character degrees and nilpotence class of finite p groups an approach via pro p groups let s be a finite set of powers of p containing 1 it is known that for some choices of s if p is a finite p group whose set of character degrees is s then the nilpotence class of p is bounded by some integer that depends on s while for some other choices of s such an integer does not exist the sets of the first type are called class bounding sets the problem of determining the class bounding sets has been studied in several papers whose results made it tempting to conjecture that a set s is class bounding if and only if p s in this article we provide a new approach to this problem our main result shows the relevance of certain p adic space groups in this problem with its help we are able to prove some results that provide new class bounding sets we also show that there exist non class bounding sets s such that p s

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

message passing for maximum weight independent set in this paper we investigate the use of message passing algorithms for the problem of finding the max weight independent set mwis in a graph first we study the performance of the classical loopy max product belief propagation we show that each fixed point estimate of max product can be mapped in a natural way to an extreme point of the linear programming lp polytope associated with the mwis problem however this extreme point may not be the one that maximizes the value of node weights the particular extreme point at final convergence depends on the initialization of max product we then show that if max product is started from the natural initialization of uninformative messages it always solves the correct lp if it converges this result is obtained via a direct analysis of the iterative algorithm and cannot be obtained by looking only at fixed points the tightness of the lp relaxation is thus necessary for max product optimality but it is not sufficient motivated by this observation we show that a simple modification of max product becomes gradient descent on a smoothed version of the dual of the lp and converges to the dual optimum we also develop a message passing algorithm that recovers the primal mwis solution from the output of the descent algorithm we show that the mwis estimate obtained using these two algorithms in conjunction is correct when the graph is bipartite and the mwis is unique finally we show that any problem of maximum a posteriori map estimation for probability distributions over finite domains can be reduced to an mwis problem we believe this reduction will yield new insights and algorithms for map estimation

character degree graphs and normal subgroups we consider the degrees of those irreducible characters of a group g whose kernels do not contain a given normal subgroup n we show that if n g and n is not perfect then the common divisor graph on this set of integers has at most two connected components also if n is solvable we obtain bounds on the diameters of the components of this graph and in the disconnected case we study the structure of n and of g

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

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

long time tails in the parabolic anderson model with bounded potential we consider the parabolic anderson problem t u u u on 0 z d with random i i d potential z z zd and the initial condition u 0 1 our main assumption is that esssup 0 0 depending on the thickness of the distribution prob 0 close to its essential supremum we identify both the asymptotics of the moments of u t 0 and the almost sure asymptotics of u t 0 as t in terms of variational problems as a by product we establish lifshitz tails for the random schrodinger operator at the bottom of its spectrum in our class of distributions the lifshitz exponent ranges from d 2 to the power law is typically accompanied by lower order corrections

random cluster dynamics in mathrm mathbb z 2 the random cluster model has been widely studied as a unifying framework for random graphs spin systems and electrical networks but its dynamics have so far largely resisted analysis in this paper we analyze the glauber dynamics of the random cluster model in the canonical case where the underlying graph is an n times n box in the cartesian lattice mathrm mathbb z 2 our main result is a o n 2 log n upper bound for the mixing time at all values of the model parameter p except the critical point p p c q and for all values of the second model parameter q ge 1 we also provide a matching lower bound proving that our result is tight our analysis takes as its starting point the recent breakthrough by beffara and duminil copin on the location of the random cluster phase transition in mathrm mathbb z 2 it is reminiscent of similar results for spin systems such as the ising and potts models but requires the reworking of several standard tools in the context of the random cluster model which is not a spin system in the usual sense

message passing algorithms for inference and optimization belief propagation and divide and concur message passing algorithms can solve a wide variety of optimization inference and constraint satisfaction problems the algorithms operate on factor graphs that visually represent and specify the structure of the problems after describing some of their applications i survey the family of belief propagation bp algorithms beginning with a detailed description of the min sum algorithm and its exactness on tree factor graphs and then turning to a variety of more sophisticated bp algorithms including free energy based bp algorithms splitting bp algorithms that generalize tree reweighted bp and the various bp algorithms that have been proposed to deal with problems with continuous variables

gibbs states of graphical representations of the potts model with external fields we consider the ferromagnetic q state potts model with each of the q spin values coupled to an external field we also introduce a generalized random cluster model which includes both the potts model in arbitrary homogeneous external fields and the noninteger q random cluster model as special cases we establish the fkg property the finite energy condition uniqueness of the infinite cluster and gibbsianness of limit states for this generalized model furthermore we develop the theory of gibbs states for the edwards sokal representation of the potts model in a field and relate the phase structure in this representation to those in the spin and random cluster representations finally we characterize the possible color s of the infinite cluster s and show that the correspondence between edwards sokal gibbs states and their random cluster marginals is bijective once the color of the infinite cluster is fixed

