a computational criterion for the kac conjecture abstract we give a criterion for the kac conjecture asserting that the free term of the polynomial counting the absolutely indecomposable representations of a quiver over a finite field of given dimension coincides with the corresponding root multiplicity of the associated kac moody algebra our criterion suits very well for computer tests

traces of harmonic functions and a new path space for the free quantum field the infinite dimensional ornstein uhlenbeck process associated with the free field of quantum field theory on rd is reconstructed as the traces of random harmonic functions these harmonic functions are the solutions of a generalised dirichlet problem with a random schwartz distribution as boundary data a detailed analysis of the associated trace space which is a subspace of a scaled sobolev space of negative fractional order leads to a new path space for the free field as well as to a convenient regularisation method because of the particular form of the trace space it is possible to give an explicit formula for the transition function of the process furthermore the dependence of the path space on the dimension d is discussed and a formulation of our result in terms of abstract wiener spaces is given

on the stability of the quenched state in mean field spin glass models while the gibbs states of spin glass models have been noted to have an erratic dependence on temperature one may expect the mean over the disorder to produce a continuously varying quenched state the assumption of such continuity in temperature implies that in the infinite volume limit the state is stable under a class of deformations of the gibbs measure the condition is satisfied by the parisi ansatz along with an even broader stationarity property the stability conditions have equivalent expressions as marginal additivity of the quenched free energy implications of the continuity assumption include constraints on the overlap distribution which are expressed as the vanishing of the expectation value for an infinite collection of multi overlap polynomials the polynomials can be computed with the aid of a real replica calculation in which the number of replicas is taken to zero

on the structure of quasi stationary competing particle systems we study point processes on the real line whose configurations x are locally finite have a maximum and evolve through increments which are functions of correlated gaussian variables the correlations are intrinsic to the points and quantified by a matrix q q ij i j a probability measure on the pair x q is said to be quasi stationary if the joint law of the gaps of x and of q is invariant under the evolution a known class of universally quasi stationary processes is given by the ruelle probability cascades rpc which are based on hierarchically nested poisson dirichlet processes it was conjectured that up to some natural superpositions these processes exhausted the class of laws which are robustly quasi stationary the main result of this work is a proof of this conjecture for the case where q ij assume only a finite number of values the result is of relevance for mean field spin glass models where the evolution corresponds to the cavity dynamics and where the hierarchical organization of the gibbs measure was first proposed as an ansatz

superpotential algebras and manifolds we study a special class of calabi yau algebras in the sense of ginzburg those arising as the fundamental group algebras of acyclic manifolds motivated partly by the usefulness of superpotential descriptions in motivic donaldson thomas theory we investigate the question of whether these algebras admit superpotential presentations we establish that the fundamental group algebras of a wide class of acyclic manifolds including all hyperbolic manifolds do not admit such descriptions disproving a conjecture of ginzburg regarding them we also describe a class of manifolds that do admit such descriptions and discuss a little their motivic donaldson thomas theory finally some links with topological field theory are described

geometric reid s recipe for dimer models crepant resolutions of three dimensional toric gorenstein singularities are derived equivalent to noncommutative algebras arising from consistent dimer models by choosing a special stability parameter and hence a distinguished crepant resolution y this derived equivalence generalises the fourier mukai transform relating the g hilbert scheme and the skew group algebra mathbb c x y z g for a finite abelian subgroup of mathrm sl 3 mathbb c we show that this equivalence sends the vertex simples to pure sheaves except for the zero vertex which is mapped to the dualising complex of the compact exceptional locus this generalises results of cautis logvinenko j reine angew math 636 193 236 2009 and cautis craw logvinenko j reine angew math arxiv 1205 3110 2014 to the dimer setting though our approach is different in each case we also describe some of these pure sheaves explicitly and compute the support of the remainder providing a dimer model analogue of results from logvinenko j algebra 324 2064 2087 2010

