universality of high dimensional spanning forests and sandpiles we prove that the wired uniform spanning forest exhibits mean field behaviour on a very large class of graphs including every transitive graph of at least quintic volume growth and every bounded degree nonamenable graph several of our results are new even in the case of mathbb z d d ge 5 in particular we prove that every tree in the forest has spectral dimension 4 3 and walk dimension 3 almost surely and that the critical exponents governing the intrinsic diameter and volume of the past of a vertex in the forest are 1 and 1 2 respectively the past of a vertex in the uniform spanning forest is the union of the vertex and the finite components that are disconnected from infinity when that vertex is deleted from the forest we obtain as a corollary that the critical exponent governing the extrinsic diameter of the past is 2 on any transitive graph of at least five dimensional polynomial growth and is 1 on any bounded degree nonamenable graph we deduce that the critical exponents describing the diameter and total number of topplings in an avalanche in the abelian sandpile model are 2 and 1 2 respectively for any transitive graph with polynomial growth of dimension at least five and are 1 and 1 2 respectively for any bounded degree nonamenable graph in the case of mathbb z d d ge 5 some of our results regarding critical exponents recover earlier results of bhupatiraju et al electron j probab 22 85 51 2017 in this case we improve upon their results by showing that the tail probabilities in question are described by the appropriate power laws to within constant order multiplicative errors rather than the polylogarithmic order multiplicative errors present in that work

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

exact large deviation bounds up to t c for the ising model in two dimensions we prove an upper large deviation bound for the block spin magnetization in the 2d ising model in the phase coexistence region the precise rate given by the wulff construction is shown to hold true for all c combined with the lower bounds derived in i those results yield an exact second order large deviation theory up to the critical temperature

on the tail of the branching random walk local time consider a critical branching random walk on mathbb z d d ge 1 started with a single particle at the origin and let l x be the total number of particles that ever visit a vertex x we study the tail of l x under suitable conditions on the offspring distribution in particular our results hold if the offspring distribution has an exponential moment

the alexander orbach conjecture holds in high dimensions we examine the incipient infinite cluster iic of critical percolation in regimes where mean field behavior has been established namely when the dimension d is large enough or when d 6 and the lattice is sufficiently spread out we find that random walk on the iic exhibits anomalous diffusion with the spectral dimension d s frac 4 3 that is p t x x t 2 3 o 1 this establishes a conjecture of alexander and orbach j phys lett paris 43 625 631 1982 en route we calculate the one arm exponent with respect to the intrinsic distance

moment analysis for localization in random schrodinger operators we study localization effects of disorder on the spectral and dynamical properties of schrodinger operators with random potentials the new results include exponentially decaying bounds on the transition amplitude and related projection kernels including in the mean these are derived through the analysis of fractional moments of the resolvent which are finite due to the resonance diffusing effects of the disorder the main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the lifshitz krein spectral shift associated with the local potential terms the difficulty is avoided here through the use of a weak l 1 estimate concerning the boundary value distribution of resolvents of maximally dissipative operators combined with standard tools of relative compactness theory

percolation on infinite graphs and isoperimetric inequalities we consider the bernoulli bond percolation process with parameter p on infinite graphs and we give a general criterion for bounded degree graphs to exhibit a non trivial percolation threshold based either on a single isoperimetric inequality if the graph has a bi infinite geodesic or two isoperimetric inequalities if the graph has not a bi infinite geodesic this new criterion extends previous criteria and brings together a large class of amenable graphs such as regular lattices and non amenable graphs such trees we also study the finite connectivity in graphs satisfying the new general criterion and show that graphs in this class with a bi infinite geodesic always have finite connectivity functions with exponential decay when p is sufficiently close to one on the other hand we show that there are graphs in the same class with no bi infinite geodesic for which the finite connectivity decays sub exponentially down to polynomially in the highly supercritical phase even for p arbitrarily close to one

the scaling limit of the incipient infinite cluster in high dimensional percolation ii integrated super brownian excursion for independent nearest neighbor bond percolation on zd with d 6 we prove that the incipient infinite cluster s two point function and three point function converge to those of integrated super brownian excursion ise in the scaling limit the proof is based on an extension of the new expansion for percolation derived in a previous paper and involves treating the magnetic field as a complex variable a special case of our result for the two point function implies that the probability that the cluster of the origin consists of n sites at the critical point is given by a multiple of n 3 2 plus an error term of order n 3 2 e with e 0 this is a strong version of the statement that the critical exponent is given by 2

mean field behavior for the survival probability and the percolation point to surface connectivity we consider the critical survival probability up to time t for oriented percolation and the contact process and the point to surface of the ball of radius t connectivity for critical percolation let t denote both quantities we prove in a unifled fashion that if t exhibits a power law and both the two point function and its certain restricted version exhibit the same mean fleld behavior then t t i1 for the time oriented models with d 4 and t t i2 for percolation with d 7

