givental formula in terms of virasoro operators we present a conjecture that the universal enveloping algebra of differential operators tk over c coincides in the origin with the universal enveloping algebra of the borel subalgebra of virasoro generators from the kontsevich model thus we can decompose any pseudo differential operator to a combination of the virasoro operators using this decomposition we present the right hand side of the givental formula math ag 0008067 as a constant part of the differential operator we introduce in the case of cp1 studied in the paper by song and song hep th 0103254 the left hand side of the givental formula is a unit which imposes certain constraints on this differential operator we explicitly check that these constraints are correct up to o q4 we also propose a conjecture of factorization modulo hirota equation of the differential operator introduced and check this conjecture with the same accuracy

s duality as fourier transform for arbitrary ϵ1 ϵ2 the alday gaiotto tachikawa relations reduce s duality to the modular transformations of conformal blocks it was recently conjectured that for the four point conformal block the modular transform up to the non perturbative contributions can be written in the form of the ordinary fourier transform when 1 2 1 here i extend this conjecture to general values of 1 2 namely i argue that for a properly normalized four point conformal block the s duality is perturbatively given by the fourier transform for arbitrary values of the deformation parameters 1 2 the conjecture is based on explicit perturbative computations in the first few orders of the string coupling constant g2 1 2 and hypermultiplet masses

on the resolution of ordering ambiguities associated with the path integral representation of stochastic processes simple path integral approaches to multiplicative stochastic processes are severely hampered by an inherent ambiguity in the underlying lagrangian associated with operator ordering problems the author shows how given the fokker planck equation the ambiguity can be resolved direct consequences for noise induced transitions reaction diffusion models and polymer dynamics are briefly discussed

two species coagulation approach to consensus by group level interactions we explore the self organization dynamics of a set of entities by considering the interactions that affect the different subgroups conforming the whole to this end we employ the widespread example of coagulation kinetics and characterize which interaction types lead to consensus formation and which do not as well as the corresponding different macroscopic patterns the crucial technical point is extending the usual one species coagulation dynamics to the two species one this is achieved by means of introducing explicitly solvable kernels which have a clear physical meaning the corresponding solutions are calculated in the long time limit in which consensus may or may not be reached the lack of consensus is characterized by means of scaling limits of the solutions the possible applications of our results to some topics in which consensus reaching is fundamental such as collective animal motion and opinion spreading dynamics are also outlined

predictive information in a nonequilibrium critical model we propose predictive information that is information between a long past of duration t and the entire infinitely long future of a time series as a general order parameter to study phase transitions in physical systems independently of the underlying dynamics it can be used in particular to study nonequilibrium transitions and other exotic transitions where a simpler order parameter cannot be identified using traditional symmetry arguments as an example we calculate predictive information for a stochastic nonequilibrium dynamics problem that forms an absorbing state under a continuous change of a parameter the information at the transition point diverges as logt and we calculate the expression for a smooth crossover to t0 away from the transition

nonequilibrium lattice models series analysis of steady states a perturbation theory for steady states of interacting particle systems is developed and applied to several lattice models with nonequilibrium critical points near an absorbing state the expansion is expressed directly in terms of the kinetic parameter creation rate rather than in powers of the interaction an algorithm for generating series expansions for local properties is described order parameter series 16 terms and precise estimates of critical properties are presented for the one dimensional contact process and several related models

on the dotsenko fateev representation of the toric conformal blocks we demonstrate that the recent ansatz of maruyoshi and yagi 2010 arxiv 1009 5553 inspired by the original remark of r dijkgraaf and c vafa reproduces the toric conformal blocks in the same sense as the spherical blocks given by the integral representation of mironov et al 2010 j high energy phys jhep02 2010 030 arxiv 0911 5721 and mironov et al 2010 arxiv 1001 0563 with a peculiar choice of open integration contours for screening insertions in other words we provide some evidence that the toric conformal blocks are reproduced by appropriate ensembles not only in the large n limit but also at finite n the check is explicitly performed at the first two levels for the 1 point toric functions generalizations to higher genera are briefly discussed