comparison and disjoint occurrence inequalities for random cluster models a principal technique for studying percolation ferromagnetic ising potts and random cluster models is the fkg inequality which implies certain stochastic comparison inequalities for the associated probability measures the first result of this paper is a new comparison inequality proved using an argument developed elsewhere in order to obtain strict inequalities for critical values as an application of this inequality we prove that the critical pointp c q of the random cluster model with cluster weighting factorq 1 is strictly monotone inq our second result is a bk inequality for the disjoint occurrence of increasing events in a weaker form than that available in percolation theory

character kernels and degree ratios in finite groups abstract let k ker be the kernel of an irreducible character of a finite group g and let s be the largest solvable normal subgroup of k we show that if the degree of is large compared to the degrees of other irreducible characters of g or if the kernel k is small compared to the kernels of other irreducible characters of g then the fitting height of s is small also we show that the derived length of a nonabelian solvable group is bounded by a logarithmic function of the ratio b c where b is the largest irreducible character degree of g and c is the smallest nonlinear irreducible character degree finally using the classification of simple groups we show that certain kernels of irreducible characters of large degree must be solvable

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

approach to equilibrium of glauber dynamics in the one phase region i the attractive case various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed in particular somefinite size conditions are discussed together with their implications for the gibbs measures and for the approach to equilibrium of glauber dynamics inarbitrarily large volumes it is shown that dobrushin shlosman s theory ofcomplete analyticity and its dynamical counterpart due to stroock and zegarlinski cannot be applied in general to the whole one phase region since it requires mixing properties for regions ofarbitrary shape an alternative approach based on previous ideas of oliveri and picco is developed which allows to establish results on rapid approach to equilibrium deeply inside the one phase region in particular in the ferromagnetic case we considerably improve some previous results by holley and aizenman and holley our results are optimal in the sene that for example they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed dimensional ising model with or without an external field in part ii we extensively consider the general case not necessarily attractive and we develop a new method based on renormalizations group ideas and on an assumption of strong mixing in a finite cube to prove hypercontractivity of the markov semigroup of the glauber dynamics

non perturbative criteria for gibbsian uniqueness for spin systems with an internal symmetry we provide sufficient conditions for unicity of the gibbs state and or complete analyticity by comparison to random cluster models

uniqueness of the infinite component in a random graph with applications to percolation and spin glasses we extend the theorem of burton and keane on uniqueness of the infinite component in dependent percolation to cover random graphs on d or d with long range edges we also study a short range percolation model related to nearest neighbor spin glasses on d or on a slab d 0 k and prove both that percolation occurs and that the infinite component is unique forv 2 0 1 or larger

power law corrections to exponential decay of connectivities and correlations in lattice models consider a translation invariant bond percolation model on the integer lattice which has exponential decay of connectivities that is the probability of a connection 0 x by a path of open bonds decreases like exp m x for some positive constant m which may depend on the direction x x in two and three dimensions it is shown that if the model has an appropriate mixing property and satisfies a special case of the fkg property then there is at most a power law correction to the exponential decay there exist a and c such that exp m x p 0 x a x c exp m x for all nonzero x in four or more dimensions a similar bound holds with x c replaced by exp c log x 2 in particular the power law lower bound holds for the fortuin kasteleyn random cluster model in two dimensions whenever the connectivity decays exponentially since the mixing property is known to hold in that case consequently a similar bound holds for correlations in the potts model at supercritical temperatures

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

agreement percolation and phase coexistence in some gibbs systems we extend some relations between percolation and the dependence of gibbs states on boundary conditions known for ising ferromagnets to other systems and investigate their general validity percolation is defined in terms of the agreement of a configuration with one of the ground states of the system this extension is studied via examples and counterexamples including the antiferromagnetic ising and hard core models on bipartite lattices potts models and many layered ising and continuum widom rowlinson models in particular our results on the hard square lattice model make rigorous observations made by hu and mak on the basis of computer simulations moreover we observe that the naturally defined clusters of the widom rowlinson model play for the wr model itself the same role that the clusters of the fortuin kasteleyn measure play for the ferromagnetic potts models the phase transition and percolation in this system can be mapped into the corresponding liquid vapor transition of a one component fluid

slab percolation and phase transitions for the ising model we prove using the random cluster model a strict inequality between site percolation and magnetization in the region of phase transition for the d dimensional ising model thus improving a result of 5 we extend this result also at the case of two plane lattices mathbb z slabs and give a characterization of phase transition in this case the general case of n slabs with n an arbitrary positive integer is partially solved and it is used to show that this characterization holds in the case of three slabs with periodic boundary conditions