neural networks with a redundant representation detecting the undetectable we consider a three layer sejnowski machine and show that features learnt via contrastive divergence have a dual representation as patterns in a dense associative memory of order p 4 the latter is known to be able to hebbian store an amount of patterns scaling as n p ensuremath 1 where n denotes the number of constituting binary neurons interacting p wisely we also prove that by keeping the dense associative network far from the saturation regime namely allowing for a number of patterns scaling only linearly with n while pg2 such a system is able to perform pattern recognition far below the standard signal to noise threshold in particular a network with p 4 is able to retrieve information whose intensity is mathcal o 1 even in the presence of a noise mathcal o sqrt n in the large n limit this striking skill stems from a redundancy representation of patterns which is afforded given the relatively low load information storage and it contributes to explain the impressive abilities in pattern recognition exhibited by new generation neural networks the whole theory is developed rigorously at the replica symmetric level of approximation and corroborated by signal to noise analysis and monte carlo simulations

about the ergodic regime in the analogical hopfield neural networks moments of the partition function in this paper we introduce and exploit the real replica approach for a minimal generalization of the hopfield model by assuming the learned patterns to be distributed according to a standard unit gaussian we consider the high storage case when the number of patterns linearly diverges with the number of neurons we study the infinite volume behavior of the normalized momenta of the partition function we find a region in the parameter space where the free energy density in the infinite volume limit self averages around its annealed approximation as well as the entropy and the internal energy density moreover we evaluate the corrections to their extensive counterparts with respect to their annealed expressions the fluctuations of properly introduced overlaps which act as order parameters are also discussed

rigorous results on the bipartite mean field model we consider a bipartite mean field model in which interaction coefficients and magnetic fields depend only on the groups the particles belong to we rigorously compute the value of the limiting pressure per particle using tail estimation techniques we study the phase space of the model in the symmetric regime without an external field and when the interaction coefficients within the two groups are identical magnetic field perturbations are considered

the parisi formula in this chapter we prove the parisi formula which gives the limiting value of the free energy per site for the sherrington kirkpatrick model at each temperature starting with the famous result of guerra that the parisi formula provided a bound for this free energy we make full use of poisson dirichlet cascades we then use the same techniques to prove in a strong sense that in presence of an external field the overlaps are positive

the aggregate path coupling method for the potts model on bipartite graph in this paper we derive the large deviation principle for the potts model on the complete bipartite graph kn n as n increases to infinity next for the potts model on kn n we provide an extension of the method of aggregate path coupling that was originally developed in the work of kovchegov otto and titus j stat phys 144 5 1009 1027 2011 for the mean field blume capel model and in kovchegov and otto j stat phys 161 3 553 576 2015 for a general mean field setting that included the generalized curie weiss potts model analyzed in the work of jahnel et al markov process relat fields 20 601 632 2014 we use the aggregate path coupling method to identify and determine the threshold value s separating the rapid and slow mixing regimes for the glauber dynamics of the potts model on kn n

the thermodynamic limit in mean field spin glass models we present a simple strategy in order to show the existence and uniqueness of the infinite volume limit of thermodynamic quantities for a large class of mean field disordered models as for example the sherrington kirkpatrick model and the derrida p spin model the main argument is based on a smooth interpolation between a large system made of n spin sites and two similar but independent subsystems made of n 1 and n 2 sites respectively with n 1 n 2 n the quenched average of the free energy turns out to be subadditive with respect to the size of the system this gives immediately convergence of the free energy per site in the infinite volume limit moreover a simple argument based on concentration of measure gives the almost sure convergence with respect to the external noise similar results hold also for the ground state energy per site

construction of a unique self adjoint generator for a symmetric local semigroup abstract a definition is given of a symmetric local semigroup of unbounded operators p t 0 t t for some t 0 on a hilbert space h such that p t is eventually densely defined as t 0 it is shown that there exists a unique unbounded below self adjoint operator h on h such that p t is a restriction of e th as an application it is proven that h0 v is essentially self adjoint where e th0 is an lp contractive semigroup and v is multiplication by a real measurable function such that v l2 e and e v l1 for some e 0