exact macroscopic dynamics in non equilibrium chemical systems non equilibrium reaction diffusion models show an exciting array of critical phenomena ranging from the continuous transitions reminiscent of equilibrium systems to the spectacular chemical oscillators transforming the underlying stochastic master equations using poisson techniques the author obtains for the first time a comprehensive description of the dynamics including in particular the problematical multitime correlation functions the formalism is ideally adapted for the use of scaling and renormalisation group ideas

characterization of jacobian varieties in terms of soliton equations pour dans le demi espace superieur de siegel h g soit x c g z g z g la variete abelienne principalement polarisee correspondante et soit z z m e z gexp 2 i t mz i t m m la fonction theta de riemann pour x les deux conditions a et b pour x sont equivalentes a 1 il existe des vecteurs a 1 a 2 a 3 c g a 1 0 et une forme quadratique q t i j 1 3 q ij t i t j q ij c tel que pour c g t t exp q t t 1 a 1 t 2 a 2 t 3 a 3 donne une solution a l equation de kadomtsev petviashvili sous la forme d hirota d 1 4 3d 2 2 4d 1 d 3 0 2 le diviseur de x est irreductible b x est isomorphe a la variete jacobienne d une courbe lisse complete de genre g sur c

equivalence between poisson representation and fock space formalism for birth death processes the equivalence between the poisson representation and the fock space formalism both used to derive coarse grained equations from the master equation for non equilibrium systems is explicitly demonstrated

integrability of hurwitz partition functions partition functions often become functions of integrable hierarchies if they are considered dependent on infinite sets of parameters called time variables the hurwitz partition functions z rd2 kr r t 1 r t k exp n ncr n depend on two types of such time variables t and kp toda integrability in t requires that k 2 and also that cr n are selected in a rather special way in particular the naive cut and join operators are not allowed for n 2 integrability in further restricts the choice of cr n forbidding for example the free cumulants it also requires that k 1 the quasi classical integrability the wdvv equations is naturally present in variables but also requires a careful definition of the generating function

on the combinatorics of several integrable hierarchies we demonstrate that statistics of certain classes of set partitions are described by generating functions related to the burgers ibragimov shabat and korteweg de vries integrable hierarchies

topological recursion in enumerative geometry and random matrices we review the method of symplectic invariants recently introduced to solve matrix models loop equations in the so called topological expansion and further extended beyond the context of matrix models for any given spectral curve one defines a sequence of differential forms and a sequence of complex numbers fg called symplectic invariants we recall the definition of fg s and we explain their main properties in particular symplectic invariance integrability modularity as well as their limits and their deformations then we give several examples of applications in particular matrix models enumeration of discrete surfaces maps algebraic geometry and topological strings and non intersecting brownian motions

high order and mass conservative methods for the conservative allen cahn equation the conservative allen cahn ac equation has been studied analytically and numerically our mathematical analysis and numerical experiment however show that previous numerical methods are not second order accurate in time and or do not conserve the initial mass the aim of this paper is to propose high order and mass conservative methods for solving the conservative ac equation in the methods we discretize the conservative ac equation by using a fourier spectral method in space and first second and third order implicit explicit runge kutta schemes in time we show that the methods inherit the mass conservation numerical experiments are presented demonstrating the accuracy and efficiency of proposed methods

intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

on a conjecture of givental these brief notes record our puzzles and findings surrounding givental s recent conjecture which expresses higher genus gromov witten invariants in terms of the genus 0 data we limit our considerations to the case of a complex projective line whose gromov witten invariants are well known and easy to compute we make some simple checks supporting his conjecture

field theoretic treatment of an effective action for a model of catalyzed autoamplification reaction diffusion models can exhibit continuous phase transitions in behaviors and their dynamics at criticality often exhibit scalings with key parameters that can be characterized by exponents while models with only a single field that transitions between absorbing and nonabsorbing states are well characterized and typically fall in the directed percolation universality class the effects of coupling multiple fields remain poorly understood we recently introduced a model which has three fields one of which relaxes exponentially one of which displays critical behavior and one of which has purely diffusive dynamics but exerts an influence on the critical field tchernookov j chem phys 130 134906 2009 simulations suggested that this model is in a universality class distinct from other reaction diffusion systems studied previously although the three fields give rise to interesting physics they complicate analysis of the model with renormalization group methods here we show how to systematically simplify the action for this model such that analytical expressions for the exponents of this universality class can be obtained by standard means we expect the approach taken here to be of general applicability in reaction diffusion systems with coupled order parameters that display qualitatively different behaviors close to criticality

