ordinary percolation model obviously belongs to this class and we also show that the ising model mentioned above belongs to it