tree graph inequalities and critical behavior in percolation models various inequalities are derived and used for the study of the critical behavior in independent percolation models in particular we consider the critical exponent associated with the expected cluster sizex and the structure of then site connection probabilities n x1 xn it is shown that quite generally 1 the upper critical dimension above which attains the bethe lattice value 1 is characterized both in terms of the geometry of incipient clusters and a diagramatic convergence condition for homogeneousd dimensional lattices with x y o x y d 2 atp p c our criterion shows that 1 if 6 d 3 the connectivity functions n are generally bounded by tree diagrams which involve the two point function we conjecture that above the critical dimension the asymptotic behavior of n in the critical regime is actually given by such tree diagrams modified by a nonsingular vertex factor other results deal with the exponential decay of the cluster size distribution and the function 2 x y

random cluster representation for the blume capel model we present in this paper a way to perform the mapping of the spin 1 blume capel model into a random cluster model and analyze thermodynamic properties of the former model in terms of geometric properties of clusters generated in the random cluster representation it is shown that there are two different relevant types of cluster and that one of them is the exact analogue of the type of cluster generated in the ising model we use this result to derive expressions for thermodynamical properties on the second order transition line which are equivalent to the ones found in the ising model the other type of cluster is responsible for the first order transitions and we may see the tricritical point as a point where both types of cluster compete on the same footing

hyperscaling inequalities for the contact process and oriented percolation the contact process and oriented percolation are expected to exhibit the same critical behavior in any dimension above their upper critical dimension dc they exhibit the same critical behavior as the branching process assuming existence of the critical exponents we prove a pair of hyperscaling inequalities which together with the results of refs 16 and 18 implies dc 4

on the upper critical dimension of lattice trees and lattice animals we give a rigorous proof of mean field critical behavior for the susceptibility 1 2 and the correlation length v 1 4 for models of lattice trees and lattice animals in two cases i for the usual model with trees or animals constructed from nearest neighbor bonds in sufficiently high dimensions and ii for a class of spread out or long range models in which trees and animals are constructed from bonds of various lengths above eight dimensions this provides further evidence that for these models the upper critical dimension is equal to eight the proof involves obtaining an infrared bound and showing that a certain square diagram is finite at the critical point and uses an expansion related to the lace expansion for the self avoiding walk

percolation and contact processes with low dimensional inhomogeneity we consider inhomogeneous nearest neighbor bernoulli bond percolation on z d where the bonds in a fixed s dimensional hyperplane 1 s d 1 have density p 1 and all other bonds have fixed density p c z d the homogeneous percolation critical value for s 2 it is natural to conjecture that there is a new critical value p c s z d for p 1 strictly between p c z d and p c z s we prove this for large d and 2 s d 3 for s 1 it is natural to conjecture that p c 1 z d 1 as shown for d 2 by zhang we prove this for large d related results for the contact process are also presented

a percolation problem for 1 valued strongly mixing random fields on ℤ d we present here an upper estimate for the probability that the origin is connected to the boundary of the cube x i k 1 i d under the condition that the expectation e r w 0 d 1 is finite where w 0 is the cluster of the origin and r w 0 is its radius r w 0 max left left x i right 1 leqq i leqq d x in w 0 right the upper estimate is given in terms of the mixing coefficient in particular if the mixing coefficient decays exponentially then our upper estimate supply an almost exponential decay of the probability in question it decays faster than exp ck log k as k for some positive constant c as an example we discuss the two dimensional ising model except at the critical point by using our result we show the above almost exponential decay for parameters h satisfying h 4 beta 1 beta c beta v0

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

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

mean field criticality for percolation on planar non amenable graphs the critical exponents and are proved to exist and to take their mean field values for independent percolation on the following classes of infinite locally finite connected transitive graphs 1 non amenable planar with one end 2 unimodular with infinitely many ends

an intermediate phase with slow decay of correlations in one dimensional 1 x y 2 percolation ising and potts models we rigorously establish the existence of an intermediate ordered phase in one dimensional 1 x y 2 percolation ising and potts models the ising model truncated two point function has a power law decay exponent which ranges from its low and high temperature value of two down to zero as the inverse temperature and nearest neighbor coupling vary similar results are obtained for percolation and potts models

constructive fractional moment criteria for localization in random operators abstract we present a family of finite volume criteria which cover the regime of exponential decay for the fractional moments of green functions of operators with random potentials such decay is a technically convenient characterization of localization for it is known to imply spectral localization absence of level repulsion dynamical localization and a related condition which plays a significant role in the quantization of the hall conductance in two dimensional fermi gases the constructive criteria also preclude fast power law decay of the green functions at mobility edges

