current fluctuations of the stationary asep and six vertex model our results in this article are twofold first we consider current fluctuations of the stationary asymmetric simple exclusion process asep run for some long time t and show that they are of order t1 3 along a characteristic line upon scaling by t1 3 we establish that these fluctuations converge to the long time height fluctuations of the stationary kardar parisi zhang kpz equation that is to the baik rains distribution this result has long been predicted under the context of kpz universality and in particular extends upon a number of results in the field including the work of ferrari and spohn from 2005 when they established the same result for the tasep and the work of balazs and seppalainen from 2010 when they established the t1 3 scaling for the general asep second we introduce a class of translation invariant gibbs measures that characterizes a one parameter family of slopes for an arbitrary ferroelectric symmetric six vertex model this family of slopes corresponds to what is known as the conical singularity or tricritical point in the free energy profile for the ferroelectric six vertex model we consider fluctuations of the height function of this model on a large grid of size t and show that they too are of order t1 3 along a certain characteristic line this confirms a prediction of bukman and shore from 1995 stating that the ferroelectric six vertex model should exhibit kpz growth at the conical singularity upon scaling the height fluctuations by t1 3 we again recover the baik rains distribution in the large t limit recasting this statement in terms of the asymmetric stochastic six vertex model confirms a prediction of gwa and spohn from 1992

the critical z invariant ising model via dimers the periodic case we study a large class of critical two dimensional ising models namely critical z invariant ising models on periodic graphs example of which are the classical mathbb z 2 triangular and honeycomb lattice at the critical temperature fisher j math phys 7 1776 1781 1966 introduced a correspondence between the ising model and the dimer model on a decorated graph thus setting dimer techniques as a powerful tool for understanding the ising model in this paper we give a full description of the dimer model corresponding to the critical z invariant ising model we prove that the dimer characteristic polynomial is equal up to a constant to the critical laplacian characteristic polynomial and defines a harnack curve of genus 0 we prove an explicit expression for the free energy and for the gibbs measure obtained as weak limit of boltzmann measures

the lee yang and polya schur programs i linear operators preserving stability in 1952 lee and yang proposed the program of analyzing phase transitions in terms of zeros of partition functions linear operators preserving non vanishing properties are essential in this program and various contexts in complex analysis probability theory combinatorics and matrix theory we characterize all linear operators on finite or infinite dimensional spaces of multivariate polynomials preserving the property of being non vanishing whenever the variables are in prescribed open circular domains in particular this solves the higher dimensional counterpart of a long standing classification problem originating from classical works of hermite laguerre hurwitz and polya schur on univariate polynomials with such properties

quadri tilings of the plane we introduce quadri tilings and show that they are in bijection with dimer models on a family of graphs r arising from rhombus tilings using two height functions we interpret a sub family of all quadri tilings called triangular quadri tilings as an interface model in dimension 2 2 assigning critical weights to edges of r we prove an explicit expression only depending on the local geometry of the graph r for the minimal free energy per fundamental domain gibbs measure this solves a conjecture of kenyon invent math 150 409 439 2002 we also show that when edges of r are asymptotically far apart the probability of their occurrence only depends on this set of edges finally we give an expression for a gibbs measure on the set of all triangular quadri tilings whose marginals are the above gibbs measures and conjecture it to be that of minimal free energy per fundamental domain

arctic boundaries of the ice model on three bundle domains in this paper we consider the six vertex model at ice point on an arbitrary three bundle domain which is a generalization of the domain wall ice model on the square or equivalently of a uniformly random alternating sign matrix we show that this model exhibits the arctic boundary phenomenon whose boundary is given by a union of explicit algebraic curves this was originally predicted by colomo and sportiello j stat phys 164 1488 1523 2016 as one of the initial applications of a general heuristic that they introduced for locating arctic boundaries called the geometric tangent method our proof uses a probabilistic analysis of non crossing directed path ensembles to provide a mathematical justification of their tangent method heuristic in this case which might be of independent interest

the arctic curve of the domain wall six vertex model the problem of the form of the arctic curve of the six vertex model with domain wall boundary conditions in its disordered regime is addressed it is well known that in the scaling limit the model exhibits phase separation with regions of order and disorder sharply separated by a smooth curve called the arctic curve to find this curve we study a multiple integral representation for the emptiness formation probability a correlation function devised to detect spatial transition from order to disorder we conjecture that the arctic curve for arbitrary choice of the vertex weights can be characterized by the condition of condensation of almost all roots of the corresponding saddle point equations at the same known value in explicit calculations we restrict to the disordered regime for which we have been able to compute the scaling limit of certain generating function entering the saddle point equations the arctic curve is obtained in parametric form and appears to be a non algebraic curve in general it turns into an algebraic one in the so called root of unity cases the arctic curve is also discussed in application to the limit shape of q enumerated with 0 q 4 large alternating sign matrices in particular as q 0 the limit shape tends to a nontrivial limiting curve given by a relatively simple equation

minimizers of convex functionals arising in random surfaces we investigate c1 regularity of minimizers to f u dx in two dimensions for certain classes of non smooth convex functionals f in particular our results apply to the surface tensions that appear in recent works on random surfaces and random tilings of kenyon okounkov and others