reconstruction of kubo martin schwinger structure from euclidean green functions the explicit form of the poisson kernel for the strip t z c 0 im z is used to derive some conditions for a state on a given c algebra dynamical system a t to be a kubo martin schwinger kms state reflection positivity and generalized periodicity of the euclidean green functions corresponding to are established and used for an analytic continuation to real time reflection positivity is also used for the construction of a cyclic and faithful representation of a t on some hilbert space a sufficient condition for the identification of the constructed dynamics with the modular group of a von neumann algebra generated by the representation is formulated in terms of the euclidean green functions

equivalence of euclidean and wightman field theories a new inversion formula for the laplace transformation of tempered distributions with supports in the closed positive semiaxis is obtained the inverse laplace transform of a tempered distribution is defined by means of a limit of a special distribution constructed from this distribution the weak spectral condition on the euclidean green s functions implies that some of the limits needed for the inversion formula exist for any euclidean green s function with an even number of variables we then prove that the initial osterwalder schrader axioms 1 and the weak spectral condition are equivalent with the wightman axioms

numerical study of the sherrington kirkpatrick model in a magnetic field we study numerically the sherrington kirkpatrick model as a function of the magnetic field h with fixed temperature t 0 6t c we investigate the finite size scaling behavior of several quantities such as the spin glass susceptibility searching for numerical evidences of the transition on the de almeida thouless line we find strong corrections to scaling which make difficult to locate the transition point this shows in a simple case the extreme difficulties of spin glass simulations in a nonzero magnetic field next we study various sum rules consequences of stochastic stability involving overlaps between three and four replicas which appear to be numerically well satisfied and in a nontrivial way finally we present data on p q for a large lattice size n 3200 at low temperature t 0 4t c where the shape predicted by the replica symmetry breaking solution of the model for a nonzero magnetic field is visible

notes on the polynomial identities in random overlap structures in these notes we review first in some detail the concept of random overlap structure rost applied to fully connected and diluted spin glasses we then sketch how to write down the general term of the expansion of the energy part from the boltzmann rost for the sherrington kirkpatrick model and the corresponding term from the ramost which is the diluted extension suitable for the viana bray model

local overlaps heterogeneities and the local fluctuation dissipation relations in this paper i introduce the probability distribution of the local overlaps in spin glasses the properties of the local overlaps are studied in detail these quantities are related to the recently proposed local version of the fluctuation dissipation relations using the general principle of stochastic stability these local fluctuation dissipation relations can be proved in a way that is very similar to the usual proof of the fluctuation dissipation relations for intensive quantities the local overlap and its probability distribution play a crucial role in this proof similar arguments can be used to prove that all sites in an ageing experiment remain at the same effective temperature at the same time

broken replica symmetry bounds in the mean field spin glass model by using a simple interpolation argument in previous work we have proven the existence of the thermodynamic limit for mean field disordered models including the sherrington kirkpatrick model and the derrida p spin model here we extend this argument in order to compare the limiting free energy with the expression given by the parisi ansatz and including full spontaneous replica symmetry breaking our main result is that the quenched average of the free energy is bounded from below by the value given in the parisi ansatz uniformly in the size of the system moreover the difference between the two expressions is given in the form of a sum rule extending our previous work on the comparison between the true free energy and its replica symmetric sherrington kirkpatrick approximation we give also a variational bound for the infinite volume limit of the ground state energy per site

w kms structure from multi time euclidean green functions an axiomatic approach to the problem of reconstruction of a dynamics from the euclidean multi time green functions is proposed the existence of a w algebra on which the reconstructed dynamics acts as the canonical group of modular automorphisms is shown a condition under which the reconstruction from the one time and multi time green functions are equivalent is formulated

free energy and complexity of spherical bipartite models we investigate both free energy and complexity of the spherical bipartite spin glass model we first prove a variational formula in high temperature for the limiting free energy based on the well known crisanti sommers representation of the mixed p spin spherical model next we show that the mean number of local minima at low levels of energy is exponentially large in the size of the system and we derive a bound on the location of the ground state energy

