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

counterexamples in percolation the site percolation critical probabilities ph and pt are unequal for a class of fully triangulated graphs a family of fully triangulated graphs is given for which the critical percolation probabilities pt and ph are unequal and the clusters per site function considered by sykes and essam 1964 is a polynomial the graphs are modifications of an example of van den berg 1981

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

percolation theory on pairs of matching lattices an important magnitude in percolation theory is the critical probability which is defined as the supremum of those values of the occupation probability p for which only finite clusters occur in 1964 sykes and essam obtained the relation p s c l p s c l 1 where l and l are a pair of matching lattices and p s c denotes the critical probability site case the proof was not complete but based on certain assumptions about the mean number of clusters though sykes and essam suggested that the above relation holds for all mosaics i e multiply connected planar graphs and decorated mosaics we have constructed a counterexample subsequently for a more restricted class of graphs an alternative derivation of the sykes essam relation is given this time based on the usual assumption that below the critical probability the mean cluster size is finite the latter assumption is also used to prove for some nontrivial subgraphs of the simple quadratic lattice s that their critical probability is equal

logarithmic corrections and finite size scaling in the two dimensional 4 state potts model we analyze the scaling and finite size scaling behavior of the two dimensional 4 state potts model we find new multiplicative logarithmic corrections for the susceptibility in addition to the already known ones for the specific heat we also find additive logarithmic corrections to scaling some of which are universal we have checked the theoretical predictions at criticality and off criticality by means of high precision monte carlo data

critical frontier of the potts and percolation models on triangular type and kagome type lattices i closed form expressions we consider the potts model and the related bond site and mixed site bond percolation problems on triangular type and kagome type lattices and derive closed form expressions for the critical frontier for triangular type lattices the critical frontier is known usually derived from a duality consideration in conjunction with the assumption of a unique transition our analysis however is rigorous and based on an established result without the need of a uniqueness assumption thus firmly establishing all derived results for kagome type lattices the exact critical frontier is not known we derive a closed form expression for the potts critical frontier by making use of a homogeneity assumption the closed form expression is unique and we apply it to a host of problems including site bond and mixed site bond percolations on various lattices it yields exact thresholds for site percolation on kagome martini and other lattices and is highly accurate numerically in other applications when compared to numerical determination

percolation threshold is not a decreasing function of the average coordination number it is commonly believed that the percolation critical probability is a monotonically decreasing function of the average coordination number for periodic lattice graphs in the same dimension this paper provides counterexamples a pair of planar lattices for which the bond percolation critical probabilities and average coordination numbers are in the same order and a pair for which the site percolation critical probabilities and average coordination numbers are in the same order these counterexamples confirm the existence of this counterintuitive phenomenon which was observed in one case in numerical estimates by van der marck

percolation transitions in two dimensions we investigate bond and site percolation models on several two dimensional lattices numerically by means of transfer matrix calculations and monte carlo simulations the lattices include the square triangular honeycomb kagome and diced lattices with nearest neighbor bonds and the square lattice with nearest and next nearest neighbor bonds results are presented for the bond percolation thresholds of the kagome and diced lattices and the site percolation thresholds of the square honeycomb and diced lattices we also include the bond and site percolation thresholds for the square lattice with nearest and next nearest neighbor bonds we find that corrections to scaling behave according to the second temperature dimension xt2 4 predicted by the coulomb gas theory and the theory of conformal invariance in several cases there is evidence for an additional term with the same exponent but modified by a logarithmic factor only for the site percolation problem on the triangular lattice does such a logarithmic term appear to be small or absent the amplitude of the power law correction associated with xt2 4 is found to be dependent on the orientation of the lattice with respect to the cylindrical geometry of the finite systems