how quickly can we sample a uniform domino tiling of the 2l times 2l square via glauber dynamics the prototypical problem we study here is the following given a 2l times 2l square there are approximately exp 4kl 2 pi ways to tile it with dominos i e with horizontal or vertical 2 times 1 rectangles where k approx 0 916 is catalan s constant kasteleyn in physica 27 1209 1225 1961 temperley and fisher in phil mag 6 1061 1063 1997 a conceptually simple even if computationally not the most efficient way of sampling uniformly one among so many tilings is to introduce a markov chain algorithm glauber dynamics where with rate 1 two adjacent horizontal dominos are flipped to vertical dominos or vice versa the unique invariant measure is the uniform one and a classical question levin et al in am math soc 2009 luby et al siam j comput 31 167 192 2001 wilson in ann appl probab 14 274 325 2004 is to estimate the time t mathrm mix it takes to approach equilibrium i e the running time of the algorithm luby et al in siam j comput 31 167 192 2001 and randall and tetali in j math phys 41 3 1598 1614 2000 fast mixing was proven t mathrm mix o l c for some finite c here we go much beyond and show that c l 2 le t mathrm mix le l 2 o 1 our result applies to rather general domain shapes not just the 2l times 2l square provided that the typical height function associated to the tiling is macroscopically planar in the large l limit under the uniform measure this is the case for instance for the temperley type boundary conditions considered in kenyon ann probab 28 759 795 2000 also our method extends to some other types of tilings of the plane for instance the tilings associated to dimer coverings of the hexagon or square hexagon lattices

dimers in piecewise temperleyan domains we study the large scale behavior of the height function in the dimer model on the square lattice richard kenyon has shown that the fluctuations of the height function on temperleyan discretizations of a planar domain converge in the scaling limit as the mesh size tends to zero to the gaussian free field with dirichlet boundary conditions we extend kenyon s result to a more general class of discretizations moreover we introduce a new factorization of the coupling function of the double dimer model into two discrete holomorphic functions which are similar to discrete fermions defined in smirnov proceedings of the international congress of mathematicians icm madrid spain 2006 ann math 2 172 1435 1467 2010 for temperleyan discretizations with appropriate boundary modifications the results of kenyon imply that the expectation of the double dimer height function converges to a harmonic function in the scaling limit we use the above factorization to extend this result to the class of all polygonal discretizations that are not necessarily temperleyan furthermore we show that quite surprisingly the expectation of the double dimer height function in the temperleyan case is exactly discrete harmonic for an appropriate choice of laplacian even before taking the scaling limit

the emergence of the electrostatic field as a feynman sum in random tilings with holes we consider random lozenge tilings on the triangular lattice with holes q 1 q n in some fixed position for each unit triangle not in a hole consider the average orientation of the lozenge covering it we show that the scaling limit of this discrete field is the electrostatic field obtained when regarding each hole q i as an electrical charge of magnitude equal to the difference between the number of unit triangles of the two different orientations inside q i this is then restated in terms of random surfaces yielding the result that the average over surfaces with prescribed height at the union of the boundaries of the holes is in the scaling limit a sum of helicoids

non colliding paths in the honeycomb dimer model and the dyson process in this paper we describe a natural family of random non intersecting discrete paths in the dimer model on the honeycomb lattice we show that when the dimer model is going to freeze this family of paths after a proper rescaling converges to the extended sine process obtained traditionally as the limit of the dyson model when the number of particles goes to infinity

arctic circles domino tilings and square young tableaux the arctic circle theorem of jockusch propp and shor asserts that uniformly random domino tilings of an aztec diamond of high order are frozen with asymptotically high probability outside the arctic circle inscribed within the diamond a similar arctic circle phenomenon has been observed in the limiting behavior of random square young tableaux in this paper we show that random domino tilings of the aztec diamond are asymptotically related to random square young tableaux in a more refined sense that looks also at the behavior inside the arctic circle this is done by giving a new derivation of the limiting shape of the height function of a random domino tiling of the aztec diamond that uses the large deviation techniques developed for the square young tableaux problem in a previous paper by pittel and the author the solution of the variational problem that arises for domino tilings is almost identical to the solution for the case of square young tableaux by pittel and the author the analytic techniques used to solve the variational problem provide a systematic guess free approach for solving problems of this type which have appeared in a number of related combinatorial probability models

algebraic arctic curves in the domain wall six vertex model the arctic curve i e the spatial curve separating ordered or frozen and disordered or temperate regions of the six vertex model with domain wall boundary conditions is discussed for the root of unity vertex weights in these cases the curve is described by algebraic equations which can be worked out explicitly from the parametric solution for this curve some interesting examples are discussed in detail the upper bound on the maximal degree of the equation in a generic root of unity case is obtained

local statistics of realizable vertex models we study planar vertex models which are probability measures on edge subsets of a planar graph satisfying certain constraints at each vertex examples including the dimer model and 1 2 model which we will define we express the local statistics of a large class of vertex models on a finite hexagonal lattice as a linear combination of the local statistics of dimers on the corresponding fisher graph with the help of a generalized holographic algorithm using an n n torus to approximate the periodic infinite graph we give an explicit integral formula for the free energy and local statistics for configurations of the vertex model on an infinite bi periodic graph as an example we simulate the 1 2 model by the technique of glauber dynamics

conformal invariance of loops in the double dimer model the dimer model is the study of random dimer covers perfect matchings of a graph a double dimer configuration on a graph mathcal g is a union of two dimer covers of mathcal g we introduce quaternion weights in the dimer model and show how they can be used to study the homotopy classes relative to a fixed set of faces of loops in the double dimer model on a planar graph as an application we prove that in the scaling limit of the uniform double dimer model on mathbb z 2 or on any other bipartite planar graph conformally approximating mathbb c the loops are conformally invariant

asymptotic shapes with free boundaries we study limit shapes for dimer models on domains of the hexagonal lattice with free boundary conditions this is equivalent to the large deviation phenomenon for a random stepped surface over domains fixed only at part of the boundary