on the uniqueness of the infinite occupied cluster in dependent two dimensional site percolation we consider dependent site percolation on the two dimensional square lattice the underlying probability measure being invariant and ergodic under each of the translations and invariant under axis reflections if this measure satisfies the fkg condition and if percolation occurs then we show that the infinite occupied cluster is unique with probability 1 and that all vacant star clusters are finite

on weak mixing in lattice models for lattice models on d weak mixing is the property that the influence of the boundary condition on a finite decays exponentially with distance from that region for a wide class of models on 2 including all finite range models we show that weak mixing is a consequence of gibbs uniqueness exponential decay of an appropriate form of connectivity and a natural coupling property in particular on 2 the fortuin kasteleyn random cluster model is weak mixing whenever uniqueness holds and the connectivity decays exponentially and the q state potts model above the critical temperature is weak mixing whenever correlations decay exponentially a hypothesis satisfied if q is sufficiently large ratio weak mixing is the property that uniformly over events a and b occurring on subsets and respectively of the lattice p a b p a p b 1 decreases exponentially in the distance between and we show that under mild hypotheses for example finite range weak mixing implies ratio weak mixing

for 2 d lattice spin systems weak mixing implies strong mixing we prove that for finite range discrete spin systems on the two dimensional latticez2 the weak mixing condition which follows for instance from the dobrushin shlosman uniqueness condition for the gibbs state implies a stronger mixing property of the gibbs state similar to the dobrushin shlosman complete analyticity condition but restricted to all squares in the lattice or more generally to all sets multiple of a large enough square the key observation leading to the proof is that a change in the boundary conditions cannot propagate either in the bulk because of the weak mixing condition or along the boundary because it is one dimensional as a consequence we obtain for ferromagnetic ising type systems proofs that several nice properties hold arbitrarily close to the critical temperature these properties include the existence of a convergent cluster expansion and uniform boundedness of the logarithmic sobolev constant and rapid convergence to equilibrium of the associated glauber dynamics on nice subsets ofz2 including the full lattice

stability of the wulff minimum and fluctuations in shape for large finite clusters in two dimensional percolation for two dimensional bernoulli percolation at densityp above the critical point there exists a natural normg determined by the rate of decay of the connectivity function in every direction ifw is the region of unit area with boundary of minimum possibleg length then it is known 4 that asn with probability approaching 1 conditionally onn c 0 0 if is a curve enclosing a region of unit area such that the hausdorff distanced h v dw for every translate v then theg lengthg g w k 2 at least for small

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

message passing algorithms and improved lp decoding linear programming lp decoding for low density parity check codes and related domains such as compressed sensing has received increased attention over recent years because of its practical performance coming close to that of iterative decoding algorithms and its amenability to finite blocklength analysis several works starting with the work of feldman showed how to analyze lp decoding using properties of expander graphs this line of analysis works for only low error rates about a couple of orders of magnitude lower than the empirically observed performance it is possible to do better for the case of random noise as shown by daskalakis and koetter and vontobel building on work of koetter and vontobel we obtain a novel understanding of lp decoding which allows us to establish a 0 05 fraction of correctable errors for rate codes this comes very close to the performance of iterative decoders and is significantly higher than the best previously noted correctable bit error rate for lp decoding our analysis exploits an explicit connection between lp decoding and message passing algorithms and unlike other techniques directly works with the primal linear program an interesting byproduct of our method is a notion of a locally optimal solution that we show to always be globally optimal i e it is the nearest codeword such a solution can in fact be found in near linear time by a reweighted version of the min sum algorithm obviating the need for lp our analysis implies in particular that this reweighted version of the min sum decoder corrects up to a 0 05 fraction of errors

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

using linear programming to decode binary linear codes a new method is given for performing approximate maximum likelihood ml decoding of an arbitrary binary linear code based on observations received from any discrete memoryless symmetric channel the decoding algorithm is based on a linear programming lp relaxation that is defined by a factor graph or parity check representation of the code the resulting lp decoder generalizes our previous work on turbo like codes a precise combinatorial characterization of when the lp decoder succeeds is provided based on pseudocodewords associated with the factor graph our definition of a pseudocodeword unifies other such notions known for iterative algorithms including stopping sets irreducible closed walks trellis cycles deviation sets and graph covers the fractional distance d sub frac of a code is introduced which is a lower bound on the classical distance it is shown that the efficient lp decoder will correct up to spl lceil d sub frac 2 spl rceil 1 errors and that there are codes with d sub frac spl omega n sup 1 spl epsi an efficient algorithm to compute the fractional distance is presented experimental evidence shows a similar performance on low density parity check ldpc codes between lp decoding and the min sum and sum product algorithms methods for tightening the lp relaxation to improve performance are also provided