toda hierarchies and their applications the 2d toda hierarchy occupies a central position in the family of integrable hierarchies of the toda type the 1d toda hierarchy and the ablowitz ladik aka relativistic toda hierarchy can be derived from the 2d toda hierarchy as reductions these integrable hierarchies have been applied to various problems of mathematics and mathematical physics since 1990s a recent example is a series of studies on models of statistical mechanics called the melting crystal model this research has revealed that the aforementioned two reductions of the 2d toda hierarchy underlie two different melting crystal models technical clues are a fermionic realization of the quantum torus algebra special algebraic relations therein called shift symmetries and a matrix factorization problem the two melting crystal models thus exhibit remarkable similarity with the hermitian and unitary matrix models for which the two reductions of the 2d toda hierarchy play the role of fundamental integrable structures

polynomial averages in the kontsevich model we obtain in closed form averages of polynomials taken over hermitian matrices with the gaussian measure involved in the kontsevich integral and prove a conjecture of witten enabling one to express analogous averages with the full cubic potential measure as derivatives of the partition function with respect to traces of inverse odd powers of the external argument the proofs are based on elementary algebraic identities involving a new set of invariant polynomials of the linear group closely related to the general schur functions

fluctuations in the limit cycle state and the problem of phase chaos abstract gaussian fluctuations and first order fluctuation corrections to the deterministic solution are investigated in the framework of the generalized ginzgurg landau type equation of motion exhibiting a hard mode transition leading to a homogeneous limit cycle state it is shown that the stationary distribution of the fluctuations around the limit cycle is not of the form of a ginzburg landau functional the nature of the further instability in the post bifurcational region resulting in the phase chaos in the deterministic problem is found to be qualitatively changed by the presence of noise

weighted hurwitz numbers and topological recursion an overview multiparametric families of hypergeometric functions of kp or toda type serve as generating functions for weighted hurwitz numbers providing weighted enumerations of branched covers of the riemann sphere a graphical interpretation of the weighting is given in terms of constellations mapped onto the covering surface the theory is placed within the framework of topological recursion with the baker function at t 0 shown to satisfy the quantum spectral curve equation whose classical limit is rational a basis for the space of formal power series in the spectral variable is generated that is adapted to the grassmannian element associated with the function multicurrent correlators are defined in terms of the function and shown to provide an alternative generating function for weighted hurwitz numbers fermionic vacuum state expectation value representations are provided for the adapted bases pair correlators and multicurrent correlators choosing the weight generating function as a polynomial and restricting the number of nonzero second kp flow parameters in the toda function to be finite implies a finite rank covariant derivative equation with rational coefficients satisfied by a finite window of adapted basis elements the pair correlator is shown to provide a christoffel darboux type finite rank integrable kernel and the wkb series coefficients of the associated adjoint system are computed recursively leading to topological recursion relations for the generators of the weighted hurwitz numbers

asymptotics of quantum weighted hurwitz numbers this work concerns both the semiclassical and zero temperature asymptotics of quantum weighted double hurwitz numbers the partition function for quantum weighted double hurwitz numbers can be interpreted in terms of the energy distribution of a quantum bose gas with vanishing fugacity we compute the leading semiclassical term of the partition function for three versions of the quantum weighted hurwitz numbers as well as lower order semiclassical corrections the classical limit is shown to reproduce the simple single and double hurwitz numbers studied by okounkov and pandharipande 2000 math res lett 7 447 53 2000 lett math phys 53 59 74 the kp toda function that serves as generating function for the quantum hurwitz numbers is shown to have the function of okounkov and pandharipande 2000 math res lett 7 447 53 2000 lett math phys 53 59 74 as its leading term in the classical limit and with suitable scaling the same holds for the partition function the weights and expectations of hurwitz numbers we also compute the zero temperature limit of the partition function and quantum weighted hurwitz numbers the kp or toda function serving as generating function for the quantum hurwitz numbers are shown to give the one for belyi curves in the zero temperature limit and with suitable scaling the same holds true for the partition function the weights and the expectations of hurwitz numbers