estimating the asymptotics of solid partitions we study the asymptotic behavior of solid partitions using transition matrix monte carlo simulations if p 3 n denotes the number of solid partitions of an integer n we show that lim n rightarrow infty n 3 4 log p 3 n sim 1 822 pm 0 001 this shows clear deviation from the value 1 7898 attained by macmahon numbers m 3 n that was conjectured to hold for solid partitions as well in addition we find estimates for other sub leading terms in log p 3 n in a pattern deviating from the asymptotics of line and plane partitions we need to add an oscillatory term in addition to the obvious sub leading terms the period of the oscillatory term is proportional to n 1 4 the natural scale in the problem this new oscillatory term might shed some insight into why partitions in dimensions greater than two do not admit a simple generating function

a periodic hexagon tiling model and non hermitian orthogonal polynomials we study a one parameter family of probability measures on lozenge tilings of large regular hexagons that interpolates between the uniform measure on all possible tilings and a particular fully frozen tiling the description of the asymptotic behavior can be separated into two regimes the low and the high temperature regime our main results are the computations of the disordered regions in both regimes and the limiting densities of the different lozenges there for low temperatures the disordered region consists of two disjoint ellipses in the high temperature regime the two ellipses merge into a single simply connected region at the transition from the low to the high temperature a tacnode appears the key to our asymptotic study is a recent approach introduced by duits and kuijlaars providing a double integral representation for the correlation kernel one of the factors in the integrand is the christoffel darboux kernel associated to polynomials that satisfy non hermitian orthogonality relations with respect to a complex valued weight on a contour in the complex plane we compute the asymptotic behavior of these orthogonal polynomials and the christoffel darboux kernel by means of a riemann hilbert analysis after substituting the resulting asymptotic formulas into the double integral we prove our main results by classical steepest descent arguments

footnotes to a paper of domokos i abstract we use regev s double centralizer theorem from 7 to study invariants of matrices with group actions this theory then has applications to trace identities and poincare series

third order phase transition in random tilings we consider the domino tilings of an aztec diamond with a cut off corner of macroscopic square shape and given size and address the bulk properties of tilings as the size is varied we observe that the free energy exhibits a third order phase transition when the cut off square increasing in size reaches the arctic ellipse the phase separation curve of the original unmodified aztec diamond we obtain this result by studying the thermodynamic limit of a certain nonlocal correlation function of the underlying six vertex model with domain wall boundary conditions the so called emptiness formation probability efp we consider efp in two different representations as a function for toda chains and as a random matrix model integral the latter has a discrete measure and a linear potential with hard walls the observed phase transition shares properties with both gross witten wadia and douglas kazakov phase transitions

hydrodynamic limit and viscosity solutions for a two dimensional growth process in the anisotropic kpz class we study a 2 1 dimensional stochastic interface growth model that is believed to belong to the so called anisotropic kpz akpz universality class borodin and ferrari 2014 it can be seen either as a two dimensional interacting particle process with drift that generalizes the one dimensional hammersley process aldous and diaconis 1995 seppalainen 1996 or as an irreversible dynamics of lozenge tilings of the plane borodin and ferrari 2014 toninelli 2015 our main result is a hydrodynamic limit the interface height profile converges after a hyperbolic scaling of space and time to the solution of a non linear first order pde of hamilton jacobi type with non convex hamiltonian non convexity of the hamiltonian is a distinguishing feature of the akpz class we prove the result in two situations i for smooth initial profiles and times smaller than the time t shock when singularities shocks appear or ii for all times including t t shock if the initial profile is convex in the latter case the height profile converges to the viscosity solution of the pde as an important ingredient we introduce a harris type graphical construction for the process

crystal melting and toric calabi yau manifolds we construct a statistical model of crystal melting to count bps bound states of d0 and d2 branes on a single d6 brane wrapping an arbitrary toric calabi yau threefold the three dimensional crystalline structure is determined by the quiver diagram and the brane tiling which characterize the low energy effective theory of d branes the crystal is composed of atoms of different colors each of which corresponds to a node of the quiver diagram and the chemical bond is dictated by the arrows of the quiver diagram bps states are constructed by removing atoms from the crystal this generalizes the earlier results on the bps state counting to an arbitrary non compact toric calabi yau manifold we point out that a proper understanding of the relation between the topological string theory and the crystal melting involves the wall crossing in the donaldson thomas theory

macroscopically separated gaps in dimer coverings of aztec rectangles in this paper we determine the interaction of diagonal defect clusters in regions of an aztec rectangle that scale to arbitrary points on its symmetry axis in earlier work we treated the case when this point was the center of the scaled aztec rectangle we use the resulting formulas to determine the asymptotics of the correlation of defects that are macroscopically separated from one another and feel the influence of the boundary in several of the treated situations this seems not to be accomplishable by previous methods our applications include the case of two long neutral strings which turn out to interact by an analog of the casimir force two families of neutral doublets that turn out to interact completely independently of one another a neutral doublet and a very long neutral string a general collection of macroscopically separated monomer and separation defects and the case of long strings consisting of consecutive monomers

a 2 1 dimensional growth process with explicit stationary measures we introduce a class of 2 1 2 1 dimensional stochastic growth processes that can be seen as irreversible random dynamics of discrete interfaces irreversible means that the interface has an average nonzero drift interface configurations correspond to height functions of dimer coverings of the infinite hexagonal or square lattice the model can also be viewed as an interacting driven particle system and in the totally asymmetric case the dynamics corresponds to an infinite collection of mutually interacting hammersley processes when the dynamical asymmetry parameter p q p q equals zero the infinite volume gibbs measures with given slope are stationary and reversible when p qp q are not reversible any more but remarkably they are still stationary in such stationary states we find that the average height function at any given point xx grows linearly with time tt with a nonzero speed eqx t e hx t hx 0 v teqx t e hx t hx 0 v t while the typical fluctuations of qx t qx t are smaller than any power of tt as t t in the totally asymmetric case of p 0 q 1p 0 q 1 and on the hexagonal lattice the dynamics coincides with the anisotropic kpz growth model introduced by a borodin and p l ferrari in j stat mech theory exp 2009 2009 p02009 comm math phys 325 603 684 for a suitably chosen integrable initial condition that is very far from the stationary state they were able to determine the hydrodynamic limit and a clt for interface fluctuations on scale logt log t exploiting the fact that in that case certain space time height correlations can be computed exactly in the same setting they proved that asymptotically for t t the local statistics of height fluctuations tends to that of a gibbs state which led to the prediction that gibbs states should be stationary