neural networks retrieving boolean patterns in a sea of gaussian ones restricted boltzmann machines are key tools in machine learning and are described by the energy function of bipartite spin glasses from a statistical mechanical perspective they share the same gibbs measure of hopfield networks for associative memory in this equivalence weights in the former play as patterns in the latter as boltzmann machines usually require real weights to be trained with gradient descent like methods while hopfield networks typically store binary patterns to be able to retrieve the investigation of a mixed hebbian network equipped with both real e g gaussian and discrete e g boolean patterns naturally arises we prove that in the challenging regime of a high storage of real patterns where retrieval is forbidden an additional load of boolean patterns can still be retrieved as long as the ratio between the overall load and the network size does not exceed a critical threshold that turns out to be the same of the standard amit gutfreund sompolinsky theory assuming replica symmetry we study the case of a low load of boolean patterns combining the stochastic stability and hamilton jacobi interpolating techniques the result can be extended to the high load by a non rigorous but standard replica computation argument

about a solvable mean field model of a gaussian spin glass in a series of papers we have studied a modified hopfield model of a neural network with learned words characterized by a gaussian distribution the model can be represented as a bipartite spin glass with one party described by dichotomic ising spins and the other party by continuous spin variables with an a priori gaussian distribution by application of standard interpolation methods we have found it useful to compare the neural network model bipartite from one side with two spin glass models each monopartite from the other side of these the first is the usual sherrington kirkpatrick model the second is a spin glass model with continuous spins and inbuilt highly nonlinear smooth cut off interactions this model is an invaluable laboratory for testing all techniques which have been useful in the study of spin glasses the purpose of this paper is to give a synthetic description of the most peculiar aspects by stressing the necessary novelties in the treatment in particular it will be shown that the control of the infinite volume limit according to the well known guerra toninelli strategy requires in addition one to consider the involvement of the cut off interaction in the interpolation procedure moreover the control of the ergodic region the annealed case cannot be directly achieved through the standard application of the borel cantelli lemma but requires previous modification of the interaction this remark could find useful application in other cases the replica symmetric expression for the free energy can be easily reached through a suitable version of the doubly stochastic interpolation technique however this model shares the unique property that the fully broken replica symmetry ansatz can be explicitly calculated a very simple sum rule connects the general expression of the fully broken free energy trial function with the replica symmetric one the definite sign of the error term shows that the replica solution is optimal then we follow a deep strategy developed by talagrand to conclude that the replica symmetric ansatz not only gives a bound for the free energy which cannot be improved by symmetry breaking but gives also the true value for the free energy for the sake of completeness we study also the fluctuations of the centered and rescaled overlaps by following a method similar to that exploited for the sherrington kirkpatrick model in the replica symmetric region

construction of pure states in mean field models for spin glasses if a mean field model for spin glasses is generic in the sense that it satisfies the extended ghirlanda guerra identities and if the law of the overlaps has a point mass at the largest point q of its support we prove that one can decompose the configuration space into a sequence of sets ak such that generically the overlap of two configurations is equal to q if and only if they belong to the same set ak for the study of the overlaps each set ak can be replaced by a single point combining this with a recent result of panchenko a connection between ghirlanda guerra identities and ultrametricity ann probab 2008 to appear this proves that if the overlaps take only finitely many values ultrametricity occurs we give an elementary self contained proof of this result based on simple inequalities and an averaging argument

on the distribution of overlaps in the sherrington kirkpatrick spin glass model this paper describes some of the analytic tools developed recently by ghirlanda and guerra in the investigation of the distribution of overlaps in the sherrington kirkpatrick spin glass model and of parisi s ultrametricity in particular we introduce to this task a simplified but also generalized model on which the gaussian analysis is made easier moments of the hamiltonian and derivatives of the free energy are expressed as polynomials of the overlaps under the essential tool of self averaging we describe with full rigour various overlap identities and replica independence that actually hold in a rather large generality the results are presented in a language accessible to probabilists and analysts