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

correlation length and its critical exponent for percolation processes some critical exponent inequalities are given involving the correlation length of site percolation processes on d in particular it is shown thatv 2 d which implies that the critical exponentv cannot take its mean field value for the three dimensional percolation processes

some general results concerning the critical exponents of percolation processes in this paper we will give some results concerning the critical exponents of percolation processes which are valid for any model these results show that in several respects the behavior which occurs for percolation on the binary tree provides bounds on one side for what happens in general these results and their proofs are closely related to their analogues for the ising model

convergence of mean field approximations in site percolation and application of cam to d 1 further neighbors percolation problem we study the mean field approximation in the site percolation problem using the analog of the simon lieb inequality we show that the mean field critical probability is convergent to the exact value when the size of clusters tends to infinity applying this approximation to the one dimensional further neighbor percolation problem and calculating some critical coefficients we prove that the asymptotic scaling relations predicted by the coherent anomaly method are satisfied

universal and dynamic locally repairable codes with maximal recoverability via sum rank codes locally repairable codes lrcs are considered with equal or unequal localities local distances and local field sizes an explicit two layer architecture with a sum rank outer code is obtained having disjoint local groups and achieving maximal recoverability mr for all families of local linear codes mds or not simultaneously up to a specified maximum locality r furthermore the local linear codes thus the localities local distances and local fields can be efficiently and dynamically modified without global recoding or changes in architecture or outer code while preserving the mr property easily adapting to new configurations in storage or new hot and cold data in addition local groups and file components can be added removed or updated without global recoding the construction requires global fields of size roughly g r for g local groups and maximum or specified locality r for equal localities these global fields are smaller than those of previous mr lrcs when r leq h global parities for unequal localities they provide an exponential field size reduction on all previous best known mr lrcs for bounded localities and a large number of local groups the global erasure correction complexity of the given construction is comparable to that of tamo barg codes or reed solomon codes with local replication while local repair is as efficient as for the cartesian product of the local codes reed solomon codes with local replication and cartesian products are recovered from the given construction when r 1 and h 0 respectively the given construction can also be adapted to provide hierarchical mr lrcs for all types of hierarchies and parameters finally subextension subcodes and sum rank alternant codes are introduced to obtain further exponential field size reductions at the expense of lower information rates

random graph asymptotics on high dimensional tori we investigate the scaling of the largest critical percolation cluster on a large d dimensional torus for nearest neighbor percolation in sufficiently high dimensions or when d 6 for sufficiently spread out percolation we use a relatively simple coupling argument to show that this largest critical cluster is with high probability bounded above by a large constant times v 2 3 and below by a small constant times v 2 3 log v 4 3 where v is the volume of the torus we also give a simple criterion in terms of the subcritical percolation two point function on mathbb z d under which the lower bound can be improved to small constant times v 2 3 i e we prove random graph asymptotics for the largest critical cluster on the high dimensional torus this establishes a conjecture by 1 apart from logarithmic corrections we discuss implications of these results on the dependence on boundary conditions for high dimensional percolation

gaussian limit for critical oriented percolation in high dimensions in this paper we consider the spread out oriented bond percolation models inz d z withd 4 and the nearest neighbor oriented bond percolation model in sufficiently high dimensions let n n 1 2 be the random measures defined onr d by eta n a sum limits x in z d 1 a x sqrt n 1 0 0 to x n the mean of n denoted by bar eta n is the measure defined by bar eta n a e p eta n a we use the lace expansion method to show that the sequence of probability measures bar eta n r d 1 bar eta n converges weakly to a gaussian limit asn for everyp in the subcritical regime as well as the critical regime of these percolation models also we show that for these models the parallel correlation length xi p p c p 1 asp pc

mean field critical behaviour for correlation length for percolation in high dimensions extending the method of 27 we prove that the corrlation length of independent bond percolation models exhibits mean field type critical behaviour i e p p c p 1 2 asp p c in two situations i for nearest neighbour independent bond percolation models on ad dimensional hypercubic lattice d withd sufficiently large and ii for a class of spread out independent bond percolation models which are believed to belong to the same universality class as the nearest neighbour model in more than six dimensions the proof is based on and extends a method developed in 27 where it was used to prove the triangle condition and hence mean field behaviour of the critical exponents and 2 for the above two cases