bond percolation on subsets of the square lattice and the threshold between one dimensional and two dimensional behaviour for each value of pi in 1 2 1 the author constructs a subgraph r pi of the square lattice z2 such that the bond percolation process on r pi has critical probability pi it is shown that the critical probability pi a of the region x y 0 or y or a ln x 1 0 or x infinity depends in a non trivial way upon the choice of the number a pi a is a continuous decreasing function of a and satisfies pi a to 1 2 as a to infinity and pi a to 1 as a down arrow 0 a family of one dimensional bond percolation processes is also constructed whose critical probabilities range over the interval 0 1 all proofs are rigorous

the lp minkowski problem and the minkowski problem in centroaffine geometry abstract the l p minkowski problem introduced by lutwak is solved for p n 1 in the smooth category the relevant monge ampere equation 0 1 is solved for all p 1 the same equation for p 1 is also studied and solved for p n 1 1 when p n 1 the equation is interpreted as a minkowski problem in centroaffine geometry a kazdan warner type obstruction for this problem is obtained

critical frontier of the potts and percolation models on triangular type and kagome type lattices ii numerical analysis in the preceding paper one of us f y wu considered the potts model and bond and site percolation on two general classes of two dimensional lattices the triangular type and kagome type lattices and obtained closed form expressions for the critical frontier with applications to various lattice models for the triangular type lattices wu s result is exact and for the kagome type lattices wu s expression is under a homogeneity assumption the purpose of the present paper is twofold first an essential step in wu s analysis is the derivation of lattice dependent constants a b c for various lattice models a process which can be tedious we present here a derivation of these constants for subnet networks using a computer algorithm second by means of a finite size scaling analysis based on numerical transfer matrix calculations we deduce critical properties and critical thresholds of various models and assess the accuracy of the homogeneity assumption specifically we analyze the q state potts model and the bond percolation on the 3 12 and kagome type subnet lattices n ifmmode times else texttimes fi n n ifmmode times else texttimes fi n n ensuremath le 4 for which the exact solution is not known our numerical determination of critical properties such as conformal anomaly and magnetic correlation length verifies that the universality principle holds to calibrate the accuracy of the finite size procedure we apply the same numerical analysis to models for which the exact critical frontiers are known the comparison of numerical and exact results shows that our numerical values are correct within errors of our finite size analysis which correspond to 7 or 8 significant digits this in turn infers that the homogeneity assumption determines critical frontiers with an accuracy of 5 decimal places or higher finally we also obtained the exact percolation thresholds for site percolation on kagome type subnet lattices 1 ifmmode times else texttimes fi 1 n ifmmode times else texttimes fi n for 1 ensuremath le n ensuremath le 6

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

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

critical behaviour of two ising models with multispin interactions abstract we have performed extensive monte carlo simulations on the two dimensional ising models with n spin interactions described recently by debierre and turban results for n 3 models with sizes up to 128 128 are analyzed by means of finite size scaling this yields a value of the magnetic exponent y h close to 15 8 direct estimates of the temperature exponent y t do not converge convincingly however assuming the presence of logarithmic corrections such as in the 4 state potts model we obtain an improved estimate of the temperature exponent close to 3 2 in agreement with 4 state potts universality this result is further supported by an exact mapping between the n 3 model and the 4 state potts model in an anisotropic limit for the n 4 model we confirm that the phase transition is first order and we estimate the discontinuities in the energy and the magnetization

counterexamples in ab percolation halley 1983 conjectured that the ab percolation probability function attains its maximum at p 1 2 and that consequently infinite ab percolation is impossible on a bipartite graph whose standard site percolation critical probability is greater than 1 2 the authors produce counterexamples that demonstrate that both conjectures are false in fact for any positive integer n a graph may be constructed which has at least n ab percolation transitions

conformal invariance and universality in finite size scaling the universal relation between critical exponents and the amplitude of the correlation length divergence as a function of finite size at the critical point of two dimensional systems is shown to be a consequence of conformal invariance both periodic and free boundary conditions are considered

the critical manifolds of inhomogeneous bond percolation on bow tie and checkerboard lattices we give a conditional derivation of the inhomogeneous critical percolation manifold of the bow tie lattice with five different probabilities a problem that does not appear at first to fall into any known solvable class although our argument is mathematically rigorous only on a region of the manifold we conjecture that the formula is correct over its entire domain and we provide a non rigorous argument for this that employs the negative probability regime of the triangular lattice critical surface we discuss how the rigorous portion of our result substantially broadens the range of lattices in the solvable class to include certain inhomogeneous and asymmetric bow tie lattices and that if it could be put on a firm foundation the negative probability portion of our method would extend this class to many further systems including f y wu s checkerboard formula for the square lattice we conclude by showing that this latter problem can in fact be proved using a recent result of grimmett and manolescu for isoradial graphs lending strong evidence in favor of our other conjectured results this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