quivers from higgs bundles over p1 and quiver gauge theory in this paper we associate a higgs bundle over p1 with a higgs quiver we need to consider the higgs representation of higgs quivers where the difference of higgs representation from the usual linear representation is putting an op1 module at each vertex in the quiver and then changing the higgs field on the underlying vector bundle is equivalent to changing the higgs representations we can also treat the space of higgs fields as a space of the representations of the so called higgs structure quiver with the action of a reductive group we produce some new quivers from the higgs structure quiver and its representations such as double quivers and bogomol nyi prasad sommerfield quivers related to quiver gauge theory and study some topology of moduli space of corresponding stable representations encoded in the structure of a higgs bundle we also introduce some algebraic structures on some vector spaces constructed from the higgs structure quiver as the generalizations of classical hall algebra and cohomological hall algebra via bipartite quivers the complexities of these algebraic structures are reflected in the nonassociativity of multiplications arising from preservation of some z2 r graded structure

the lee yang and polya schur programs ii theory of stable polynomials and applications in the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being nonvanishing in products of open circular domains for such sets this completes the multivariate generalization of the classification program initiated by polya and schur for univariate real polynomials we build on these classification theorems to develop here a theory of multivariate stable polynomials applications and examples show that this theory provides a natural framework for dealing in a uniform way with lee yang type problems in statistical mechanics combinatorics and geometric function theory in one or several variables in particular we answer a question of hinkkanen on multivariate apolarity

a formal inverse to the cayley hamilton theorem abstract we prove that a q algebra r with formal trace can be realized as n n matrices if and only if it satisfies the cayley hamilton identity of degree n

a note on dimer models and mckay quivers we give one formulation of a procedure of hanany and vegh j high energy phys 0710 029 35 2007 which takes a lattice polygon as an input and produces a set of isoradial dimer models we study the case of lattice triangles in detail and discuss the relation with coamoebas following feng et al adv theor math phys 12 3 489 545 2008

brownian gibbs property for airy line ensembles consider a collection of n brownian bridges b i n n to mathbb r b i n b i n 0 1 i n conditioned not to intersect the edge scaling limit of this system is obtained by taking a weak limit as n of the collection of curves scaled so that the point 0 21 2 n is fixed and space is squeezed horizontally by a factor of n 2 3 and vertically by n 1 3 if a parabola is added to each of the curves of this scaling limit an x translation invariant process sometimes called the multi line airy process is obtained we prove the existence of a version of this process which we call the airy line ensemble in which the curves are almost surely everywhere continuous and non intersecting this process naturally arises in the study of growth processes and random matrix ensembles as do related processes with wanderers and outliers we formulate our results to treat these relatives as well note that the law of the finite collection of brownian bridges above has the property called the brownian gibbs property of being invariant under the following action select an index 1 k n and erase b k on a fixed time interval a b n n then replace this erased curve with a new curve on a b according to the law of a brownian bridge between the two existing endpoints a b k a and b b k b conditioned to intersect neither the curve above nor the one below we show that this property is preserved under the edge scaling limit and thus establish that the airy line ensemble has the brownian gibbs property an immediate consequence of the brownian gibbs property is a confirmation of the prediction of m prahofer and h spohn that each line of the airy line ensemble is locally absolutely continuous with respect to brownian motion we also obtain a proof of the long standing conjecture of k johansson that the top line of the airy line ensemble minus a parabola attains its maximum at a unique point this establishes the asymptotic law of the transversal fluctuation of last passage percolation with geometric weights our probabilistic approach complements the perspective of exactly solvable systems which is often taken in studying the multi line airy process and readily yields several other interesting properties of this process

loop statistics in the toroidal honeycomb dimer model the dimer model on a graph embedded in the torus can be interpreted as a collection of random self avoiding loops in this paper we consider the uniform toroidal honeycomb dimer model we prove that when the mesh of the graph tends to zero and the aspect of the torus is fixed the winding number of the collection of loops converges in law to a two dimensional discrete gaussian distribution this is known to physicists in more generality from their analysis of toroidal two dimensional critical loop models and their mapping to the massless free field on the torus this paper contains the first mathematical proof of this more general physics result in the specific case of the loop model induced by a toroidal dimer model

probability distributions related to tilings of non convex polygons in this paper we study random lozenge tilings of non convex polygonal regions the interaction of the non convexities cuts leads to new kernels and thus new statistics for the tiling fluctuations near these regions this paper gives new probability distributions and joint probability distributions for the fluctuation of tiles along lines in between the cuts

asymptotics of height change on toroidal temperleyan dimer models the dimer model is an exactly solvable model of planar statistical mechanics in its critical phase various aspects of its scaling limit are known to be described by the gaussian free field for periodic graphs criticality is an algebraic condition on the spectral curve of the model determined by the edge weights kenyon et al in ann math 2 163 3 1019 1056 2006 isoradial graphs provide another class of critical dimer models in which the edge weights are determined by the local geometry in the present article we consider another class of graphs general temperleyan graphs i e graphs arising in the generalized temperley bijection between spanning trees and dimer models building in particular on forman s formula and representations of laplacian determinants in terms of poisson operators and under a minimal assumption viz that the underlying random walk converges to brownian motion we show that the natural topological observable on macroscopic tori converges in law to its universal limit i e the law of the periods of the dimer height function converges to that of the periods of a compactified free field