random walk on the incipient infinite cluster for oriented percolation in high dimensions we consider simple random walk on the incipient infinite cluster for the spread out model of oriented percolation on mathbb z d times mathbb z in dimensions d 6 we obtain bounds on exit times transition probabilities and the range of the random walk which establish that the spectral dimension of the incipient infinite cluster is frac 4 3 and thereby prove a version of the alexander orbach conjecture in this setting the proof divides into two parts one part establishes general estimates for simple random walk on an arbitrary infinite random graph given suitable bounds on volume and effective resistance for the random graph a second part then provides these bounds on volume and effective resistance for the incipient infinite cluster in dimensions d 6 by extending results about critical oriented percolation obtained previously via the lace expansion

discontinuity of the percolation density in one dimensional 1 x y 2 percolation models we consider one dimensional percolation models for which the occupation probability of a bond kx y has a slow power decay as a function of the bond s length for independent models and with suitable reformulations also for more general classes of models it is shown that i no percolation is possible if for short bondskx y p 1 this dichotomy resembles one for the magnetization in 1 x y 2 ising models which was first proposed by thouless and further supported by the renormalization group flow equations of anderson yuval and hamann the proofs of the above percolation phenomena involve rigorous renormalization type arguments of a different sort

rigorous studies of critical behavior the task addressed in the talk is the analysis of the critical behavior in statistical mechanical systems and related problems in the theory of random fields while it would have been very useful to have mathematical means of approaching this subject at the level of generality presented here in professor dobrushin s analysis of the high temperature phases we find that the richness of the behavior exhibited at low temperatures calls for more restricted treatments of special models nevertheless it was found that the somewhat delicate issues of the phase structure and the critical behavior may be approached by methods which apply to a number of systems in particular the picture offered by random walk representations led to very useful insights for the study of ferromagnetic systems 4 d field theory and percolation models it is hoped that random surface expansions might play a somewhat similar role for gauge fields and some models of spin glasses

the cluster structure in collapsing animals in this paper i revisit the connection between edge percolation and the collapse transition in lattice animals it was shown by domb 1976 j phys a math gen 9 l141 that the critical percolation point is a transition in a certain model of self interacting animals i extend this result by showing that the free energy of lattice animals in the cycle contact model is non analytic in contact and cycle activities at points other than the critical percolation point this correspondence between percolation and collapsing animals suggests that the collapse transition in this model may be related to percolation contact collapse in animals is then studied from a percolation of clusters of contacts perspective i first argue that there is a critical activity for percolation of clusters of contacts and then investigate this numerically

phase diagram of inhomogeneous percolation with a defect plane let mathbb l be the d dimensional hypercubic lattice and let mathbb l 0 be an s dimensional sublattice with 2 le s p c d and a surface supercritical regime for p sigma p we show that sigma p is a strictly decreasing function for p in 0 p c d with a jump discontinuity at p c d we extend the aizenman barsky differential inequalities for homogeneous percolation to the inhomogeneous model and use them to prove that the susceptibility is finite inside the subcritical phase we prove that the cluster size distribution decays exponentially in the subcritical phase and sub exponentially in the supercritical phases for a model of lattice animals with a defect plane the free energy is related to functions of the inhomogeneous percolation model and we show how the percolation transition implies a non analyticity in the free energy of the animal model finally we present simulation estimates of the critical curve sigma p

supercritical percolation on nonamenable graphs isoperimetry analyticity and exponential decay of the cluster size distribution let g be a connected locally finite transitive graph and consider bernoulli bond percolation on g we prove that if g is nonamenable and p p c g then there exists a positive constant c p such that begin aligned mathbf p p n le k infty le e c p n end aligned for every n ge 1 where k is the cluster of the origin we deduce the following two corollaries

optimal locally repairable codes of distance 3 and 4 via cyclic codes like classical block codes a locally repairable code also obeys the singleton type bound we call a locally repairable code optimal if it achieves the singleton type bound in the breakthrough work of tamo and barg several classes of optimal locally repairable codes were constructed via subcodes of reed solomon codes thus the lengths of the codes given by tamo and barg are upper bounded by the code alphabet size q recently it was proved through the extension of construction by tamo and barg that the length of q ary optimal locally repairable codes can be q 1 by jin et al surprisingly barg et al presented a few examples of q ary optimal locally repairable codes of small distance and locality with code length achieving roughly q 2 very recently it was further shown in the work of li et al that there exist q ary optimal locally repairable codes with the length bigger than q 1 and the distance proportional to n thus it becomes an interesting and challenging problem to construct new families of q ary optimal locally repairable codes of length bigger than q 1 in this paper we construct a class of optimal locally repairable codes of distances 3 and 4 with unbounded length i e length of the codes is independent of the code alphabet size our technique is through cyclic codes with particular generator and parity check polynomials that are carefully chosen