the response of glassy systems to random perturbations a bridge between equilibrium and off equilibrium we discuss the response of aging systems with short range interactions to a class of random perturbations although these systems are out of equilibrium the limit value of the free energy at long times is equal to the equilibrium free energy by exploiting this fact we define a new order parameter function and we relate it to the ratio between response and fluctuation which is in principle measurable in an aging experiment for a class of systems possessing stochastic stability we show that this new order parameter function is intimately related to the static order parameter function describing the distribution of overlaps between clustering states the same method is applied to investigate the geometrical organization of pure states we show that the ultrametric organization in the dynamics implies static ultrametricity and we relate these properties to static separability i e the property that the measure of the overlap between pure states is essentially unique our results especially relevant for spin glasses pave the way to an experimental determination of the order parameter function

stochastically positive structures on weyl algebras the case of quasi free states we consider quasi free stochastically positive ground and thermal states on weyl algebras in the imaginary time formulation in particular we obtain a new derivation of a general form of thermal quasi free state and give conditions when such a state is stochastically positive i e when it defines a periodic stochastic process with respect to imaginary time a so called thermal process then we show that the thermal process completely determines modular structure canonically associated with the quasi free thermal state on weyl algebra we discuss a variety of examples connected with free quantum field theories on globally hyperbolic stationary space times and models of quantum statistical mechanics

noncommutative mirror symmetry for punctured surfaces in 2013 abouzaid auroux efimov katzarkov and orlov showed that the wrapped fukaya categories of punctured spheres and finite unbranched covers of punctured spheres are derived equivalent to the categories of singularities of a superpotential on certain crepant resolutions of toric 3 dimensional singularities we generalize this result to other punctured riemann surfaces and reformulate it in terms of certain noncommutative algebras coming from dimer models in particular given any consistent dimer model we can look at a subcategory of noncommutative matrix factorizations and show that this category is mathtt a infty isomorphic to a subcategory of the wrapped fukaya category of a punctured riemann surface the connection between the dimer model and the punctured riemann surface then has a nice interpretation in terms of a duality on dimer models

mean field behaviour of spin systems with orthogonal interaction matrix for the long range deterministic spin models with glassy behaviour of marinari parisi and ritort we prove weighted factorization properties of the correlation functions which represent the natural generalization of the factorization rules valid for the curie weiss case

a connection between the ghirlanda guerra identities and ultrametricity we consider a symmetric positive definite weakly exchangeable infinite random matrix and show that under the technical condition that its elements take a finite number of values the ghirlanda guerra identities imply ultrametricity

the extremal process of critical points of the pure p spin spherical spin glass model recently sharp results concerning the critical points of the hamiltonian of the p spin spherical spin glass model have been obtained by means of moments computations in particular these moments computations allow for the evaluation of the leading term of the ground state i e of the global minimum in this paper we study the extremal point process of critical points that is the point process associated to all critical values in the vicinity of the ground state we show that the latter converges in distribution to a poisson point process of exponential intensity in particular we identify the correct centering of the ground state and prove the convergence in distribution of the centered minimum to a minus gumbel variable these results are identical to what one obtains for a sequence of i i d variables correctly normalized namely we show that the model is in the universality class of rem

stochastic processes associated with kms states abstract a general mathematical framework is presented for the connection between quantum statistical mechanics and periodic stochastic processes stochastically positive kms systems periodic os positive stochastic processes periodic positive semigroup structures are defined and shown to be equivalent the objects of the tomita takesaki theory are explicitly exhibited in terms of the associated stochastic process and the statements of the theory are proven a two sided markov property for the stochastic process is related to a cyclicity property of the kms state perturbations are constructed by the feynman kac nelson formula the general framework is applied to kms states given by density matrices

the double wedge algebra for quantum fields on schwarzschild and minkowski spacetimes we consider the klein gordon equation m 0 on the double schwarzschild wedge of the kruskal spacetime and construct the hartle hawking state h as a thermal state relative to the boulware quantization we prove that on the double wedge h is a pure state and in the corresponding representation the left and right wedgec algebras each have the reeh schlieder property while the corresponding von neumann algebras are typeiii1 factors which are dual to i e commutants of each other we discuss the extent to which these properties may generalize to non quasi free field theories