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

percolation on hypergraphs with four edges we study percolation on self dual hypergraphs that contain hyperedges with four bounding vertices or four edges using three different generators each containing bonds or sites with three distinct probabilities p r and t connecting the four vertices we find explicit values of these probabilities that satisfy the self duality conditions discussed by bollobas and riordan this demonstrates that explicit solutions of the self duality conditions can be found using generators containing bonds and sites with independent probabilities these solutions also provide new examples of lattices where exact percolation critical points are known one of the generators exhibits three distinct criticality solutions p r t we carry out monte carlo simulations of two of the generators on two different hypergraphs to confirm the critical values for the case of the hypergraph and uniform generator studied by wierman et al we also determine the threshold p 0 441 374 0 000 001 which falls within the tight bounds that they derived furthermore we consider a generator in which all or none of the vertices can connect and find a soluble inhomogeneous percolation system that interpolates between site percolation on the union jack lattice and bond percolation on the square lattice

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

percolation and critical o n loop configurations we study a percolation problem based on critical loop configurations of the text o n loop model on the honeycomb lattice we define dual clusters as groups of sites on the dual triangular lattice that are not separated by a loop and investigate the bond percolation properties of these dual clusters the universal properties at the percolation threshold are argued to match those of kasteleyn fortuin random clusters in the critical potts model this relation is checked numerically by means of cluster simulations of several text o n models in the range 1 ensuremath le n ensuremath le 2 the simulation results include the percolation threshold for several values of n as well as the universal exponents associated with bond dilution and the size distribution of the diluted clusters at the percolation threshold our numerical results for the exponents are in agreement with existing coulomb gas results for the random cluster model which confirms the relation between both models we discuss the renormalization flow of the bond dilution parameter p as a function of n and provide an expression that accurately describes a line of unstable fixed points as a function of n corresponding with the percolation threshold furthermore the renormalization scenario indicates the existence in a p versus n diagram of another line of fixed points at p 1 which is stable with respect to p

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

site percolation thresholds for archimedean lattices precise thresholds for site percolation on eight archimedean lattices are determined by the hull walk gradient percolation simulation method with the results p c 0 697 043 honeycomb or 6 3 0 807 904 3 12 2 0 747 806 4 6 12 0 729 724 4 8 2 0 579 498 3 4 6 0 621 819 3 4 6 4 0 550 213 3 3 4 2 and 0 550 806 3 2 4 3 4 with errors of about 3 x 10 6 the remaining archimedean lattices are the square 4 4 triangular 3 6 and kagome 3 6 3 6 for which p c is already known exactly or to a high degree of accuracy the numerical result for the 3 12 2 lattice is consistent with the exact value 1 2 sin pi 18 1 2 the values of p c for all 11 archimedean lattices as well as a number of nonuniform lattices are found to be well correlated by a nearly linear function of a generalized scher zallen filling factor this correlation is much more accurate than recently proposed correlations based solely upon coordination number

on convergence to sle6 i conformal invariance for certain models of the bond triangular type we establish convergence to sle6 of the exploration processes for the correlated bond triangular type models studied by two of us in an earlier work chayes and lei in rev math phys 19 511 565 2007 this rigorously establishes that these models are in the same universality class as the standard site percolation model on the triangular lattice in the context of these models the result is proven for all domains with boundary minkowski dimension less than two moreover the proof of convergence applies in the context of general critical 2d percolation models and for general domains under the stipulation that cardy s formula can be established for domains in this generality