appearance of correlations and symmetry breaking in non equilibrium reaction diffusion systems using the reaction diffusion master equation of chemical kinetics the authors study fluctuations and correlation functions around non equilibrium steady states they demonstrate that the underlying lagrangian description in the poisson representation possesses in the equilibrium limit an exact symmetry which ensures that statistical mechanics is recovered and that the intrinsically non equilibrium long range correlations in these systems vanish an explicit form for the two time single particle correlation function including both the long range and short range contributions is constructed using mean field arguments

phase ordering with a global conservation law ostwald ripening and coalescence globally conserved phase ordering dynamics is investigated in systems with short range correlations at t 0 a ginzburg landau equation with a global conservation law is employed as the phase field model the conditions are found under which the sharp interface limit of this equation is reducible to the area preserving motion by curvature numerical simulations show that for both critical and off critical quench the equal time pair correlation function exhibits dynamic scaling and the characteristic coarsening length obeys l t ensuremath sim t 1 2 for the critical quench our results are in excellent agreement with earlier results for off critical quench ostwald ripening we investigate the dynamics of the size distribution function of the minority phase domains the simulations show that at large times this distribution function has a self similar form with growth exponent 1 2 the scaled distribution however strongly differs from the classical wagner distribution we attribute this difference to coalescence of domains a theory of ostwald ripening is developed that takes into account binary coalescence events the theoretical scaled distribution function agrees well with that obtained in the simulations

field theories for kinetic growth models field theories for kinetic growth models such as dla or the eden model are formulated using simple reaction diffusion master equations the approach based on poisson transform techniques specifies the stochastic fluctuations exactly the author studies the mean field approximations fluctuations and the renormalisation group

laplacian growth in a channel and hurwitz numbers we study the integrable structure of the 2d laplacian growth problem with zero surface tension in an infinite channel with periodic boundary conditions in a transverse direction similarly to the laplacian growth in radial geometry this problem can be embedded into the 2d toda lattice hierarchy in the zero dispersion limit however the relevant solution to the hierarchy is different we characterize this solution by the string equations and construct the corresponding dispersionless tau function this tau function is shown to coincide with the genus zero part of the generating function for double hurwitz numbers

matrix models and a proof of the open analog of witten s conjecture in a recent work r pandharipande j p solomon and the second author have initiated a study of the intersection theory on the moduli space of riemann surfaces with boundary they conjectured that the generating series of the intersection numbers satisfies the open kdv equations in this paper we prove this conjecture our proof goes through a matrix model and is based on a kontsevich type combinatorial formula for the intersection numbers that was found by the second author

critical fluctuations around non equilibrium steady states scaling and renormalisation group concepts are used to derive phenomenological langevin equations for non equilibrium steady states for the schlogl model the authors unambiguously identify the critical langevin equation in 4 epsilon dimensions simply by requiring that the correct mean field limit be reproduced the approach is applicable to models for which only the mean field behaviour is well established

instantons and merons in matrix models abstract various branches of matrix model partition functions can be represented as intertwined products of universal elementary constituents gaussian partition functions z g and kontsevich functions z k in physical terms this decomposition is the matrix model version of multi instanton and multi meron configurations in yang mills theories technically decomposition formulas are related to the representation theory of algebras of krichever novikov type on families of spectral curves with additional seiberg witten structure representations of these algebras are encoded in terms of the global partition functions they interpolate between z g and z k associated with different singularities on spectral riemann surfaces this construction is nothing but m theory like unification of various matrix models with explicit and representative realization of dualities

hurwitz numbers and integrable hierarchy of volterra type a generating function of the single hurwitz numbers of the riemann sphere is a tau function of the lattice kp hierarchy the associated lax operator l turns out to be expressed as where is a difference differential operator of the form satisfies a set of lax equations that form a continuum version of the bogoyavlensky itoh aka hungry lotka volterra hierarchies emergence of this underlying integrable structure is further explained in the language of generalized string equations for the lax and orlov schulman operators of the 2d toda hierarchy this leads to logarithmic string equations which are confirmed with the help of a factorization problem of operators