stochastically stable quenched measures we analyze a class of stochastically stable quenched measures we prove that stochastic stability is fully characterized by an infinite family of zero average polynomials in the covariance matrix entries

rhombic embeddings of planar quad graphs given a finite or infinite planar graph all of whose faces have degree 4 we study embeddings in the plane in which all edges have length 1 that is in which every face is a rhombus we give a necessary and sucient condition for the existence of such an embedding as well as a description of the set of all such embeddings r etant donne un graphe planaire fini ou infini dont toutes les faces sont de degre 4 onetudie ses plongements dans le plan dont toutes les aretes sont de longueur 1 c estdire dont toutes les faces sont des losanges on donne une condition necessaire et susante pour l existence d un tel plongement et on decrit l ensemble de ces plongements

structure of finite rsb asymptotic gibbs measures in the diluted spin glass models we suggest a possible approach to proving the mezard parisi formula for the free energy in the diluted spin glass models such as diluted k spin or random k sat model at any positive temperature in the main contribution of the paper we show that a certain small modification of the hamiltonian in any of these models forces all finite rsb asymptotic gibbs measures in the sense of the overlaps to satisfy the mezard parisi ansatz for the distribution of spins unfortunately what is still missing is a description of the general full rsb asymptotic gibbs measures if one could show that the general case can be approximated by finite rsb case in the right sense then one could a posteriori remove the small modification of the hamiltonian to recover the mezard parisi formula for the original model

wall crossing free fermions and crystal melting we describe wall crossing for local toric calabi yau manifolds without compact four cycles in terms of free fermions vertex operators and crystal melting firstly to each such manifold we associate two states in the free fermion hilbert space the overlap of these states reproduces the bps partition function corresponding to the non commutative donaldson thomas invariants given by the modulus square of the topological string partition function secondly we introduce the wall crossing operators which represent crossing the walls of marginal stability associated to changes of the b field through each two cycle in the manifold bps partition functions in non trivial chambers are given by the expectation values of these operators thirdly we discuss crystal interpretation of such correlators for this whole class of manifolds we describe evolution of these crystals upon a change of the moduli and find crystal interpretation of the flop transition and the dt pt transition the crystals which we find generalize and unify various other calabi yau crystal models which appeared in literature in recent years

the orbifold topological vertex abstract we develop a topological vertex formalism for computing the donaldson thomas invariants of calabi yau orbifolds the basic combinatorial object is the orbifold vertex v g a generating function for the number of 3d partitions asymptotic to 2d partitions and colored by representations of a finite abelian group g acting on c 3 in the case where g z n acting on c 3 with transverse a n 1 quotient singularities we give an explicit formula for v g in terms of schur functions we discuss applications of our formalism to the donaldson thomas crepant resolution conjecture and to the orbifold donaldson thomas gromov witten correspondence we also explicitly compute the donaldson thomas partition function for some simple orbifold geometries the local football p a b 1 and the local b z 2 gerbe

phase diagram of restricted boltzmann machines and generalized hopfield networks with arbitrary priors restricted boltzmann machines are described by the gibbs measure of a bipartite spin glass which in turn can be seen as a generalized hopfield network this equivalence allows us to characterize the state of these systems in terms of their retrieval capabilities both at low and high load of pure states we study the paramagnetic spin glass and the spin glass retrieval phase transitions as the pattern i e weight distribution and spin i e unit priors vary smoothly from gaussian real variables to boolean discrete variables our analysis shows that the presence of a retrieval phase is robust and not peculiar to the standard hopfield model with boolean patterns the retrieval region becomes larger when the pattern entries and retrieval units get more peaked and conversely when the hidden units acquire a broader prior and therefore have a stronger response to high fields moreover at low load retrieval always exists below some critical temperature for every pattern distribution ranging from the boolean to the gaussian case

rescaled magnetization for critical bipartite curie weiss models we consider a bipartite generalization of the curie weiss model in a critical regime in order to study the asymptotic behavior of the random vector of the total magnetization we apply the change of variables that diagonalizes the hessian matrix of the pressure functional associated to the model we obtain a new vector that suitably rescaled weakly converges to the product of a gaussian distribution and a distribution proportional to exp xi x 4 where the positive constant xi can be computed from the pressure functional