generalized cell dual cell transformation and exact thresholds for percolation suggested by scullard s recent star triangle relation for correlated bond systems we propose a general cell dual cell transformation which allows in principle an infinite variety of lattices with exact percolation thresholds to be generated we directly verify scullard s site percolation thresholds and derive the bond thresholds for his martini lattice p c 1 sqrt 2 and the a lattice p c 0 625457 dots solution to p 5 ensuremath 4 p 4 3 p 3 2 p 2 ensuremath 1 0 we also present a precise monte carlo test of the site threshold for the a lattice

on polya random walks lattice green functions and the bond percolation threshold based on numerical evidence the authors conjecture a connection between the bond percolation threshold of bravais lattices in three or more dimensions and the value at the origin of a lattice green function related to the probability of return to the origin for a polya random walk

convex solutions to the mean curvature flow in this paper we study the classication of ancient convex solutions to the mean curvature ow in r n 1 an open problem related to the classication of type ii singularities is whether a convex translating solution is k rotationally symmetric for some integer 2 k n namely whether its level set is a sphere or cylinder s k 1 r n k in this paper we give an afrmative answer for entire solutions in dimension 2 in high dimensions we prove that there exist nonrotationally symmetric entire convex translating solutions but the blow down in space of any entire convex translating solution is k rotationally symmetric we also prove that the blow down in space time of an ancient convex solution which sweeps the whole space r n 1 is a shrinking sphere or cylinder

critical percolation on the torus we compute the various crossing probabilities defined by r langlands p pouliot and y saint aubin for the critical percolation on the torus

shape dependent universality in percolation abstract the shape dependent universality of the excess percolation cluster number and cross configuration probability on a torus is examined besides the aspect ratio of the torus the universality class depends upon the twist in the periodic boundary conditions which may be introduced when triangular lattices are used in simulations

on the range of bond percolation thresholds for fully triangulated graphs this paper constructs a sequence of planar periodic fully triangulated lattices which have bond percolation thresholds tending to zero as a consequence we see that bond percolation thresholds of such graphs can range from 0 to 2 sin 18 0 3473 which is the threshold of the triangular lattice rigorous bounds and simulations for other fully triangulated lattices suggest that the threshold for the triangular lattice is the largest for this class of graphs

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

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

robust game theory we present a distribution free model of incomplete information games both with and without private information in which the players use a robust optimization approach to contend with payoff uncertainty our robust game model relaxes the assumptions of harsanyi s bayesian game model and provides an alternative distribution free equilibrium concept which we call robust optimization equilibrium to that of the ex post equilibrium we prove that the robust optimization equilibria of an incomplete information game subsume the ex post equilibria of the game and are unlike the latter guaranteed to exist when the game is finite and has bounded payoff uncertainty set for arbitrary robust finite games with bounded polyhedral payoff uncertainty sets we show that we can compute a robust optimization equilibrium by methods analogous to those for identifying a nash equilibrium of a finite game with complete information in addition we present computational results

liouville property and regularity of a hessian quotient equation we are concerned with properties of convex solutions to the hessian quotient equation sn c d2 p 1 k n as our first main result we prove some regularity of strong solutions while the second states that for p 1 an entire convex solution with a quadratic growth bound must be a quadratic polynomial

exact site percolation thresholds using a site to bond transformation and the star triangle transformation i construct a two dimensional lattice on which the inhomogeneous site percolation threshold is exactly calculable and use this result to find two more lattices on which the site thresholds can be determined the primary lattice studied here the martini lattice is a hexagonal lattice with every second site transformed into a triangle the site threshold of this lattice is found to be 0 764826 i e the solution to p4 3p3 1 0 while the others have square root 5 1 2 the inverse of the golden ratio and 1 square root 2 this last solution suggests a possible approach to establishing the bound for the hexagonal site threshold pc 1 square root 2 to derive these results i solve a correlated bond problem on the hexagonal lattice by use of the star triangle transformation and then by a particular choice of correlations derived from a site to bond transformation solve the site problem on the martini lattice