complete analyticity of the 2d potts model above the critical temperature we investigate the complete analyticity ca of the two dimensional q state potts model for large values of q we are able to prove it for every temperature provided we restrict ourselves to nice subsets their niceness depending on the temperature t contrary to this restricted complete analyticity rca the full ca is known to fail for some values of the temperature above

random cluster representation of the blume capel model the so called diluted random cluster model may be viewed as a random cluster representation of the blume capel model it has three parameters a vertex parameter a an edge parameter p and a cluster weighting factor q stochastic comparisons of measures are developed for the vertex marginal when q 1 2 and the edge marginal when q 1 taken in conjunction with arguments used earlier for the random cluster model these permit a rigorous study of part of the phase diagram of the blume capel model

strict inequality for critical values of potts models and random cluster processes we prove that the critical value c of a ferromagnetic potts model is astrictly decreasing function of the strengths of interaction of the process this is achieved in the more general context of the random cluster representation of fortuin and kasteleyn by deriving and utilizing a formula which generalizes the technique known in percolation theory as russo s formula as a byproduct of the method we present a general argument for showing that at any given point on the critical surface of a multiparameter process the values of a certain critical exponent do not depend on the direction of approach of that point our results apply to all random cluster processes satisfying the fkg inequality

random cluster representation of the ashkin teller model we show that a class of spin models containing the ashkin teller model admits a generalized random cluster grc representation moreover we show that basic properties of the usual representation such as fkg inequalities and comparison inequalities still hold for this generalized random cluster model some elementary consequences are given we also consider the duality transformations in the spin representation and in the grc model and show that they commute

completely analytical interactions constructive description an interactionu is called a completely analytical ca interaction if it satisfies one of 12 given conditions formulated in terms of analyticity properties of the partition functions zv u or correlation decay or truncated correlation bounds or asymptotic behavior of ln zv u v the 12 conditions are presented together with part of the proof of their equivalence the main result of the paper is that each condition is constructive in the following sense instead of checking it in all finite volumesv v it is enough to consider only a finite amount of volumes with restricted size in particular the partition functions z v u for the complex perturbationsu ofu do not vanish for all v v and all with provided this is true only forv with diam v c and but with

an answer to a question of isaacs on character degree graphs abstract let n be a normal subgroup of a finite group g we consider the graph g n whose vertices are the prime divisors of the degrees of the irreducible characters of g whose kernel does not contain n and two vertices are joined by an edge if the product of the two primes divides the degree of some of the characters of g whose kernel does not contain n we prove that if g n is disconnected then g n is solvable this proves a strong form of a conjecture of isaacs

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

nonsolvable normal subgroups and irreducible character degrees abstract let the nonsolvable n be a normal subgroup of the finite group g and cd g n denote the irreducible character degrees of g such that there exist respectively corresponding character kernels not containing n write cd g n to stand for the cardinality of cd g n suppose that cd g n 5 in this paper we prove that if n is a minimal normal subgroup then n is a simple group of lie type when n is a normal subgroup we prove that g has a normal series 1 v u n g such that v is solvable u v is a simple group of lie type and the cardinality cd g u 3 as an application we investigate the structure of g when 5 cd g 6 here cd g denotes the set of irreducible character degrees of g

complete analyticity for 2d ising completed we study the behavior of the two dimensional nearest neighbor ferromagnetic ising model under an external magnetic fieldh we extend to every subcritical value of the temperature a result previously proven by martirosyan at low enough temperature and which roughly states that for finite systems with boundary conditions under a positive external field the boundary effect dominates in the bulk if the linear size of the system is of orderb h withb small enough while ifb is large enough then the external field dominates in the bulk as a consequence we are able to complete the proof that complete analyticity for nice sets holds for every value of the temperature and external field in the interior of the uniqueness region in the phase diagram of the model the main tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field and recently extended to all temperatures below the critical one by ioffe

on the unicity of discontinuous transitions in the two dimensional potts and ashkin teller models we investigate the uniqueness of the discontinuous phase transition of the q state potts model and the symmetric q s s cubic model generalized ashkin teller model on the integer square lattice with q 1 for the ferromagnetic cases when there is a discontinuous phase transition we show that the discontinuity can only occur at a point of self duality