bulk universality for random lozenge tilings near straight boundaries and for tensor products we prove that the asymptotic of the bulk local statistics in models of random lozenge tilings is universal in the vicinity of straight boundaries of the tiled domains the result applies to uniformly random lozenge tilings of large polygonal domains on triangular lattice and to the probability measures describing the decomposition in gelfand tsetlin bases of tensor products of representations of unitary groups in a weaker form our theorem also applies to random domino tilings

classification and construction of quasisimple lie algebras abstract we study a class of possibly infinite dimensional lie algebras called the quasisimple lie algebras qsla s and generalizing semisimple and affine kac moody lie algebras they are characterized by the existence of a finite dimensional cartan subalgebra a non degenerate symmetric ad invariant killing form and nilpotent rootspaces attached to non isotropic roots we are then able to derive a classification theorem for the possible irreducible elliptic quasisimple root systems moreover we construct explicit realizations of some of them as untwisted and twisted current algebras generalizing the affine loop algebras

tilings of non convex polygons skew young tableaux and determinantal processes this paper studies random lozenge tilings of general non convex polygonal regions we show that the pairwise interaction of the non convexities leads asymptotically to new kernels and thus to new statistics for the tiling fluctuations the precise geometrical figure here consists of a hexagon with cuts along opposite edges for this model we take limits when the size of the hexagon and the cuts tend to infinity while keeping certain geometric data fixed in order to guarantee sufficient interaction between the cuts in the limit we show in this paper that the kernel for the finite tiling model can be expressed as a multiple integral where the number of integrations is related to the fixed geometric data above the limiting kernel is believed to be a universal master kernel

the representations of wreath products via double centralizing theorems abstract double centralizing theorems for wreath products and lie groups or superalgebras are given they extend a famous theorem of i schur gesammelte abhandlungen iii pp 68 85 springer verlag berlin 1973 and enable one to obtain the representations of wreath products very easily they also imply a multibook theorem for direct sum of lie superalgebras pl k i l i

the alternative torus and the structure of elliptic quasi simple lie algebras of type we present the complete classification of the tame irreducible elliptic quasi simple lie algebras of type a2 and in particular specialize on the case where the coordinates are not associative here the coordinates are cayley dickson algebras over laurent polynomial rings in v 3 variables which we call alternative tori in giving our classification we need to present much information on these alternative tori and the lie algebras coordinatized by them

degenerations of amoebae and berkovich spaces we prove a continuity result for the fibers of the berkovich analytification of a complex algebraic variety with respect to the maximum of the archimedean norm and the trivial norm as a consequence we obtain generalizations of a result of mikhalkin and rullgard about degenerations of amoebae onto tropical varieties

identities of bilinear mappings and graded polynomial identities of matrices we describe the polynomial identities of the bilinear mappings v w k and vr v r mr k where v w vr are finite dimensional vector spaces over a field k of characteristic 0 dim vr r and mr k is the r r matrix algebra we show that these identities follow from the commutativity of the values in k of the bilinear forms v w k and v r vr k and capelli identities we apply these results to the g graded polynomial identities of the algebra mr k with the elementary g grading associated with the rtuple g g h where g is an arbitrary group and g h g are such that g 1 h 2 e i n

critical temperature of periodic ising models a periodic ising model has interactions which are invariant under translations of a full rank sublattice mathfrak l of mathbb z 2 we prove an exact quantitative characterization of the critical temperature defined as the supremum of temperatures for which the spontaneous magnetization is strictly positive for the ferromagnetic model the critical temperature is the solution of a certain algebraic equation resulting from the condition that the spectral curve of the corresponding dimer model on the fisher graph has a real zero on the unit torus with our technique we provide a simple proof for the exponential decay of spin spin correlations above the critical temperature as well as the exponential decay of the edge edge correlations for all non critical edge weights of the corresponding dimer model on periodic fisher graphs

limit shapes and local statistics for the stochastic six vertex model in this paper we consider the stochastic six vertex model on a cylinder with arbitrary initial data first we show that it exhibits a limit shape in the thermodynamic limit whose density profile is given by the entropy solution to an explicit non linear conservation law that was predicted by gwa spohn phys rev lett 68 725 728 1992 and by reshetikhin sridhar commun math phys 363 741 765 2018 then we show that the local statistics of this model around any continuity point of its limit shape are given by an infinite volume translation invariant gibbs measure of the appropriate slope

dimer problem on the cylinder and torus institute of mathematics academia sinica nsfc 10771086 10671162 fujian province university nsc nsc96 2115 m 001 005

corner percolation on ℤ2 and the square root of 17 we consider a four vertex model introduced by balint toth a dependent bond percolation model on 2 in which every edge is present with probability 1 2 and each vertex has exactly two incident edges perpendicular to each other we prove that all components are finite cycles almost surely but the expected diameter of the cycle containing the origin is infinite moreover we derive the following critical exponents the tail probability diameter of the cycle of the origin n n and the expectation mathbb e length of a typical cycle with diameter n n with gamma 5 sqrt 17 4 0 219 ldots and delta sqrt 17 1 4 1 28 ldots the value of comes from a singular sixth order ode while the relation 3 2 corresponds to the fact that the scaling limit of the natural height function in the model is the additive brownian motion whose level sets have hausdorff dimension 3 2 we also include many open problems for example on the conformal invariance of certain linear entropy models

