edge scaling limits for a family of non hermitian random matrix ensembles a family of random matrix ensembles interpolating between the ginibre ensemble of n n matrices with iid centered complex gaussian entries and the gaussian unitary ensemble gue is considered the asymptotic spectral distribution in these models is uniform in an ellipse in the complex plane which collapses to an interval of the real line as the degree of non hermiticity diminishes scaling limit theorems are proven for the eigenvalue point process at the rightmost edge of the spectrum and it is shown that a non trivial transition occurs between poisson and airy point process statistics when the ratio of the axes of the supporting ellipse is of order n 1 3 in this regime the family of limiting probability distributions of the maximum of the real parts of the eigenvalues interpolates between the gumbel and tracy widom distributions

random matrices in 2d laplacian growth and operator theory since it was first applied to the study of nuclear interactions by wigner and dyson almost 60 years ago random matrix theory rmt has developed into a field of its own within applied mathematics and is now essential to many parts of theoretical physics from condensed matter to high energy the fundamental results obtained so far rely mostly on the theory of random matrices in one dimension the dimensionality of the spectrum or equilibrium probability density in the last few years this theory has been extended to the case where the spectrum is two dimensional or even fractal with dimensions between 1 and 2 in this paper we review these recent developments and indicate some physical problems where the theory can be applied

a lattice boltzmann equation for diffusion the formulation of lattice gas automata lga for given partial differential equations is not straightforward and still requires some sort of magic lattice boltzmann equation lbe models are much more flexible than lga because of the freedom in choosing equilibrium distributions with free parameters which can be set after a multiscale expansion according to certain requirements here a lbe is presented for diffusion in an arbitrary number of dimensions the model is probably the simplest lbe which can be formulated it is shown that the resulting algorithm with relaxation parameter 1 is identical to an explicit finite difference efd formulation at its stability limit underrelaxation 0 1 allows stable integration beyond the stability limit of efd the time step of the explicit lbe integration is limited by accuracy and not by stability requirements

building reliable lattice monte carlo models for real drift and diffusion problems we revisit the well known issue of representing an overdamped drift and diffusion system by an equivalent lattice random walk model we demonstrate that commonly used monte carlo algorithms do not conserve the diffusion coefficient when a driving field of arbitrary amplitude is present and that such algorithms would actually require fluctuating jumping times and one clock per cartesian direction to work properly although it is in principle possible to construct valid algorithms with fixed time steps we show that no such algorithm can be used in more than two dimensions if the jumps are made along only one axis at each time step

lattices for the lattice boltzmann method a recently introduced theory of higher order lattice boltzmann models chikatamarla and karlin phys rev lett 97 190601 2006 is elaborated in detail a general theory of the construction of lattice boltzmann models as an approximation to the boltzmann equation is presented new lattices are found in all three dimensions and are classified according to their accuracy degree of approximation of the boltzmann equation the numerical stability of these lattices is argued based on the entropy principle the efficiency and accuracy of many new lattices are demonstrated via simulations in all three dimensions

straight velocity boundaries in the lattice boltzmann method various ways of implementing boundary conditions for the numerical solution of the navier stokes equations by a lattice boltzmann method are discussed five commonly adopted approaches are reviewed analyzed and compared including local and nonlocal methods the discussion is restricted to velocity dirichlet boundary conditions and to straight on lattice boundaries which are aligned with the horizontal and vertical lattice directions the boundary conditions are first inspected analytically by applying systematically the results of a multiscale analysis to boundary nodes this procedure makes it possible to compare boundary conditions on an equal footing although they were originally derived from very different principles it is concluded that all five boundary conditions exhibit second order accuracy consistent with the accuracy of the lattice boltzmann method the five methods are then compared numerically for accuracy and stability through benchmarks of two dimensional and three dimensional flows none of the methods is found to be throughout superior to the others instead the choice of a best boundary condition depends on the flow geometry and on the desired trade off between accuracy and stability from the findings of the benchmarks the boundary conditions can be classified into two major groups the first group comprehends boundary conditions that preserve the information streaming from the bulk into boundary nodes and complete the missing information through closure relations boundary conditions in this group are found to be exceptionally accurate at low reynolds number boundary conditions of the second group replace all variables on boundary nodes by new values they exhibit generally much better numerical stability and are therefore dedicated for use in high reynolds number flows