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

random subgraphs of finite graphs ii the lace expansion and the triangle condition in a previous paper we defined a version of the percolation triangle condition that is suitable for the analysis of bond percolation on a finite connected transitive graph and showed that this triangle condition implies that the percolation phase transition has many features in common with the phase transition on the complete graph in this paper we use a new and simplified approach to the lace expansion to prove quite generally that for finite graphs that are tori the triangle condition for percolation is implied by a certain triangle condition for simple random walks on the graph the latter is readily verified for several graphs with vertex set 0 1 r 1 n including the hamming cube on an alphabet of r letters the n cube for r 2 the n dimensional torus with nearest neighbor bonds and n sufficiently large and the n dimensional torus with n 6 and sufficiently spread out long range bonds the conclusions of our previous paper thus apply to the percolation phase transition for each of the above examples

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

feature importance in nonlinear embeddings fine applications in digital pathology quantitative histomorphometry qh refers to the process of computationally modeling disease appearance on digital pathology images by extracting hundreds of image features and using them to predict disease presence or outcome since constructing a robust and interpretable classifier is challenging in a high dimensional feature space dimensionality reduction dr is often implemented prior to classifier construction however when dr is performed it can be challenging to quantify the contribution of each of the original features to the final classification result we have previously presented a method for scoring features based on their importance for classification on an embedding derived via principal components analysis pca however nonlinear dr involves the eigen decomposition of a kernel matrix rather than the data itself compounding the issue of classifier interpretability in this paper we present feature importance in nonlinear embeddings fine an extension of our pca based feature scoring method to kernel pca kpca as well as several nldr algorithms that can be cast as variants of kpca fine is applied to four digital pathology datasets to identify key qh features for predicting the risk of breast and prostate cancer recurrence measures of nuclear and glandular architecture and clusteredness were found to play an important role in predicting the likelihood of recurrence of both breast and prostate cancers compared to the t test fisher score and gini index fine was able to identify a stable set of features that provide good classification accuracy on four publicly available datasets from the nips 2003 feature selection challenge

explicit construction of optimal locally recoverable codes of distance 5 and 6 via binary constant weight codes in a paper by guruswami et al it was shown that the length n of a q ary linear locally recoverable code with distance d geqslant 5 is upper bounded by o dq 3 thus it is a challenging problem to construct q ary locally recoverable codes with distance d geqslant 5 and length approaching the upper bound the same paper also gave an algorithmic construction of q ary locally recoverable codes with locality r and length n omega r q 2 for d 5 and 6 where omega r means that the implicit constant depends on locality r in this paper we present an explicit construction of q ary locally recoverable codes of distance d 5 and 6 via binary constant weight codes it turns out that 1 our construction is simpler and more explicit and 2 the length of our codes is greater than previously known

gap exponents for percolation processes with triangle condition a study is made of the gap exponents for percolation processes with the triangle condition in the subcritical region it is show that the gaps are given by t 2 fort 2 3 scaling theory predicts thatp p c 0 s p p c p ande p 1 c 0 c 0 s p p c p 3 wheres p is the typical cluster size it is found that p c p p p c 0 s p 1 p c p 1 2 and p c p 3 e p 1 c 0 c 0 s p 1 p c p 3 4

decay properties of the connectivity for mixed long range percolation models on z d in this short note we consider mixed short long range independent bond percolation models on k d let puv be the probability that the edge u v will be open allowing a x y dependent length scale and using a multi scale analysis due to aizenman and newman we show that the long distance behavior of the connectivity xy is governed by the probability pxy the result holds up to the critical point

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

a least squares framework for component analysis over the last century component analysis ca methods such as principal component analysis pca linear discriminant analysis lda canonical correlation analysis cca locality preserving projections lpp and spectral clustering sc have been extensively used as a feature extraction step for modeling classification visualization and clustering ca techniques are appealing because many can be formulated as eigen problems offering great potential for learning linear and nonlinear representations of data in closed form however the eigen formulation often conceals important analytic and computational drawbacks of ca techniques such as solving generalized eigen problems with rank deficient matrices e g small sample size problem lacking intuitive interpretation of normalization factors and understanding commonalities and differences between ca methods this paper proposes a unified least squares framework to formulate many ca methods we show how pca lda cca lpp sc and its kernel and regularized extensions correspond to a particular instance of least squares weighted kernel reduced rank regression ls wkrrr the ls wkrrr formulation of ca methods has several benefits 1 provides a clean connection between many ca techniques and an intuitive framework to understand normalization factors 2 yields efficient numerical schemes to solve ca techniques 3 overcomes the small sample size problem 4 provides a framework to easily extend ca methods we derive weighted generalizations of pca lda sc and cca and several new ca techniques