pattern theorems ratio limit theorems and gumbel maximal clusters for random fields we study occurrences of patterns on clusters of size n in random fields on d we prove that for a given pattern there is a constant a 0 such that the probability that this pattern occurs at most na times on a cluster of size n is exponentially small moreover for random fields obeying a certain markov property we show that the ratio between the numbers of occurrences of two distinct patterns on a cluster is concentrated around a constant value this leads to an elegant and simple proof of the ratio limit theorem for these random fields which states that the ratio of the probabilities that the cluster of the origin has sizes n 1 and n converges as n implications for the maximal cluster in a finite box are discussed

on thompson s theorem let p be a prime divisor of the order of a finite group g thompson 1970 j algebra14 129 134 has proved the following remarkable result a finite group g is p nilpotent if the degrees of all its nonlinear irreducible characters are divisible by p in fact in that case g is solvable in this note we prove that a group g having only one nonlinear irreducible character of p degree is a cyclic extension of thompson s group this result is a consequence of the following theorem a nonabelian simple group possesses two nonlinear irreducible characters 1 and 2 of distinct degrees such that p does not divide 1 1 2 1 here p is arbitrary but fixed our proof depends on the classification of finite simple groups some properties of solvable groups possessing exactly two nonlinear irreducible characters of p degree are proved some open questions are posed

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

phase separation in random cluster models ii the droplet at equilibrium and local deviation lower bounds we study the droplet that results from conditioning the planar subcritical fortuin kasteleyn random cluster model on the presence of an open circuit 0 encircling the origin and enclosing an area of at least or exactly n2 we consider local deviation of the droplet boundary measured in a radial sense by the maximum local roughness mlr 0 this being the maximum distance from a point in the circuit 0 to the boundary conv 0 of the circuit s convex hull and in a longitudinal sense by what we term maximum facet length namely the length of the longest line segment of which the polygon conv 0 is formed we prove that there exists a constant c 0 such that the conditional probability that the normalised quantity n 1 3 log n 2 3 mlr 0 exceeds c tends to 1 in the high n limit and that the same statement holds for n 2 3 log n 1 3 mfl 0 to obtain these bounds we exhibit the random cluster measure conditional on the presence of an open circuit trapping high area as the invariant measure of a markov chain that resamples sections of the circuit boundary we analyse the chain at equilibrium to prove the local roughness lower bounds alongside complementary upper bounds provided in 14 the fluctuations mlr 0 and mfl 0 are determined up to a constant factor

on the uniqueness of the infinite cluster in the percolation model we simplify the recent proof by aizenman kesten and newman of the uniqueness of the infinite open cluster in the percolation model our new proof is more suitable for generalization in the direction of percolation type processes with dependent site variables

density and uniqueness in percolation two results on site percolation on thed dimensional lattice d 1 arbitrary are presented in the first theorem we show that for stationary underlying probability measures each infinite cluster has a well defined density with probability one the second theorem states that if in addition the probability measure satisfies the finite energy condition of newman and schulman then there can be at most one infinite cluster with probability one the simple arguments extend to a broad class of finite dimensional models including bond percolation and regular lattices

phase separation in random cluster models iii circuit regularity we study the droplet that results from conditioning the subcritical fortuin kasteleyn random cluster model on the presence of an open circuit encircling the origin and enclosing an area of at least or exactly n 2 in this paper we prove that the resulting circuit is highly regular we define a notion of a regeneration site in such a way that for any such site v the circuit cuts through the radial line segment through v only at v we show that provided that the conditioned circuit is centred at the origin in a natural sense the set of regeneration sites reaches into all parts of the circuit with maximal distance from one such site to the next being at most logarithmic in n with high probability the result provides a flexible control on the conditioned circuit that permits the use of surgical techniques to bound its fluctuations and as such it plays a crucial role in the derivation of bounds on the local fluctuation of the circuit carried out in hammond arxiv 1001 1527 2009 arxiv 1001 1528 2009

a particular bit of universality scaling limits of some dependent percolation models we study families of dependent site percolation models on the triangular lattice and hexagonal lattice that arise by applying certain cellular automata to independent percolation configurations we analyze the scaling limit of such models and show that the distance between macroscopic portions of cluster boundaries of any two percolation models within one of our families goes to zero almost surely in the scaling limit it follows that each of these cellular automaton generated dependent percolation models has the same scaling limit in the sense of aizenman burchard 3 as independent site percolation on