the spectrum of coupled random matrices the study of the spectrum of coupled random matrices has received rather little attention to the best of our knowledge coupled random matrices have been studied to some extent by mehta in this work we explain how the integrable technology can be brought to bear to gain insight into the nature of the distribution of the spectrum of coupled hermitean random matrices and the equations the associated probabilities satisfy in particular the two toda lattice its algebra of symmetries and its vertex operators will play a prominent role in this interaction namely the method is to introduce time parameters in an artificial way and to dress up a certain matrix integral with a vertex integral operator for which we find virasoro like differential equations these methods lead to very simple nonlinear third order partial differential equations for the joint statistics of the spectra of two coupled gaussian random matrices comment 56 pages published version abstract added in migration

conformal maps and integrable hierarchies we show that conformal maps of simply connected domains with an analytic boundary to a unit disk have an intimate relation to the dispersionless 2d toda integrable hierarchy the maps are determined by a particular solution to the hierarchy singled out by the conditions known as string equations the same hierarchy locally solves the 2d inverse potential problem i e reconstruction of the domain out of a set of its harmonic moments this is the same solution which is known to describe 2d gravity coupled to c matter we also introduce a concept of the function for analytic curves

a knudsen layer theory for lattice gases abstract a knudsen layer theory is presented for lattice gases with arbitrary boundary conditions analytical results are obtained for two special orientations these exhibit anisotropic knudsen layers provided suitable conditions are satisfied however the standard boundary conditions used in previous simulations are shown to be isotropic the bulk steady state extending everywhere in the gas this theory allows a more accurate localization of the obstacle with respect to the lattice nodes these results are in good agreement with the numerical simulations

topology of quadrature domains we address the problem of topology of quadrature domains namely we give upper bounds on the connectivity of the domain in terms of the number of nodes and their multiplicities in the quadrature identity

on the structure of correlation functions in the normal matrix model we study the structure of the normal matrix model nmm we show that all correlation functions of the model with an axially symmetric potential can be expressed in terms of a single holomorphic function of one variable this observation is used to demonstrate the exact solvability of the model the two point correlation function is calculated in the continuum limit the answer is proven to be universal i e potential independent up to a change of the scale in connection with nmm with a general polynomial potential we have developed a two dimensional free fermion formalism and constructed a family of completely integrable hierarchies of non linear differential equations which we call the extended kp n hierarchies the well known kp hierarchy is a lower dimensional reduction of this family the extended kp 1 hierarchy contains the 2 1 dimensional burgers equations the partition function of the n n nmm is the function of the extended kp n hierarchy which is invariant with respect to a subalgebra of an algebra of all infinitesimal diffeomorphisms of the plane

vortex patch equilibria of the euler equation and random normal matrices a new analogy is drawn between vortex patch or v state equilibria of the euler equations for ideal fluids and the planar limit of random normal matrix models physically the former is a quite different fluid dynamical problem to hele shaw flow or laplacian growth to which an analogy with matrix models has become well known in recent years the connection of random matrices with vortex dynamics is made via the so called modified schwarz potential this theoretical link while interesting in itself has immediate ramifications for random matrix theory by virtue of a transfer of mathematical technology already well developed for vortex dynamics hence for multi support matrix models in the planar limit we describe a constructive approach to the inverse problem of finding the shapes of the eigenvalue supports for a given potential using automorphic conformal mappings within a schottky model of the underlying algebraic curves and use of the schottky klein prime function two support eigenvalue distributions in a quartic potential are given as an illustrative example

correlation functions of eigenvalues of multi matrix models and the limit of a time dependent matrix the universality of correlation functions of eigenvalues of large random matrices has been observed in various physical systems and proved in some particular cases as the hermitian one matrix model with polynomial potential here we consider the more difficult case of a unidimensional chain of hermitian matrices with first neighbour couplings and polynomial potentials an asymptotic expression of the orthogonal polynomials and a generalization of the darboux christoffel theorem allow us to find new results for the correlations of eigenvalues of different matrices of the chain eventually we consider the limit of the infinite chain of matrices which can be interpreted as a time dependent hermitian one matrix model and give the correlation functions of eigenvalues at different times