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

phase transition for the speed of the biased random walk on the supercritical percolation cluster we prove the sharpness of the phase transition for the speed in biased random walk on the supercritical percolation cluster on d that is for each d 2 and for any supercritical parameter p pc we prove the existence of a critical strength for the bias such that below this value the speed is positive and above the value it is zero we identify the value of the critical bias explicitly and in the subballistic regime we find the polynomial order of the distance moved by the particle each of these conclusions is obtained by investigating the geometry of the traps that are most effective at delaying the walk a key element in proving our results is to understand that on large scales the particle trajectory is essentially one dimensional we prove such a dynamic renormalization statement in a much stronger form than was previously known 2013 wiley periodicals inc

surface order large deviations for ising potts and percolation models we derive uniform surface order large deviation estimates for the block magnetization in finite volume ising or potts models with plus or free or a combination of both boundary conditions in the phase coexistence regime ford 3 the results are valid up to a limit of slab thresholds conjectured to agree with the critical temperature our arguments are based on the renormalization of the random cluster model withq 1 andd 3 and on corresponding large deviation estimates for the occurrence in a box of a largest cluster with density close to the percolation probability the results are new even for the case of independent percolation q 1 as a byproduct of our methods we obtain further results in the fk model concerning semicontinuity inp andq of the percolation probability the second largest cluster in a box and the tail of the finite cluster size distribution

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

potts models and random cluster processes with many body interactions known differential inequalities for certain ferromagnetic potts models with pair interactions may be extended to potts models with many body interactions as a major application of such differential inequalities we obtain necessary and sufficient conditions on the set of interactions of such a potts model in order that its critical point be astrictly monotonic function of the strengths of interactions the method yields some ancillary information concerning the equality of certain critical exponents for potts models this amounts to a small amount of rigorous universality these results are achieved in the context of a fortuin kasteleyn representation of potts models with many body interactions for such a potts model the corresponding random cluster process is a random hypergraph

construction of the incipient infinite cluster for spread out oriented percolation above 4 1 dimensions we construct the incipient infinite cluster measure iic for sufficiently spread out oriented percolation on d for d 1 4 1 we consider two different constructions for the first construction we define n e by taking the probability of the intersection of an event e with the event that the origin is connected to x n d summing this probability over x d and normalising the sum to get a probability measure we let n and prove existence of a limiting measure the iic for the second construction we condition the connected cluster of the origin in critical oriented percolation to survive to time n and let n under the assumption that the critical survival probability is asymptotic to a multiple of n 1 we prove existence of a limiting measure with in addition we study the asymptotic behaviour of the size of the level set of the cluster of the origin and the dimension of the cluster of the origin under our methods involve minor extensions of the lace expansion methods used in a previous paper to relate critical oriented percolation to super brownian motion for d 1 4 1

constructions of optimal r delta locally repairable codes via constacyclic codes locally repairable codes lrcs are introduced in distributed storage systems due to their low repair overhead an lrc is called optimal if its minimum distance attains the singleton like upper bound chen et al 2018 recently studied the constructions of optimal r delta lrcs with length n mid q 1 and r delta 1 mid n where many classes of optimal cyclic constructions were obtained in this paper by employing constacyclic mds codes we construct seven classes of optimal r delta lrcs with new parameters after adding these new optimal lrcs via constacyclic codes we have completely obtained all optimal r delta lrcs with length n mid q 1 and r delta 1 mid n for all possible parameters for the completeness in the coding theory it is worth noting that the optimal constacyclic lrcs with new parameters provide more alternatives to cyclic lrcs in the practical demands of distributed storage systems where specific values of n k r and delta are required moreover constacyclic lrcs also possess the encoding and decoding efficiency as cyclic lrcs

finite volume fractional moment criteria for anderson localization a technically convenient signature of localization exhibited by discrete operators with random potentials is exponential decay of the fractional moments of the green function within the appropriate energy ranges known implications include spectral localization absence of level repulsion strong form of dynamical localization and a related condition which plays a significant role in the quantization of the hall conductance in two dimensional fermi gases we present a family of finite volume criteria which under some mild restrictions on the distribution of the potential cover the regime where the fractional moment decay condition holds the constructive criteria permit to establish this condition at spectral band edges provided there are sufficient lifshitz tail estimates on the density of states they are also used here to conclude that the fractional moment condition and thus the other manifestations of localization are valid throughout the regime covered by the multiscale analysis in the converse direction the analysis rules out fast power law decay of the green functions at mobility edges

gaussian fluctuations of connectivities in the subcritical regime of percolation we consider thed dimensional bernoulli bond percolation model and prove the following results for allp pc the finite volume conditional measures defined by requiring the existence of a dual path between opposing faces of the boundary converge in the infinite volume limit to the standard bernoulli measure