phase transition and uniqueness of levelset percolation the main purpose of this paper is to introduce and establish basic results of a natural extension of the classical boolean percolation model also known as the gilbert disc model we replace the balls of that model by a positive non increasing attenuation function l 0 0 to create the random field y x l x y where is a homogeneous poisson process in rd the field is then a random potential field with infinite range dependencies whenever the support of the function l is unbounded in particular we study the level sets h y containing the points y rd such that y h in the case where l has unbounded support we give for any d 2 a necessary and sufficient condition on l for h y to have a percolative phase transition as a function of h we also prove that when l is continuous then so is almost surely moreover in this case and for d 2 we prove uniqueness of the infinite component of h when such exists and we also show that the so called percolation function is continuous below the critical value hc

lebowitz inequalities for ashkin teller systems abstract we consider the ashkin teller model with negative four spin coupling but still in the region where the ground state is ferromagnetic we establish the standard lebowitz inequality as well as the extension that is necessary to prove a divergent susceptibility

max product for maximum weight matching convergence correctness and lp duality max product belief propagation bp is an iterative message passing algorithm for finding the maximum a posteriori map assignment of a discrete probability distribution specified by a graphical model despite the spectacular success of the algorithm in many application areas such as iterative decoding and combinatorial optimization which involve graphs with many cycles theoretical results about both the correctness and convergence of the algorithm are known in only a few cases see section i for references in this paper we prove the correctness and convergence of max product for finding the maximum weight matching mwm in bipartite graphs even though the underlying graph of the mwm problem has many cycles somewhat surprisingly we show that the max product algorithm converges to the correct mwm as long as the mwm is unique we provide a bound on the number of iterations required and show that for a graph of size n the computational cost of the algorithm scales as o n3 which is the same as the computational cost of the best known algorithms for finding the mwm we also provide an interesting relation between the dynamics of the max product algorithm and the auction algorithm which is a well known distributed algorithm for solving the mwm problem

coexistence of infinite clusters ii ising percolation in two dimensions we show a strong type of conditionally mixing property for the gibbs states ofd dimensional ising model when the temperature is above the critical one by using this property we show that there is always coexistence of infinite and clusters when is smaller than c andh 0 in two dimensions it is also possible to show that this coexistence region extends to some non zero external field case i e for every c there exists someh c 0 such that h h c implies the coexistence of infinite clusters with respect to the gibbs state for h

analysis of the min sum algorithm for packing and covering problems via linear programming message passing algorithms based on belief propagation bp are successfully used in many applications including decoding error correcting codes and solving constraint satisfaction and inference problems the bp based algorithms operate over graph representations called factor graphs that are used to model the input although in many cases the bp based algorithms exhibit impressive empirical results not much has been proved when the factor graphs have cycles this paper deals with packing and covering integer programs in which the constraint matrix is zero one the constraint vector is integral and the variables are subject to box constraints we study the performance of the min sum algorithm when applied to the corresponding factor graph models of packing and covering linear programmings lps we compare the solutions computed by the min sum algorithm for packing and covering problems to the optimal solutions of the corresponding lp relaxations in particular we prove that if the lp has an optimal fractional solution then for each fractional component the min sum algorithm either computes multiple solutions or the solution oscillates below and above the fraction this implies that the min sum algorithm computes the optimal integral solution only if the lp has a unique optimal solution that is integral the converse is not true in general for a special case of packing and covering problems we prove that if the lp has a unique optimal solution that is integral and on the boundary of the box constraints then the min sum algorithm computes the optimal solution in pseudopolynomial time our results unify and extend recent results for the maximum weight matching problem and for the maximum weight independent set problem

the asymmetric random cluster model and comparison of ising and potts models we introduce the asymmetric random cluster or arc model which is a graphical representation of the potts lattice gas and establish its basic properties the arc model allows a rich variety of comparisons in the fkg sense between models with different parameter values we give for example values h for which the 0 s configuration in the potts lattice gas is dominated by the configuration of the h ising model the potts model with possibly an external field applied to one of the spins is a special case of the potts lattice gas which allows our comparisons to yield rigorous bounds on the critical temperatures of potts models for example we obtain 0 571 1 exp c 0 600 for the 9 state potts model on the hexagonal lattice another comparison bounds the movement of the critical line when a small potts interaction is added to a lattice gas which otherwise has only interparticle attraction arc models can also be compared to related models such as the partial fk model obtained by deleting a fraction of the nonsingleton clusters from a realization of the fortuin kasteleyn random cluster model this comparison leads to bounds on the effects of small annealed site dilution on the critical temperature of the potts model