fermionic approach to weighted hurwitz numbers and topological recursion a fermionic representation is given for all the quantities entering in the generating function approach to weighted hurwitz numbers and topological recursion this includes kp and 2d toda tau functions of hypergeometric type which serve as generating functions for weighted single and double hurwitz numbers the baker function which is expanded in an adapted basis obtained by applying the same dressing transformation to all vacuum basis elements the multipair correlators and the multicurrent correlators multiplicative recursion relations and a linear differential system are deduced for the adapted bases and their duals and a christoffel darboux type formula is derived for the pair correlator the quantum and classical spectral curves linking this theory with the topological recursion program are derived as well as the generalized cut and join equations the results are detailed for four special cases the simple single and double hurwitz numbers the weakly monotone case corresponding to signed enumeration of coverings the strongly monotone case corresponding to belyi curves and the simplest version of quantum weighted hurwitz numbers

breakdown of universality in multi cut matrix models we solve the puzzle of the disagreement between orthogonal polynomials methods and mean field calculations for random n n matrices with a disconnected eigenvalue support we show that the difference does not stem from a 2 symmetry breaking but from the discreteness of the number of eigenvalues this leads to additional terms quasiperiodic in n which must be added to the naive mean field expressions our result invalidates the existence of a smooth topological large n expansion and some postulated universality properties of correlators we derive the large n expansion of the free energy for the general two cut case from it we rederive by a direct and easy mean field like method the two point correlators and the asymptotic orthogonal polynomials we extend our results to any number of cuts and to non real potentials

an explicit form for kerov s character polynomials kerov considered the normalized characters of irreducible representations of the symmetric group evaluated on a cycle as a polynomial in free cumulants biane has proved that this polynomial has integer coefficients and made various conjectures recently sniady has proved biane s conjectured explicit form for the first family of nontrivial terms in this polynomial in this paper we give an explicit expression for all terms in kerov s character polynomials our method is through lagrange inversion

identification of the givental formula with the spectral curve topological recursion procedure we identify the givental formula for the ancestor formal gromov witten potential with a version of the topological recursion procedure for a collection of isolated local germs of the spectral curve as an application we prove a conjecture of norbury and scott on the reconstruction of the stationary sector of the gromov witten potential of mathbb c rm p 1 via a particular spectral curve

polynomiality of hurwitz numbers bouchard marino conjecture and a new proof of the elsv formula in this paper we give a new proof of the elsv formula first we refine an argument of okounkov and pandharipande in order to prove quasi polynomiality of hurwitz numbers without using the elsv formula the only way to do that before used the elsv formula then using this polynomiality we give a new proof of the bouchard marino conjecture after that using the correspondence between the givental group action and the topological recursion coming from matrix models we prove the equivalence of the bouchard marino conjecture and the elsv formula it is a refinement of an argument by eynard c 2015 elsevier inc all rights reserved

smoluchowski and beyond a field theoretical study of coagulation fragmentation processes adopting simple field theoretical techniques the author demonstrates how the smoluchowski approach to reaction diffusion models can be systematically corrected focusing on coagulation fragmentation processes he identifies the upper critical dimension dc 2 below which finite mobility corrections are important and full universality lattice independence etc can be achieved trends seem by direct simulation of simple lattice models for d 2 are confirmed whilst problems of interpretation exist for d 2

scaling in reaction diffusion systems the authors study irreversible reaction diffusion models of the form px from k1 from mx to k2 to qx sx from k3 from nx m dc 2 n m 1 n 1 which they study for d dc using the renormalisation group in conjunction with then epsilon expansion and exact techniques for m 1 and any n the transition remains continuous they present epsilon expansions for the case n 3 by contrast for m 2 and any n the system shows a fluctuation induced first order transition

pole expansion determination of the pair correlation function of a simple fluid abstract an explicit expression for the pair correlation function is given in terms of an expansion in the poles of the structure factor s q in the complex q plane this is derived when the direct correlation function or structure factor is of known analytical form an illustrative example is given for a hard sphere yukawa fluid in the mean spherical approximation

non equilibrium scaling in the schlogl model from the reaction diffusion master equation formulation of the schlogl model the authors develop a comprehensive description of the non equilibrium dynamics using the poisson transformation and the dynamical renormalisation group they firstly show how the equilibrium limit is recovered and secondly describe the low frequency low momenta scaling behaviour of the concentration fluctuations the approach extends simplifies or corrects previous studies and is easily generalised to treat more complicated systems