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

multiplicity of phase transitions and mean field criticality on highly non amenable graphs we consider independent percolation ising and potts models and the contact process on infinite locally finite connected graphs

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

the scaling limit of lattice trees in high dimensions we prove that above eight dimensions the scaling limit of sufficiently spread out lattice trees is the variant of super brownian motion known as integrated super brownian excursion ise as conjectured by aldous the same is true for nearest neighbour lattice trees in sufficiently high dimensions the proof uses the lace expansion

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

laplacian eigenmaps for dimensionality reduction and data representation one of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data we consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space drawing on the correspondence between the graph laplacian the laplace beltrami operator on the manifold and the connections to the heat equation we propose a geometrically motivated algorithm for representing the high dimensional data the algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering some potential applications and illustrative examples are discussed

subspace learning for unsupervised feature selection via matrix factorization dimensionality reduction is an important and challenging task in machine learning and data mining feature selection and feature extraction are two commonly used techniques for decreasing dimensionality of the data and increasing efficiency of learning algorithms specifically feature selection realized in the absence of class labels namely unsupervised feature selection is challenging and interesting in this paper we propose a new unsupervised feature selection criterion developed from the viewpoint of subspace learning which is treated as a matrix factorization problem the advantages of this work are four fold first dwelling on the technique of matrix factorization a unified framework is established for feature selection feature extraction and clustering second an iterative update algorithm is provided via matrix factorization which is an efficient technique to deal with high dimensional data third an effective method for feature selection with numeric data is put forward instead of drawing support from the discretization process fourth this new criterion provides a sound foundation for embedding kernel tricks into feature selection with this regard an algorithm based on kernel methods is also proposed the algorithms are compared with four state of the art feature selection methods using six publicly available datasets experimental results demonstrate that in terms of clustering results the proposed two algorithms come with better performance than the others for almost all datasets we experimented with here highlightspropose a new feature selection based on matrix factorization present a fast convergent algorithm for matrix factorization on certain constraints incorporate kernel tricks into feature selection problems construct a unified framework for feature extraction feature selection and clustering

critical exponents for the contact process under the triangle condition we show that a continuous time version of the triangle condition for percolation implies mean field values for several contact process critical exponents our results support the belief that the upper critical spatial dimension for the contact process is four

critical two point functions and the lace expansion for spread out high dimensional percolation and related models we consider spread out models of self avoiding walk bond percolation lattice trees and bond lattice animals on mathbb z d having long finite range connections above their upper critical dimensions d 4 self avoiding walk d 6 percolation and d 8 trees and animals the two point functions for these models are respectively the generating function for self avoiding walks from the origin to x in mathbb z d the probability of a connection from 0 to x and the generating function for lattice trees or lattice animals containing 0 and x we use the lace expansion to prove that for sufficiently spread out models above the upper critical dimension the two point function of each model decays at the critical point as a multiple of x 2 d as x to infty we use a new unified method to prove convergence of the lace expansion the method is based on x space methods rather than the fourier transform our results also yield unified and simplified proofs of the bubble condition for self avoiding walk the triangle condition for percolation and the square condition for lattice trees and lattice animals for sufficiently spread out models above the upper critical dimension

on the range of validity of the 6 e expansion for percolation in 6 epsilon dimensions the critical properties of percolation can be computed starting from a phi 3 theory having the same symmetry of the one state potts model the authors argue that this fixed point must be unstable with respect to a phi 4 interaction in two dimensions therefore a phi 4 perturbation is relevant this suggestion is confirmed by an explicit two loops computation

percolation of the site random cluster model by monte carlo method we propose a site random cluster model by introducing an additional cluster weight in the partition function of the traditional site percolation to simulate the model on a square lattice we combine the color assignation and the swendsen wang methods to design a highly efficient cluster algorithm with a small critical slowing down phenomenon to verify whether or not it is consistent with the bond random cluster model we measure several quantities such as the wrapping probability re the percolating cluster density p and the magnetic susceptibility per site p as well as two exponents such as the thermal exponent yt and the fractal dimension yh of the percolating cluster we find that for different exponents of cluster weight q 1 5 2 2 5 3 3 5 and 4 the numerical estimation of the exponents yt and yh are consistent with the theoretical values the universalities of the site random cluster model and the bond random cluster model are completely identical for larger values of q we find obvious signatures of the first order percolation transition by the histograms and the hysteresis loops of percolating cluster density and the energy per site our results are helpful for the understanding of the percolation of traditional statistical models

geometric analysis of φ4 fields and ising models parts i and ii we provide here the details of the proof announced in 1 that ind 4 dimensions the even 4 euclidean field theory with a lattice cut off is inevitably free in the continuum limit in the single phase regime the analysis is nonperturbative and is based on a representation of the field variables or spins in ising systems as source sink creation operators in a system of random currents which may be viewed as the mediators of correlations in this dual representation the onset of long range order is attributed to percolation in an ensemble of sourceless currents and the physical interaction in the 4 field and other aspects of the critical behavior in ising models are directly related to the intersection properties of long current clusters an insight into the criticality of the dimensiond 4 is derived from an analogy foreseen by k symanzik with the intersection properties of paths of brownian motion other results include the proof that in certain respect the critical behavior in ising models is in exact agreement with the mean field approximation in high dimensionsd 4 but not in the low dimensiond 2 for which we establish the universality of hyperscaling