on decoding irregular tanner codes with local optimality guarantees we consider decoding of binary linear tanner codes using message passing iterative decoding and linear programming lp decoding in memoryless binary input output symmetric mbios channels we present new certificates that are based on a combinatorial characterization for the local optimality of a codeword in irregular tanner codes with respect to any mbios channel this characterization is a generalization of arora proc acm symp theory of computing 2009 and vontobel proc inf theory and appl workshop 2010 and is based on a conical combination of normalized weighted subtrees in the computation trees of the tanner graph these subtrees may have any finite height h even equal or greater than half of the girth of the tanner graph in addition the degrees of local code nodes in these subtrees are not restricted to two i e these subtrees are not restricted to skinny trees we prove that local optimality in this new characterization implies maximum likelihood ml optimality and lp optimality and show that a certificate can be computed efficiently we also present a new message passing iterative decoding algorithm called normalized weighted min sum nwms nwms decoding is a belief propagation bp type algorithm that applies to any irregular binary tanner code with single parity check local codes e g low density and high density parity check codes we prove that if a locally optimal codeword with respect to height parameter h exists whereby notably h is not limited by the girth of the tanner graph then nwms decoding finds this codeword in h iterations the decoding guarantee of the nwms decoding algorithm applies whenever there exists a locally optimal codeword because local optimality of a codeword implies that it is the unique ml codeword the decoding guarantee also provides an ml certificate for this codeword finally we apply the new local optimality characterization to regular tanner codes and prove lower bounds on the noise thresholds of lp decoding in mbios channels when the noise is below these lower bounds the probability that lp decoding fails to decode the transmitted codeword decays doubly exponentially in the girth of the tanner graph

phase separation in random cluster models i uniform upper bounds on local deviation this is the first in a series of three papers that addresses the behaviour of the droplet that results in the percolating phase from conditioning the planar fortuin kasteleyn random cluster model on the presence of an open dual circuit 0 encircling the origin and enclosing an area of at least or exactly n2 by the fortuin kasteleyn representation the model is a close relative of the droplet formed by conditioning the potts model on an excess of spins of a given type we consider local deviation of the droplet boundary measured in a radial sense by the maximum local roughness mlr 0 this being the maximum distance from a point in the circuit 0 to the boundary of the circuit s convex hull and in a longitudinal sense by what we term maximum facet length mlf 0 namely the length of the longest line segment of which the polygon is formed the principal conclusion of the series of papers is the following uniform control on local deviation that there are constants 0 c c such that the conditional probability that the normalized quantity n 1 3 log n 2 3mlr lies in the interval c c tends to 1 in the high n limit and that the same statement holds for n 2 3 log n 1 3 mlf in this way we confirm the anticipated n1 3 scaling of maximum local roughness and provide a sharp logarithmic power law correction this local deviation behaviour occurs by means of locally gaussian effects constrained globally by curvature and we believe that it arises in many radially defined stochastic interface models including growth models belonging to the kardar parisi zhang universality class

random cluster models on the triangular lattice we study percolation and the random cluster model on the triangular lattice with 3 body interactions starting with percolation we generalize the star triangle transformation we introduce a new parameter the 3 body term and identify configurations on the triangles solely by their connectivity in this new setup necessary and sufficient conditions are found for positive correlations and this is used to establish regions of percolation and non percolation next we apply this set of ideas to the q 1 random cluster model we derive duality relations for the suitable random cluster measures prove necessary and sufficient conditions for them to have positive correlations and finally prove some rigorous theorems concerning phase transitions

percolation phenomena in low and high density systems we consider the 2d quenched disordered q state potts ferromagnets and show that in the translation invariant measure averaged over the disorder at self dual points any amalgamation of q 1 species will fail to percolate despite an overall high density of 1 q 1 further in the dilute bond version of these systems if the system is just above threshold then throughout the low temperature phase there is percolation of a single species despite a correspondingly small density finally we demonstrate both phenomena in a single model by considering a perturbation of the dilute model that has a self dual point we also demonstrate that these phenomena occur by a similar mechanism in a simple coloring model invented by o haggstrom

critical ising on the square lattice mixes in polynomial time the ising model is widely regarded as the most studied model of spin systems in statistical physics the focus of this paper is its dynamic stochastic version the glauber dynamics introduced in 1963 and by now the most popular means of sampling the ising measure intensive study throughout the last three decades has yielded a rigorous understanding of the spectral gap of the dynamics on mathbb z 2 everywhere except at criticality while the critical behavior of the ising model has long been the focus for physicists mathematicians have only recently developed an understanding of its critical geometry with the advent of sle cle and new tools to study conformally invariant systems