replica equivalence in the edwards anderson model after introducing and discussing the link overlap between spin configurations we show that the edwards anderson model has a replica equivalent quenched equilibrium state a property introduced by parisi in the description of the mean field spin glass phase which generalizes ultrametricity our method is based on the control of fluctuations through the property of stochastic stability and works for all the finite dimensional spin glass models

gentle perturbations of the free bose gas i it is demonstrated that the thermal structure of the noncritical free bose gas is completely described by certain periodic generalized gaussian stochastic process or equivalently by a certain periodic generalized gaussian random field elementary properties of this gaussian stochastic thermal structure are established gentle perturbations of several types of the free thermal stochastic structure are studied in particular new models of non gaussian thermal structures are constructed and a new functional integral representation of the corresponding euclidean time green functions is obtained rigorously

consistency conditions for brane tilings abstract given a brane tiling on a torus we provide a new way to prove and generalise the recent results of szendr i mozgovoy and reineke regarding the donaldson thomas theory of the moduli space of framed cyclic representations of the associated algebra using only a natural cancellation type consistency condition we show that the algebras are 3 calabi yau and calculate donaldson thomas type invariants of the moduli spaces two new ingredients to our proofs are a grading of the algebra by the path category of the associated quiver modulo relations and a way of assigning winding numbers to pairs of paths in the lift of the brane tiling to the universal cover these ideas allow us to generalise the above results to all consistent brane tilings on k 1 surfaces we also prove a converse no consistent brane tiling on a sphere gives rise to a 3 calabi yau algebra

finite range spin glasses in the kac limit free energy and local observables we study a finite range spin glass model in arbitrary dimension where the intensity of the coupling between spins decays to zero over some distance 1 we prove that under a positivity condition for the interaction potential the infinite volume free energy of the system converges to that of the sherrington kirkpatrick model in the kac limit 0 we study the implication of this convergence for the local order parameter i e the local overlap distribution function and a family of susceptibilities associated with it and we show that locally the system behaves like its mean field analogue similar results are obtained for models with p spin interactions finally we discuss a possible approach to the problem of the existence of long range order for finite based on a large deviation functional for overlap profiles this will be developed in future work

a note on mixed matrix moments for the complex ginibre ensemble we consider the mixed matrix moments for the complex ginibre ensemble these are well known we consider the relation to the expected overlap functions of chalker and mehlig this leads to new asymptotic problems for the overlap we obtain some results but we also state some remaining open problems

uniqueness and clustering properties of gibbs states for classical and quantum unbounded spin systems we consider quantum unbounded spin systems lattice boson systems in dimensional lattice space z under appropriate conditions on the interactions we prove that in a region of high temperatures the gibbs state is unique is translationally invariant and has clustering properties the main methods we use are the wiener integral representation the cluster expansions for zero boundary conditions and for general gibbs state and explicitly dependent probability estimates for one dimensional systems we show the uniqueness of gibbs states for any value of temperature by using the method of perturbed states we also consider classical unbounded spin systems we derive necessary estimates so that all of the results for the quantum systems hold for the classical systems by straightforward applications of the methods used in the quantum case

on the noncommutative donaldson thomas invariants arising from brane tilings abstract given a brane tiling that is a bipartite graph on a torus we can associate with it a quiver potential and a quiver potential algebra under certain consistency conditions on a brane tiling we prove a formula for the donaldson thomas type invariants of the moduli space of framed cyclic modules over the corresponding quiver potential algebra we relate this formula with the counting of perfect matchings of the periodic plane tiling corresponding to the brane tiling we prove that the same consistency conditions imply that the quiver potential algebra is a 3 calabi yau algebra we also formulate a rationality conjecture for the generating functions of the donaldson thomas type invariants