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

optimal locally repairable codes via elliptic curves constructing locally repairable codes achieving singleton type bound we call them optimal codes in this paper is a challenging task and has attracted great attention in the last few years tamo and barg first gave a breakthrough result in this topic by cleverly considering subcodes of reed solomon codes thus q ary optimal locally repairable codes from subcodes of reed solomon codes given by tamo and barg have length upper bounded by q recently it was shown through extension of construction by tamo and barg that length of q ary optimal locally repairable codes can be q 1 by jin et al surprisingly it was shown by barg et al that unlike classical mds codes q ary optimal locally repairable codes could have length bigger than q 1 thus it becomes an interesting and challenging problem to construct q ary optimal locally repairable codes of length bigger than q 1 in this paper we make use of rich algebraic structures of elliptic curves to construct a family of q ary optimal locally repairable codes of length up to q 2 sqrt q it turns out that locality of our codes can be as big as 23 and distance can be linear in length

the scaling limit of the incipient infinite cluster in high dimensional percolation i critical exponents this is the first of two papers on the critical behavior of bond percolation models in high dimensions in this paper we obtain strong joint control of the critical exponents and for the nearest neighbor model in very high dimensions d 6 and for sufficiently spread out models in all dimensions d 6 the exponent describes the low frequency behavior of the fourier transform of the critical two point connectivity function while describes the behavior of the magnetization at the critical point our main result is an asymptotic relation showing that in a joint sense 0 and 2 the proof uses a major extension of our earlier expansion method for percolation this result provides evidence that the scaling limit of the incipient infinite cluster is the random probability measure on d known as integrated super brownian excursion ise in dimensions above 6 in the sequel to this paper we extend our methods to prove that the scaling limits of the incipient infinite cluster s two point and three point functions are those of ise for the nearest neighbor model in dimensions d 6

dynamic critical behavior of a swendsen wang type algorithm for the ashkin teller model we study the dynamic critical behavior of a swendsen wang type algorithm for the ashkin teller model we find that the li sokal bound on the autocorrelation time int const xc h holds along the self dual curve of the symmetric ashkin teller model and is almost but not quite sharp the ratio int c h appears to tend to infinity either as a logarithm or as a small power 0 05 p 0 12 in an appendix we discuss the problem of extracting estimates of the exponential autocorrelation time

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

quenched invariance principles for random walks and elliptic diffusions in random media with boundary via a dirichlet form extension theorem and making full use of two sided heat kernel estimates we establish quenched invariance principles for random walks in random environments with a boundary in particular we prove that the random walk on a supercritical percolation cluster or amongst random conductances bounded uniformly from below in a half space quarter space etc converges when rescaled diffusively to a reflecting brownian motion which has been one of the important open problems in this area we establish a similar result for the random conductance model in a box which allows us to improve existing asymptotic estimates for the relevant mixing time furthermore in the uniformly elliptic case we present quenched invariance principles for domains with more general boundaries

generalized approach to the non backtracking lace expansion the lace expansion is a powerful perturbative technique to analyze the critical behavior of random spatial processes such as the self avoiding walk percolation and lattice trees and animals the non backtracking lace expansion noble is a modification that allows us to improve its applicability in the nearest neighbor setting on the mathbb z d lattice for percolation lattice trees and lattice animals the noble gives rise to a recursive formula that we study in this paper at a general level we state assumptions that guarantee that the solution of this recursive formula satisfies the infrared bound in two related papers we show that these conditions are satisfied for percolation in d ge 11 for lattice trees in d ge 16 and for lattice animals in d ge 18

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

high dimensional incipient infinite clusters revisited the incipient infinite cluster iic measure is the percolation measure at criticality conditioned on the cluster of the origin to be infinite using the lace expansion we construct the iic measure for high dimensional percolation models in three different ways extending previous work by the second named author and jarai we show that each construction yields the same measure indicating that the iic is a robust object furthermore our constructions apply to spread out versions of both finite range and long range percolation models we also get estimates on structural properties of the iic such as the volume of the intersection between the iic and euclidean balls