random walks and percolation on trees there is a way to define an average number of branches per vertex for an arbitrary infinite locally finite tree it equals the exponential of the hausdorff dimension of the boundary in an appropriate metric its importance for probabilistic processes on a tree is shown in several ways including random walk and percolation where it provides points of phase transition

existence of entire solutions to the monge ampere equation we prove the existence of infinitely many entire convex solutions to the monge amp ere equation rm det d 2 u f in bbb r n assuming that the inhomogeneous term f ge 0 and is of polynomial growth at infinity when f satisfies the doubling condition we show that solution is of polynomial growth as an application we resolve the existence of translating solutions to a class of gauss curvature flow

bond percolation critical probability bounds derived by edge contraction a contraction principle is valid for bond percolation models if gc is obtained from g by contraction of a set of edges then the bond percolation critical probabilities satisfy pc gc or pc g the contraction principle provides relationships between graphs which do not follow from the inclusion principle application of the principle provides the following bounds 0 5000 or pc pentagon lattice or 0 6527 pc kagome lattice or 0 6180 0 3820 or pc dice lattice

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

universal formulas for percolation thresholds a power law is postulated for both site and bond percolation thresholds the formula is mathit p mathit c mathit p 0 d 1 q 1 mathrm ensuremath mathit a mathit d mathit b where d is the space dimension and q the coordination number all thresholds up to d ensuremath rightarrow ensuremath infty are found to belong to only three universality classes for two classes b 0 for site dilution while b a for bond dilution the remaining class associated with high dimensions is characterized by b 2a 1 for both sites and bonds classes are defined by a set of value for mathit p 0 a deviations from available numerical estimates at d ensuremath le 7 are within ifmmode pm else textpm fi 0 008 and ifmmode pm else textpm fi 0 0004 for high dimensional hypercubic expansions at d ensuremath ge 8 the formula is found to be also valid for ising critical temperatures textcopyright 1996 the american physical society

some exact results on bond percolation we present some exact results on bond percolation we derive a relation that specifies the consequences for bond percolation quantities of replacing each bond of a lattice by l bonds connecting the same adjacent vertices thereby yielding the lattice l this relation is used to calculate the bond percolation threshold on l we show that this bond inflation leaves the universality class of the percolation transition invariant on a lattice of dimensionality d 2 but changes it on a one dimensional lattice and quasi one dimensional infinite length strips we also present analytic expressions for the average cluster number per vertex and correlation length for the bond percolation problem on the n limits of several families of n vertex graphs finally we explore the effect of bond vacancies on families of graphs with the property of bounded diameter as n

ab percolation on close packed graphs for a class of close packed graphs the ab percolation critical probability is equal to the classical site percolation critical probability of the related graph obtained by inserting next nearest neighbour bonds as a consequence infinite ab percolation occurs for many fully triangulated graphs

site percolation thresholds in all dimensions abstract site percolation thresholds are reproduced in all dimensions for all lattices using a linear combination of two analytic terms one is the well known cayley tree percolation threshold which is believed to be exact at infinite dimension the other one is obtained from a new approach to two dimensional site dilution using a ferromagnetic ising system these two formulas are then combined with weighting factors a and 1 a respectively where a is a fitting parameter which depends only on space dimension it is equal to 0 047 at d 2 to 0 924 at d 6 and 1 at d our results agree with exact numerical estimates for any lattice in any dimension within few percent

an extension to a theorem of jorgens calabi and pogorelov 1 1 det d2u 1 in rn must be a quadratic polynomial for n 2 a classical solution is either convex or concave the result holds without the convexity hypothesis a simpler and more analytical proof along the lines of affine geometry of the theorem was later given by cheng and yau 9 the first author extended the result for classical solutions to viscosity solutions 4 it was proven by trudinger and wang in 19 that the only open convex subset of rn which admits a convex c2 solution of det d2u 1 in with limx u x is rn in this paper we give the following extension to the theorem of jorgens calabi and pogorelov let u be a convex viscosity solution of det d2u 1 outside a bounded subset of