fixed point subalgebras of extended affine lie algebras abstract it is a well known result that the fixed point subalgebra of a finite dimensional complex simple lie algebra under a finite order automorphism is a reductive lie algebra so it is a direct sum of finite dimensional simple lie subalgebras and an abelian subalgebra we consider this for the class of extended affine lie algebras and are able to show that the fixed point subalgebra of an extended affine lie algebra under a finite order automorphism which satisfies certain natural properties is a sum of extended affine lie algebras up to existence of some isolated root spaces a subspace of the center and a subspace which is contained in the centralizer of the core moreover we show that the core of the fixed point subalgebra modulo its center is isomorphic to the direct sum of the cores modulo centers of the involved summands

quantum tori and the structure of elliptic quasi simple lie algebras abstract we study and classify those tame irreducible elliptic quasi simple lie algebras which are simply laced and of rankl 3 the first step is to identify the core of such an algebra up to central isogeny by identifying the coordinates when the type isdorethe coordinates are laurent polynomials in variables while for typeathe coordinates can be any quantum torus in variables the next step is to study the universal central extension as well as the derivation algebra of the core these are related to the first connes cyclic homology group of the coordinates the final step is to use this information to give constructions of lie algebras which we then prove yield representatives of all isomorphism classes of the above types of algebras

a variational principle for domino tilings 1 1 description of results a domino is a 1 x 2 or 2 x 1 rectangle and a tiling of a region by dominos is a way of covering that region with dominos so that there are no gaps or overlaps in 1961 kasteleyn kal found a formula for the number of domino tilings of an m x n rectangle with mn even as shown in figure 1 for rm n 68 temperley and fisher tf used a differenit method and arrived at the same result at almost exactly the same time both lines of calculation showed that the logarithm of number of tilings divided by the number of dominos in a tiling that is mn 2 converges to 2g 7r 0 58 here g is catalan s constant on the other hand in 1992 elkies et al eklp studied domino tilings of regions they called aztec diamonds figure 2 shows an aztec diamond of order 48 and showed that the logarithm of the number of tilings divided by the number of dominos converges to the smaller number log 2 2 0 35 thus even though the region in figure 1 has slightly smaller area than the region in figure 2 the former has far more domino tilings for regions with other shapes neither of these asymptotic formulas may apply in the present paper we consider simply connected regions of arbitrary shape we give an exact formula for the limiting value of the logarithm of the number of tilings per unit area as a function of the shape of the boundary of the region as the size of the region goes to infinity in particular we show that computation of this limit is intimately linked with an understanding of long range variations in the local statistics of random domino tilings such variations can be seen by comparing figures 1 and 2 each of the two tilings is random in the sense that the algorithm pwi that was used to create it generates each of the possible tilings of the region being tiled with the same probability hence one can expect each tiling to be qualitatively typical of the overwhelming majority of tilings of the region in

dimers and amoebae we study random surfaces which arise as height functions of random perfect matchings a k a dimer configurations on a weighted bipartite doubly periodic graph g embedded in the plane we derive explicit formulas for the surface tension and local gibbs measure probabilities of these models the answers involve a certain plane algebraic curve which is the spectral curve of the kasteleyn operator of the graph for example the surface tension is the legendre dual of the ronkin function of the spectral curve the amoeba of the spectral curve represents the phase diagram of the dimer model further we prove that the spectral curve of a dimer model is always a real curve of special type namely it is a harnack curve this implies many qualitative and quantitative statement about the behavior of the dimer model such as existence of smooth phases decay rate of correlations growth rate of height function fluctuations etc

arctic curves of the twenty vertex model with domain wall boundaries we use the tangent method to compute the arctic curve of the twenty vertex 20v model with particular domain wall boundary conditions for a wide set of integrable weights to this end we extend to the finite geometry of domain wall boundary conditions the standard connection between the bulk 20v and 6v models via the kagome lattice ice model this allows to express refined partition functions of the 20v model in terms of their 6v counterparts leading to explicit parametric expressions for the various portions of its arctic curve the latter displays a large variety of shapes depending on the weights and separates a central liquid phase from up to six different frozen phases a number of numerical simulations are also presented which highlight the arctic curve phenomenon and corroborate perfectly the analytic predictions of the tangent method we finally compute the arctic curve of the quarter turn symmetric holey aztec domino tiling qthadt model a problem closely related to the 20v model and whose asymptotics may be analyzed via a similar tangent method approach again results for the qthadt model are found to be in perfect agreement with our numerical simulations

beyond aztec castles toric cascades in the dp 3 quiver given one of an infinite class of supersymmetric quiver gauge theories string theorists can associate a corresponding toric variety which is a calabi yau 3 fold as well as an associated combinatorial model known as a brane tiling in combinatorial language a brane tiling is a bipartite graph on a torus and its perfect matchings are of interest to both combinatorialists and physicists alike a cluster algebra may also be associated to such quivers and in this paper we study the generators of this algebra known as cluster variables for the quiver associated to the cone over the del pezzo surface d p 3 in particular mutation sequences involving mutations exclusively at vertices with two in coming arrows and two out going arrows are referred to as toric cascades in the string theory literature such toric cascades give rise to interesting discrete integrable systems on the level of cluster variable dynamics we provide an explicit algebraic formula for all cluster variables that are reachable by toric cascades as well as a combinatorial interpretation involving perfect matchings of subgraphs of the d p 3 brane tiling for these formulas in most cases