random normal matrices and ward identities we consider the random normal matrix ensemble associated with a potential in the plane of sufficient growth near infinity it is known that asymptotically as the order of the random matrix increases indefinitely the eigenvalues approach a certain equilibrium density given in terms of frostman s solution to the minimum energy problem of weighted logarithmic potential theory at a finer scale we may consider fluctuations of eigenvalues about the equilibrium in the present paper we give the correction to the expectation of the fluctuations and we show that the potential field of the corrected fluctuations converge on smooth test functions to a gaussian free field with free boundary conditions on the droplet associated with the potential

two matrix model with semiclassical potentials and extended whitham hierarchy we consider the two matrix model with potentials whose derivatives are arbitrary rational functions of fixed pole structure and the support of the spectra of the matrices are union of intervals hard edges we derive an explicit formula for the planar limit of the free energy and we derive a calculus which allows us to compute derivatives of arbitrarily high order by extending classical rauch s variational formulae the four point correlation functions are explicitly worked out the formalism extends naturally to the computation of residue formulae for the tau function of the so called universal whitham hierarchy studied mainly by i krichever our setting extends the moduli space in that there are certain extra data

conformal mappings and dispersionless toda hierarchy ii general string equations in this article we classify the solutions of the dispersionless toda hierarchy into degenerate and non degenerate cases we show that every non degenerate solution is determined by a function mathcal h z 1 z 2 of two variables we interpret these non degenerate solutions as defining evolutions on the space mathfrak d of pairs of conformal mappings g f where g is a univalent function on the exterior of the unit disc f is a univalent function on the unit disc normalized such that g f 0 0 and f 0 g 1 for each solution we show how to define the natural time variables t n n in mathbb z as complex coordinates on the space mathfrak d we also find explicit formulas for the tau function of the dispersionless toda hierarchy in terms of mathcal h z 1 z 2 imposing some conditions on the function mathcal h z 1 z 2 we show that the dispersionless toda flows can be naturally restricted to the subspace of mathfrak d defined by f w 1 overline g 1 bar w this recovers the result of zabrodin theor math phy 12 1511 1525 2001

integrable structure of the dirichlet boundary problem in two dimensions we study how the solution of the two dimensional dirichlet boundary problem for smooth simply connected domains depends upon variations of the data of the problem we show that the hadamard formula for the variation of the dirichlet green function under deformations of the domain reveals an integrable structure the independent variables corresponding to the infinite set of commuting flows are identified with harmonic moments of the domain the solution to the dirichlet boundary problem is expressed through the tau function of the dispersionless toda hierarchy we also discuss a degenerate case of the dirichlet problem on the plane with a gap in this case the tau function is identical to the partition function of the planar large n limit of the hermitian one matrix model

optimizing the accuracy of lattice monte carlo algorithms for simulating diffusion the behavior of a lattice monte carlo lmc algorithm if it is designed correctly must approach that of the continuum system that it is designed to simulate as the time step and the mesh step tend to zero however we show for an algorithm for unbiased particle diffusion that if one of these two parameters remains fixed the accuracy of the algorithm is optimal for a certain finite value of the other parameter in one dimension the optimal algorithm with moves to the two nearest neighbor sites reproduces the correct second and fourth moments and minimizes the error for the higher moments at large times of the particle distribution and preserves the first two moments of the first passage time distributions in two and three dimensions the same level of accuracy requires simultaneous moves along two axes diagonal moves such moves attempting to cross an impenetrable boundary should be projected along the boundary rather than simply rejected we also treat the case of absorbing boundaries we discuss the relation between optimally accurate lmc algorithms and a particular case of lattice boltzmann lb algorithms for simulating diffusion and compare the computational efficiency of optimal lmc and optimal lb algorithms