universal formulas for percolation thresholds ii extension to anisotropic and aperiodic lattices in a recent paper we reported a universal power law for both site and bond percolation thresholds in any bravais lattice with q equivalent nearest neighbors in dimension d we now extend it to three different classes of lattices which are respectively anisotropic lattices without equivalent nearest neighbors non bravais lattices with two atom unit cells and quasicrystals the investigation is focused on d 2 and 3 due to the lack of experimental data at higher dimensions the extension to these lattices requires the substitution of q by an effective non integer value q mathrm eff in the universal law for each of the 17 lattices which constitute our sample we argue for the existence of one q mathrm eff which reproduces both the site and the percolation threshold with a deviation with respect to numerical estimates which does not exceed ensuremath mp 0 01

a localization property of viscosity solutions to the monge ampere equation and their strict convexity the purpose of this note is to show a localization property of convex viscosity solutions to the monge ampere inequality 0 1 2 n are strictly convex

duality with real space renormalization and its application to bond percolation we obtain the exact solution of the bond percolation thresholds with inhomogeneous probabilities on the square lattice our method is based on the duality analysis with real space renormalization which is a profound technique invented in the spin glass theory our formulation is a more straightforward way than that of a very recent study on the same problem r m ziff et al j phys a math theor 45 494005 2012 the resultant generic formulas from our derivation can give several estimations for the bond percolation thresholds on other lattices rather than the square lattice

exact bond percolation thresholds in two dimensions recent work in percolation has led to exact solutions for the site and bond critical thresholds of many new lattices here we show how these results can be extended to other classes of graphs significantly increasing the number and variety of solved problems any graph that can be decomposed into a certain arrangement of triangles which we call self dual gives a class of lattices whose percolation thresholds can be found exactly by a recently introduced triangle triangle transformation we use this method to generalize wierman s solution of the bow tie lattice to yield several new solutions we also give another example of a self dual arrangement of triangles that leads to a further class of solvable problems there are certainly many more such classes

scaling and universality in the spanning probability for percolation we discuss the spanning percolation probability function for three different spanning rules in general dimensions with both free and periodic boundary conditions our discussion relies on the renormalization group theory which indicates that apart from a few scale factors the scaling functions for the spanning probability are determined by the fixed point and therefore are universal for every system with the same dimensionality spanning rule aspect ratio and boundary conditions for square and rectangular systems we combine this theory with simple relations between the spanning rules and with duality arguments and find strong relations among different derivatives of the spanning function with respect to the scaling variables thus yielding several universal amplitude ratios and allowing a systematic study of the corrections to scaling both singular and analytic in the system size the theoretical predictions are numerically confirmed with excellent accuracy textcopyright 1996 the american physical society

on the number of entangled clusters we prove that the number of entangled clusters with n edges in the simple cubic lattice grows exponentially in n this answers an open question posed by grimmett and holroyd proc lond math soc 81 485 512 2000 our result has immediate implications for entanglement percolation we obtain an improved rigorous lower bound on the critical probability and we prove that the radius of the entangled component of the origin has exponentially decaying tail when p is small

simulation algorithms for the random cluster model we compare the performance of monte carlo algorithms for the simulation of the random cluster representation of the q state potts model for continuous values of q in particular we consider a local bond update method a statistical reweighting method of percolation configurations and a cluster algorithm all of which generate boltzmann statistics the dynamic exponent z of the cluster algorithm appears to be quite small and to assume the values of the swendsen wang algorithm for q 2 and 3 the cluster algorithm appears to be much more efficient than our versions of the other two methods for the simulation of the random cluster model the higher efficiency of the cluster method with respect to the local method is primarily due to the fact that the computer time usage of the local method increases more rapidly with system size the difference between the dynamic exponents is less important