arctic curves in path models from the tangent method recently colomo and sportiello introduced a powerful method known as the tangent method for computing the arctic curve in statistical models which have a non or weakly intersecting lattice path formulation we apply the tangent method to compute arctic curves in various models the domino tiling of the aztec diamond for which we recover the celebrated arctic circle a model of dyck paths equivalent to the rhombus tiling of a half hexagon for which we find an arctic half ellipse another rhombus tiling model with an arctic parabola the vertically symmetric alternating sign matrices where we find the same arctic curve as for unconstrained alternating sign matrices the latter case involves lattice paths that are non intersecting but that are allowed to have osculating contact points for which the tangent method was argued to still apply for each problem we estimate the large size asymptotics of a certain one point function using lu decomposition of the corresponding gessel viennot matrices and a reformulation of the result amenable to asymptotic analysis

on the double random current nesting field we relate the planar random current representation introduced by griffiths hurst and sherman to the dimer model more precisely we provide a measure preserving map between double random currents obtained as the sum of two independent random currents on a planar graph and dimers on an associated bipartite graph we also define a nesting field for the double random current which under this map corresponds to the height function of the dimer model as applications we provide an alternative derivation of some of the bozonization rules obtained recently by dubedat and show that the spontaneous magnetization of the ising model on a planar biperiodic graph vanishes at criticality

coupling functions for domino tilings of aztec diamonds the inverse kasteleyn matrix of a bipartite graph holds much information about the perfect matchings of the system such as local statistics which can be used to compute local and global asymptotics in this paper we consider three different weightings of domino tilings of the aztec diamond and show using recurrence relations that we can compute the inverse kasteleyn matrix these weights are the one periodic weighting where the horizontal edges have one weight and the vertical edges have another weight the qvol weighting which corresponds to multiplying the product of tile weights by q if we add a box to the height function and the two periodic weighting which exhibits a flat region with defects in the center

uniqueness of maximal entropy measure on essential spanning forests an essential spanning forest of an infinite graph g is a spanning forest of g in which all trees have infinitely many vertices let gn be an increasing sequence of finite connected subgraphs of g for which gn g pemantle s arguments imply that the uniform measures on spanning trees of gn converge weakly to an aut g invariant measure g on essential spanning forests of g we show that if g is a connected amenable graph and aut g acts quasitransitively on g then g is the unique invariant measure on essential spanning forests of g for which the specific entropy is maximal this result originated with burton and pemantle who gave a short but incorrect proof in the case d lyons discovered the error and asked about the more general statement that we prove

arctic curves of the six vertex model on generic domains the tangent method we revisit the problem of determining the arctic curve in the six vertex model with domain wall boundary conditions we describe an alternative method by which we recover the previously conjectured analytic expression in the square domain we adapt the method to work for a large class of domains and for other models exhibiting limit shape phenomena we study in detail some examples and derive in particular the arctic curve of the six vertex model in a triangoloid domain at the ice point

mixing times of monotone surfaces and sos interfaces a mean curvature approach we consider stochastic spin flip dynamics for i monotone discrete surfaces in mathbb z 3 with planar boundary height and ii the one dimensional discrete solid on solid sos model confined to a box in both cases we show almost optimal bounds o l2polylog l for the mixing time of the chain where l is the natural size of the system the dynamics at a macroscopic scale should be described by a deterministic mean curvature motion such that each point of the surface feels a drift which tends to minimize the local surface tension spohn in j stat phys 71 1081 1132 1993 inspired by this heuristics our approach consists in bounding the dynamics with an auxiliary one which with very high probability follows quite closely the deterministic mean curvature evolution key technical ingredients are monotonicity coupling and an argument due to wilson ann appl probab 14 274 325 2004 in the framework of lozenge tiling markov chains our approach works equally well for both models despite the fact that their equilibrium maximal height fluctuations occur on very different scales log l for monotone surfaces and sqrt l for the sos model finally combining techniques from kinetically constrained spin systems cancrini et al in probab th rel fields 140 459 504 2008 together with the above mixing time result we prove an almost diffusive lower bound of order 1 l2polylog l for the spectral gap of the sos model with horizontal size l and unbounded heights

log gases on quadratic lattices via discrete loop equations and q boxed plane partitions abstract we study a general class of log gas ensembles on shifted quadratic lattices we prove that the corresponding empirical measures satisfy a law of large numbers and that their global fluctuations are gaussian with a universal covariance we apply our general results to analyze the asymptotic behavior of a q boxed plane partition model introduced by borodin gorin and rains in particular we show that the global fluctuations of the height function on a fixed slice are described by a one dimensional section of a pullback of the two dimensional gaussian free field our approach is based on a q analogue of the schwinger dyson or loop equations which originate in the work of nekrasov and his collaborators and extends the methods developed by borodin gorin and guionnet to quadratic lattices

fluctuations of an atomic ledge bordering a crystalline facet when a high symmetry facet joins the rounded part of a crystal the step line density vanishes as square root of r with r denoting the distance from the facet edge this means that the ledge bordering the facet has a lot of space to meander as caused by thermal activation we investigate the statistical properties of the border ledge fluctuations in the scaling regime they turn out to be non gaussian and related to the edge statistics of gaussian unitary ensemble multimatrix models

maximum matchings in scale free networks with identical degree distribution the size and number of maximum matchings in a network have found a large variety of applications in many fields as a ubiquitous property of diverse real systems power law degree distribution was shown to have a profound influence on size of maximum matchings in scale free networks where the size of maximum matchings is small and a perfect matching often does not exist in this paper we study analytically the maximum matchings in two scale free networks with identical degree sequence and show that the first network has no perfect matchings while the second one has many for the first network we determine explicitly the size of maximum matchings and provide an exact recursive solution for the number of maximum matchings for the second one we design an orientation and prove that it is pfaffian based on which we derive a closed form expression for the number of perfect matchings moreover we demonstrate that the entropy for perfect matchings is equal to that corresponding to the extended sierpiski graph with the same average degree as both studied scale free networks our results indicate that power law degree distribution alone is not sufficient to characterize the size and number of maximum matchings in scale free networks