cellular resolutions of noncommutative toric algebras from superpotentials abstract this paper constructs cellular resolutions for classes of noncommutative algebras analogous to those introduced by bayer and sturmfels 1998 2 in the commutative case to achieve this we generalise the dimer model construction of noncommutative crepant resolutions of three dimensional toric algebras by associating a superpotential and a notion of consistency to toric algebras of arbitrary dimension for abelian skew group algebras and algebraically consistent dimer model algebras we introduce a cell complex in a real torus whose cells describe uniformly all maps in the minimal projective bimodule resolution of a we illustrate the general construction of for an example in dimension four arising from a tilting bundle on a smooth toric fano threefold to highlight the importance of the incidence function on

morita equivalences and azumaya loci from higgsing dimer algebras abstract let a a be a cyclic contraction of dimer algebras with a non cancellative and a cancellative a is then prime noetherian and a finitely generated module over its center in contrast a is often not prime nonnoetherian and an infinitely generated module over its center we present certain morita equivalences that relate the representation theory of a with that of a we then characterize the azumaya locus of a in terms of the azumaya locus of a and give an explicit classification of the simple a modules parameterized by the azumaya locus furthermore we show that if the smooth and azumaya loci of a coincide then the smooth and azumaya loci of a coincide this provides the first known class of algebras that are nonnoetherian and infinitely generated modules over their centers with the property that their smooth and azumaya loci coincide

overlap fluctuations from the boltzmann random overlap structure we investigate overlap fluctuations of the sherrington kirkpatrick mean field spin glass model in the framework of the random overlap structure rost the concept of rost has been introduced recently by aizenman and co workers who developed a variational approach to the sherrington kirkpatrick model here we propose an iterative procedure to show that in the so called boltzmann rost aizenman contucci polynomials naturally arise for almost all values of the inverse temperature not in average over some interval only these polynomials impose restrictions on the overlap fluctuations in agreement with parisi theory

cohomological hall algebras vertex algebras and instantons we define an action of the double of cohomological hall algebra of kontsevich and soibelman on the cohomology of the moduli space of spiked instantons of nekrasov we identify this action with the one of the affine yangian of mathfrak gl 1 based on that we derive the vertex algebra at the corner mathcal w r 1 r 2 r 3 of gaiotto and rapcak we conjecture that our approach works for a big class of calabi yau categories including those associated with toric calabi yau 3 folds

analogue neural networks on correlated random graphs we consider a generalization of the hopfield model where the entries of patterns are gaussian and diluted we focus on the high storage regime and we investigate analytically the topological properties of the emergent network as well as the thermodynamic properties of the model we find that by properly tuning the dilution in the pattern entries the network can recover different topological regimes characterized by peculiar scalings of the average coordination number with respect to the system size the structure is also shown to exhibit a large degree of cliquishness even when very sparse moreover we obtain explicitly the replica symmetric free energy and the self consistency equations for the overlaps order parameters of the theory which turn out to be classical weighted sums of sub overlaps defined on all possible sub graphs finally a study of criticality is performed through a small overlap expansion of the self consistencies and through a whole fluctuation theory developed for their rescaled correlations both approaches show that the net effect of dilution in pattern entries is to rescale the critical noise level at which ergodicity breaks down

equilibrium statistical mechanics of bipartite spin systems the aim of this paper is to give an extensive treatment of bipartite mean field spin systems pure and disordered at first bipartite ferromagnets are investigated and an explicit expression for the free energy is achieved through a new minimax variational principle then via the hamilton jacobi technique the same structure of the free energy is obtained together with the existence of its thermodynamic limit and the minimax principle is connected to a standard max one the same is investigated for bipartite spin glasses by the borel cantelli lemma we obtain the control of the high temperature regime while via the double stochastic stability technique we also obtain the explicit expression of the free energy in the replica symmetric approximation uniquely defined by a minimax variational principle again we also obtain a general result that states that the free energies of these systems are convex linear combinations of their independent one party model counterparts for the sake of completeness we show further that at zero temperature the replica symmetric entropy becomes negative and consequently such a symmetry must be broken the treatment of the fully broken replica symmetry case is deferred to a forthcoming paper as a first step in this direction we start deriving the linear and quadratic constraints to overlap fluctuations