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

two terminal routing games with unknown active players we analyze 2 terminal routing games with linear cost functions and with unknown number of active players we deal with both splittable and unsplittable models we prove the existence and uniqueness of a symmetric safety level equilibrium in such games and show that in many cases every player benefits from the common ignorance about the number of players furthermore we prove new theorems on existence and uniqueness of equilibrium in 2 terminal convex routing games with complete information

critical phenomena for spitzer s reversible nearest particle systems motivated by several results and open problems concerning harris basic contact process we consider the relationship between the critical behavior of the finite and infinite versions of spitzer s reversible nearest particle systems we show that the critical values for the finite and infinite systems agree but that the behavior of the two systems at the common critical value can differ the nash williams recurrence criterion for reversible markov chains is an important tool used in the proofs of the main results and we give a new treatment of that theory finally we compute several critical exponents for the nearest particle systems

random surface correlation functions truncated pair functions for free random surface models and bernoulli ensembles are examined in both cases the pair function is shown to obey ornstein zernike scaling whenever various correlation lengths of the system satisfy a nonperturbative criterion under the same conditions the transverse displacement of surfaces contributing to the pair function is shown to be normally distributed a new type of transition which concerns the width of typical surfaces is introduced and studied whenever the system is below the melting transition temperature of a related lower dimensional model the width of typical surfaces is shown to be finite a thermodynamic formalism for free random surface models is developed the formalism is used to obtain sharp estimates of the entropy of surfaces contributing to the pair function

percolation thresholds on two dimensional voronoi networks and delaunay triangulations the site percolation threshold for the random voronoi network is determined numerically with the result pc 0 714 10 0 000 02 using monte carlo simulation on periodic systems of up to 40 000 sites the result is very close to the recent theoretical estimate pc approximately 0 7151 of neher for the bond threshold on the voronoi network we find pc 0 666 931 0 000 005 implying that for its dual the delaunay triangulation pc 0 333 069 0 000 005 these results rule out the conjecture by hsu and huang that the bond thresholds are 2 3 and 1 3 respectively but support the conjecture of wierman that for fully triangulated lattices other than the regular triangular lattice the bond threshold is less than 2 sin pi 18 approximately 0 3473

conformal metrics with prescribed curvature functions on manifolds with boundary we study the dirichlet problem for a class of fully nonlinear elliptic equations related to conformal deformations of metrics on riemannian manifolds with boundary as a consequence we prove the existence of a conformal metric given its value on the boundary as a prescribed metric conformal to the induced background metric with a prescribed curvature function of the schouten tensor

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

the wetting transition in a random surface model we continue our analysis of the phase diagram of a discrete random surface with no downward fingers lying above a flat two dimensional substrate the surface is closely related to the 2d ising model and its free energy is exactly solvable in much but not all of the phase diagram there is a transition at temperaturet w from a high t infinite height or wet phase to a low t finite height or partially wet phase previously it was shown that when a parameterb related to the contact interaction is positive t w is independent ofb and there is a logarithmic specific heat divergence ast w is approached fromeither side here we show that forb 0 t w does depend onb and there isno thermodynamic singularity from the wet phase the partially wet phases forb 0 andb 0 differ in the absence or presence of a monolayer covering the entire substrate this results in a first order transition across the lineb 0 t t w

criteria for evaluation of universal formulas for percolation thresholds several universal formulas that predict approximate values for percolation thresholds of all periodic graphs have been proposed in the physics and engineering literature the existing universal formulas have been found to have substantial errors in their predictions for some lattices this paper proposes a set of desirable criteria for universal formulas to satisfy and investigates which criteria are satisfied by two bond threshold formulas and two site threshold formulas most cited in the literature the analysis is limited to lattices in two dimensions

competitive routing in networks with polynomial costs we study a class of noncooperative general topology networks shared by n users each user has a given flow which it has to ship from a source to a destination we consider a class of polynomial link cost functions adopted originally in the context of road traffic modeling and show that these costs have appealing properties that lead to predictable and efficient network flows in particular we show that the nash equilibrium is unique and is moreover efficient these properties make the polynomial cost structure attractive for traffic regulation and link pricing in telecommunication networks we finally discuss the computation of the equilibrium in the special case of the affine cost structure for a topology of parallel links