pattern densities in non frozen planar dimer models in this paper we introduce a family of observables for the dimer model on a bi periodic bipartite planar graph called pattern density fields we study the scaling limit of these objects for non frozen gibbs measures of the dimer model and prove that they converge to a linear combination of a derivative of the gaussian massless free field and an independent white noise

mixing time of markov chains for the 1 2 model a 1 2 model configuration is a subset of edges of a hexagonal lattice satisfying the constraint that each vertex is incident to 1 or 2 edges we introduce markov chains to sample the 1 2 model configurations on the 2d hexagonal lattice and prove that the mixing time of these chains is polynomial in the sizes of the graphs for a large class of probability measures

domino statistics of the two periodic aztec diamond random domino tilings of the aztec diamond shape exhibit interesting features and some of the statistical properties seen in random matrix theory as a statistical mechanical model it can be thought of as a dimer model or as a certain random surface we consider the aztec diamond with a two periodic weighting which exhibits all three possible phases that occur in these types of models often referred to as solid liquid and gas to analyze this model we use entries of the inverse kasteleyn matrix which give the probability of any configuration of dominoes a formula for these entries for this particular model was derived by chhita and young 2014 in this paper we find a major simplification of this formula expressing entries of the inverse kasteleyn matrix by double contour integrals which makes it possible to investigate their asymptotics in a part of the aztec diamond where the asymptotic analysis is simpler we use this formula to show that the entries of the inverse kasteleyn matrix converge to the known entries of the full plane inverse kasteleyn matrices for the different phases we also study the detailed asymptotics of the inverse kasteleyn matrix at both the liquid solid and liquid gas boundaries and find the extended airy kernel in the next order asymptotics finally we provide a potential candidate for a combinatorial description of the liquid gas boundary

exact solution of the six vertex model with domain wall boundary conditions critical line between ferroelectric and disordered phases this is a continuation of the papers of bleher and fokin commun math phys 268 223 284 2006 and of bleher and liechty commun math phys 286 777 801 2009 in which the large n asymptotics is obtained for the partition function z n of the six vertex model with domain wall boundary conditions in the disordered and ferroelectric phases respectively in the present paper we obtain the large n asymptotics of z n on the critical line between these two phases

the arctic curve for aztec rectangles with defects via the tangent method the tangent method of colomo and sportiello is applied to the study of the asymptotics of domino tilings of large aztec rectangles with some fixed distribution of defects along a boundary the associated non intersecting lattice path configurations are made of schroder paths whose weights involve two parameters gamma and q keeping track respectively of one particular type of step and of the area below the paths we predict the arctic curve for an arbitrary distribution of defects and illustrate our result with a number of examples involving different classes of boundary defects

dimers on surface graphs and spin structures ii partition functions for dimers on closed oriented surfaces are known to be alternating sums of pfaffians of kasteleyn matrices in this paper we obtain the formula for the coefficients in terms of discrete spin structures

dynamics of 2 1 dimensional sos surfaces above a wall slow mixing induced by entropic repulsion we study the glauber dynamics for the 2 1 d solid on solid model above a hard wall and below a far away ceiling on an l times l box of z 2 with zero boundary conditions at large inverse temperature beta it was shown by bricmont el mellouki and frohlich 1986 that the floor constraint induces an entropic repulsion effect which lifts the surface to an average height h asymp 1 beta log l as an essential step in understanding the effect of entropic repulsion on the glauber dynamics we determine the equilibrium height h to within an additive constant h 1 4 beta log l o 1 we then show that starting from zero initial conditions the surface rises to its final height h through a sequence of metastable transitions between consecutive levels the time for a transition from height h ah a in 0 1 to height h 1 is roughly exp c l a for some constant c 0 in particular the mixing time of the dynamics is exponentially large in l i e t mix geq e c l we also provide the matching upper bound t mix leq e c l requiring a challenging analysis of the statistics of height contours at low temperature and new coupling ideas and techniques finally to emphasize the role of entropic repulsion we show that without a floor constraint at height zero the mixing time is no longer exponentially large in l

lower tail of the kpz equation we provide the first tight bounds on the lower tail probability of the one point distribution of the kardar parisi zhang kpz equation with narrow wedge initial data our bounds hold for all sufficiently large times t and demonstrates a crossover between superexponential decay with exponent 52 and leading prefactor 415 t1 3 for tail depth greater than t2 3 and exponent 3 with leading prefactor at least 112 for tail depth less than t2 3

planar dimers and harnack curves in this article we study the connection between dimers and harnack curves discovered in 15 we prove that every harnack curve arises as a spectral curve of some dimer model we also prove that the space of harnack curves of given degree is homeomorphic to a closed octant and that the areas of the amoeba holes and the distances between the amoeba tentacles give these global coordinates we characterize harnack curves of genus zero as spectral curves of isoradial dimers and also as minimizers of the volume under their ronkin function with given boundary conditions

a tangent method derivation of the arctic curve for q weighted paths with arbitrary starting points we use a tangent method approach to obtain the arctic curve in a model of non intersecting lattice paths within the first quadrant including a q dependent weight associated with the area delimited by the paths our model is characterized by an arbitrary sequence of starting points along the positive horizontal axis whose distribution involves an arbitrary piecewise differentiable function we give an explicit expression for the arctic curve in terms of this arbitrary function and of the parameter q a particular emphasis is put on the deformation of the arctic curve upon varying q and on its limiting shapes when q tends to 0 or infinity our analytic results are illustrated by a number of detailed examples

hook young diagrams with applications to combinatorics and to representations of lie superalgebras definition et etude des tableaux k l semi standards applications a la combinatoire et aux representations des superalgebres de lie des diagrammes de young a crochets