random matrices close to hermitian or unitary overview of methods and results the paper discusses recent progress in understanding statistical properties of eigenvalues of weakly non hermitian and non unitary random matrices the first type of ensembles is of the form i with being a large random n n hermitian matrix with independent entries deformed by a certain anti hermitian n n matrix i satisfying in the limit of large dimension n the condition tr 2 n tr 2 here can be either a random or just a fixed given hermitian matrix ensembles of such a type with 0 emerge naturally when describing quantum scattering in systems with chaotic dynamics and serve to describe resonance statistics related models are used to mimic complex spectra of the dirac operator with chemical potential in the context of quantum chromodynamics ensembles of the second type arising naturally in scattering theory of discrete time systems are formed by n n matrices with complex entries such that i for 0 this coincides with the circular unitary ensemble and 0 i describes deviation from unitarity our result amounts to answering statistically the following old question given the singular values of a matrix describe the locus of its eigenvalues we systematically show that the obtained expressions for the correlation functions of complex eigenvalues describe a non trivial crossover from wigner dyson statistics of real unimodular eigenvalues typical of hermitian unitary matrices to ginibre statistics in the complex plane typical of ensembles with strong non hermiticity tr 2 tr 2 when n finally we discuss scarce results available on eigenvector statistics for weakly non hermitian random matrices

almost hermitian random matrices crossover from wigner dyson to ginibre eigenvalue statistics by using the method of orthogonal polynomials we analyze the statistical properties of complex eigenvalues of random matrices describing a crossover from hermitian matrices characterized by the wigner dyson statistics of real eigenvalues to strongly non hermitian ones whose complex eigenvalues were studied by ginibre two point statistical measures as e g spectral form factor number variance and small distance behavior of the nearest neighbor distance distribution p s are studied in more detail in particular we found that the latter function may exhibit unusual behavior p s ensuremath propto s 5 2 for some parameter values

large n expansion for the 2d dyson gas we discuss the 1 n expansion of the free energy of n logarithmically interacting charges in the plane in an external field for some particular values of the inverse temperature this system is equivalent to the eigenvalue version of certain random matrix models where it is referred to as the dyson gas of eigenvalues to find the free energy at large n and the structure of 1 n corrections we first use the effective action approach and then confirm the results by solving the loop equation the results obtained give some new representations of the mathematical objects related to the dirichlet boundary value problem complex analysis and spectral geometry of exterior domains they also suggest interesting links with bosonic field theory on riemann surfaces gravitational anomalies and topological field theories

large scale correlations in normal non hermitian matrix ensembles we compute the large scale macroscopic correlations in ensembles of normal random matrices with a general non gaussian measure and in ensembles of general non hermitian matrices with a class of non gaussian measures in both cases the eigenvalues are complex and in the large n limit they occupy a domain in the complex plane for the case when the support of eigenvalues is a connected compact domain we compute two three and four point connected correlation functions in the first non vanishing order in 1 n in a manner that the algorithm of computing higher correlations becomes clear the correlation functions are expressed through the solution of the dirichlet boundary problem in the domain complementary to the support of eigenvalues

generic critical points of normal matrix ensembles the evolution of the degenerate complex curve associated with the ensemble at a generic critical point is related to the finite time singularities of laplacian growth it is shown that the scaling behaviour at a critical point of singular geometry x3 y2 is described by the first painleve transcendent the regularization of the curve resulting from discretization is discussed

on coupled longitudinal and lateral vibrations of elastic rods abstract in this paper the coupled longitudinal and lateral vibrations of a rectangular parallelepiped composed of a uniform isotropic elastic body are examined the one dimensional model for the body was developed by green naghdi and several of their co workers it is based on a cosserat or directed rod theory the frequency spectrum is found to divide naturally into three regions which are separated by degenerate cases when the cross section of the body is square a natural splitting of the modes occurs one half of this split involves motions coupling the axial extension and symmetric lateral deformation of the body the other half involves asymmetric lateral deformations the cases of a free free rod and a cantilevered rod are discussed and the results compared to existing works in the literature which use the three dimensional theory of elasticity to model the vibrations

theory of stochastic laplacian growth we generalize the diffusion limited aggregation by issuing many randomly walking particles which stick to a cluster at the discrete time unit providing its growth using simple combinatorial arguments we determine probabilities of different growth scenarios and prove that the most probable evolution is governed by the deterministic laplacian growth equation a potential theoretical analysis of the growth probabilities reveals connections with the tau function of the integrable dispersionless limit of the two dimensional toda hierarchy normal matrix ensembles and the two dimensional dyson gas confined in a non uniform magnetic field we introduce the time dependent hamiltonian which generates transitions between different classes of equivalence of closed curves and prove the hamiltonian structure of the interface dynamics finally we propose a relation between probabilities of growth scenarios and the semi classical limit of certain correlation functions of light exponential operators in the liouville conformal field theory on a pseudosphere