inhomogeneous bond percolation on square triangular and hexagonal lattices the star triangle transformation is used to obtain an equivalence extending over the set of all in homogeneous bond percolation models on the square triangular and hexagonal lattices among the consequences are box crossing rsw inequalities for such models with parameter values at which the transformation is valid this is a step toward proving the universality and conformality of these processes it implies criticality of such values thereby providing a new proof of the critical point of inhomogeneous systems the proofs extend to certain isoradial models to which previous methods do not apply

percolation thresholds and universal formulas a calculation of percolation thresholds of 11 two dimensional and 18 three dimensional lattices is presented among the three dimensional ones are a random lattice and its dual plus a number of aniso tropic regular lattices the results are used to test universal formulas that relate the percolation thresholds of lattices to their dimension and coordination number the evidence suggests that dimension and coordination number are not sufficient to predict percolation thresholds

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

critical probability bounds for two dimensional site percolation models the authors present three techniques for determining rigorous bounds for site percolation critical probabilities of two dimensional lattices a technique for finding lower bounds for critical probabilities of bipartite graphs is used to show that pc d or 0 5020 for the dice lattice d combining this method with kesten s duality result simplifies toth s derivation of the lower bound pc s or 0 5034 for the square lattice s the authors also present a technique for deriving upper bounds for bipartite graphs a technique of grouping sites is used to derive upper bounds for the critical probability of the hexagonal lattice h pc h or 0 8079 and pc h or pc s the grouping technique is applied to the dice lattice to find the upper bound pc d or 0 7937

a bond percolation critical probability determination based on the star triangle transformation the bond percolation critical probability of a planar graph with square and triangular faces obtained by inserting a diagonal in every other face of the square lattice is the root of 1 p 6p2 6p3 p5 0 in 0 1 which is approximately 0 404 518 the proof uses the star triangle transformation to determine the parameter value for which the percolative behaviour of the lattice and its dual lattice are identical

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

r extreme signalling for congestion control abstractin many smart city applications congestion arises in part due to the nature of signals received by individuals from a central authority in the model of marecek et al each agent uses one out of multiple resources at each time instant the per use cost of a resource depends on the number of concurrent users a central authority has up to date knowledge of the congestion across all resources and uses randomisation to provide a scalar or an interval for each resource at each time in this paper the interval to broadcast per resource is obtained by taking the minima and maxima of costs observed within a time window of length r rather than by randomisation we show that the resulting distribution of agents across resources also converges in distribution under plausible assumptions about the evolution of the population over time

potts and percolation models on bowtie lattices we give the exact critical frontier of the potts model on bowtie lattices for the case of q 1 the critical frontier yields the thresholds of bond percolation on these lattices which are exactly consistent with the results given by ziff et al j phys a 39 15083 2006 for the q 2 potts model on a bowtie a lattice the critical point is in agreement with that of the ising model on this lattice which has been exactly solved furthermore we do extensive monte carlo simulations of the potts model on a bowtie a lattice with noninteger q our numerical results which are accurate up to seven significant digits are consistent with the theoretical predictions we also simulate the site percolation on a bowtie a lattice and the threshold is s c 0 5479148 7 in the simulations of bond percolation and site percolation we find that the shape dependent properties of the percolation model on a bowtie a lattice are somewhat different from those of an isotropic lattice which may be caused by the anisotropy of the lattice

percolation in networks with voids and bottlenecks a general method is proposed for predicting the asymptotic percolation threshold of networks with bottlenecks in the limit that the subnet mesh size goes to zero the validity of this method is tested for bond percolation on filled checkerboard and stack of triangle lattices thresholds for the checkerboard lattices of different mesh sizes are estimated using the gradient percolation method while for the triangular system they are found exactly using the triangle triangle transformation the values of the thresholds approach the asymptotic values of 0 64222 and 0 53993 respectively as the mesh is made finer consistent with a direct determination based upon the predicted critical corner connection probability

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