mixing properties and exponential decay for lattice systems in finite volumes an infinite volume mixing or exponential decay property in a spin system or percolation model reflects the inability of the influence of the configuration in one region to propagate to distant regions but in some circumstances where such properties hold propagation can nonetheless occur in finite volumes endowed with boundary conditions we establish the absense sic of such propagation particularly in two dimensions in finite volumes which are simply connected under a variety of conditions mainly for the potts model and the fortuin kasteleyn fk random cluster model allowing external fields for example for the fk model in two dimensions we show that exponential decay of connectivity in infinite volume implies exponential decay in simply connected finite volumes uniformly over all such volumes and all boundary conditions and implies a strong mixing property for such volumes with certain types of boundary conditions for the potts model in two dimensions we show that exponential decay of correlations in infinite volume implies a strong mixing property in simply connected finite volumes which includes exponential decay of correlations in simply connected finite volumes uniformly over all such volumes and all boundary conditions

belief propagation and lp relaxation for weighted matching in general graphs loopy belief propagation has been employed in a wide variety of applications with great empirical success but it comes with few theoretical guarantees in this paper we analyze the performance of the max product form of belief propagation for the weighted matching problem on general graphs we show that the performance of max product is exactly characterized by the natural linear programming lp relaxation of the problem in particular we first show that if the lp relaxation has no fractional optima then max product always converges to the correct answer this establishes the extension of the recent result by bayati shah and sharma which considered bipartite graphs to general graphs perhaps more interestingly we also establish a tight converse namely that the presence of any fractional lp optimum implies that max product will fail to yield useful estimates on some of the edges we extend our results to the weighted b matching and r edge cover problems we also demonstrate how to simplify the max product message update equations for weighted matching making it easily deployable in distributed settings like wireless or sensor networks

discontinuity of the spin wave stiffness in the two dimensional xy model using a graphical representation based on the wolff algorithm the classical d dimensional xy model and some related spin systems are studied it is proved that in d 2 the predicted discontinuity in the spin wave stiffness indeed occurs further the critical properties of the spin system are related to percolation properties of the graphical representation in particular a suitably defined notion of percolation in the graphical representation is proved to be the necessary and sufficient condition for positivity of the spontaneous magnetization

fitting heights of odd order groups with few character degrees abstract if g is any finite solvable group having a normal sylow 2 subgroup in particular if g is odd and satisfying cd g 5 where cd g is the set of ordinary irreducible character degrees of g we show that the fitting height of g does not exceed cd g 2 in case cd g 5 this upper bound on the fitting height of g is best possible

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

non coexistence of infinite clusters in two dimensional dependent site percolation this paper presents three results on dependent site percolation on the square lattice first there exists no positively associated probability measure on 0 1 mathbb z 2 with the following properties a a single infinite 0cluster exists almost surely b at most one infinite 1 cluster exists almost surely c certain probabilities regarding 1 clusters are bounded away from zero second the coexistence of an infinite 1 cluster and an infinite 0cluster has probability zero when the underlying probability measure is ergodic with respect to translations positively associated and satisfies the finite energy condition the third result analyzes the typical structure of infinite clusters of both types in the absence of positive association namely under a slightly sharpened finite energy condition the existence of infinitely many disjoint infinite self avoiding 1 paths follows from the existence of an infinite 1 cluster the same holds with respect to 0paths and 0clusters

graphical representations and cluster algorithms ii we continue the study initiated in part i of graphical representations and cluster algorithms for various models in or related to statistical mechanics for certain models e g the blume emery griffths model and various generalizations we develop fortuin kasteleyn type representations which lead immediately to swendsen wang type algorithms for other models e g the random cluster model that are defined by a graphical representation we develop cluster algorithms without reference to an underlying spin system in all cases phase transitions are related to percolation or incipient percolation in the graphical representation which via the ic algorithm allows for the rapid simulation of these systems at the transition point pertinent examples include the continuum widom rowlinson model the restricted 1 step solid on solid model and the xy model

the wulff construction and asymptotics of the finite cluster distribution for two dimensional bernoulli percolation we consider two dimensional bernoulli percolation at densityp p c and establish the following results

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 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

the percolation transition for the zero temperature stochastic ising model on the hexagonal lattice on the planar hexagonal lattice mathbb h we analyze the markov process whose state t in 1 1 mathbb h updates each site v asynchronously in continuous time t 0 so that v t agrees with a majority of its three neighbors the initial v 0 s are i i d with p v 0 1 p 0 1 we study both rigorously and by monte carlo simulation the existence and nature of the percolation transition as t and p 1 2 denoting by t p the expected size of the plus cluster containing the origin we 1 prove that 1 2 and 2 study numerically critical exponents associated with the divergence of p as p 1 2 a detailed finite size scaling analysis suggests that the exponents and of this t dependent percolation model have the same values 4 3 and 43 18 as standard two dimensional independent percolation we also present numerical evidence that the rate at which t as t is exponential

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