universal amplitude ratios for self avoiding walks on the kagome lattice we have calculated exactly the number the mean square end to end distance the mean square radius of gyration and the mean square distance of a monomer from the origin for n step self avoiding walks on the kagome lattice up to 30 steps we have estimated the connective constant and the critical amplitudes our numerical results are consistent with the theoretical prediction by cardy and saleur 1989 on the universality for certain amplitude ratios

cavity approach to the spectral density of sparse symmetric random matrices the spectral density of various ensembles of sparse symmetric random matrices is analyzed using the cavity method we consider two cases matrices whose associated graphs are locally treelike and sparse covariance matrices we derive a closed set of equations from which the density of eigenvalues can be efficiently calculated within this approach the wigner semicircle law for gaussian matrices and the marcenko pastur law for covariance matrices are recovered easily our results are compared with numerical diagonalization showing excellent agreement

spectra of complex networks we propose a general approach to the description of spectra of complex networks for the spectra of networks with uncorrelated vertices and a local treelike structure exact equations are derived these equations are generalized to the case of networks with correlations between neighboring vertices the tail of the density of eigenvalues ensuremath rho ensuremath lambda at large ensuremath lambda is related to the behavior of the vertex degree distribution p k at large k in particular as p k ensuremath sim k ensuremath ensuremath gamma ensuremath rho ensuremath lambda ensuremath sim ensuremath lambda 1 ensuremath 2 ensuremath gamma we propose a simple approximation which enables us to calculate spectra of various graphs analytically we analyze spectra of various complex networks and discuss the role of vertices of low degree we show that spectra of locally treelike random graphs may serve as a starting point in the analysis of spectral properties of real world networks e g of the internet

spectra of sparse random matrices we compute the spectral density for ensembles of sparse symmetric random matrices using replica our formulation of the replica symmetric ansatz shares the symmetries of that suggested in a seminal paper by rodgers and bray symmetry with respect to permutation of replica and rotation symmetry in the space of replica but uses a different representation in terms of superpositions of gaussians it gives rise to a pair of integral equations which can be solved by a stochastic population dynamics algorithm remarkably our representation allows us to identify pure point contributions to the spectral density related to the existence of normalizable eigenstates our approach is not restricted to matrices defined on graphs with poissonian degree distribution matrices defined on regular random graphs or on scale free graphs are easily handled we also look at matrices with row constraints such as discrete graph laplacians our approach naturally allows us to unfold the total density of states into contributions coming from vertices of different local coordinations and an example of such an unfolding is presented our results are well corroborated by numerical diagonalization studies of large finite random matrices

outliers in the spectrum of iid matrices with bounded rank perturbations it is known that if one perturbs a large iid random matrix by a bounded rank error then the majority of the eigenvalues will remain distributed according to the circular law however the bounded rank perturbation may also create one or more outlier eigenvalues we show that if the perturbation is small then the outlier eigenvalues are created next to the outlier eigenvalues of the bounded rank perturbation but if the perturbation is large then many more outliers can be created and their law is governed by the zeroes of a random laurent series with gaussian coefficients on the other hand these outliers may be eliminated by enforcing a row sum condition on the final matrix

correlations of eigenvectors for non hermitian random matrix models we establish a general relation between the diagonal correlator of eigenvectors and the spectral green close quote s function for non hermitian random matrix models in the large n limit we apply this result to a number of non hermitian random matrix models and show that the outcome is in good agreement with numerical results copyright ital 1999 ital the american physical society

spectral density of random graphs with topological constraints the spectral density of random graphs with topological constraints is analysed using the replica method we consider graph ensembles featuring generalised degree degree correlations as well as those with a community structure in each case an exact solution is found for the spectral density in the form of consistency equations depending on the statistical properties of the graph ensemble in question we highlight the effect of these topological constraints on the resulting spectral density

how many eigenvalues of a random matrix are real let a be an n x n matrix whose elements are independent random variables with standard normal distributions as n oo the expected number of real eigenvalues is asymptotic to v 7r we obtain a closed form expres sion for the expected number of real eigenvalues for finite n and a formula for the density of a real eigenvalue for finite n asymptotically a real normalized eigenvalue al ii of such a random matrix is uniformly distributed on the in terval 1 1 analogous but strikingly different results are presented for the real generalized eigenvalues we report on numerical experiments confirming these results and suggesting that the assumption of normality is not important for the asymptotic results department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address edelmanfmath mit edu arts and sciences kapi olani community college 4303 diamond head road honolulu hawaii 98616 e mail address kostlanfkahuna math hawaii edu tj watson rsearch center 32 2 ibm yorktown heights new york 10598 0218 e mail address shubqwatson ibm com this content downloaded from 157 55 39 29 on tue 12 apr 2016 08 54 01 utc all use subject to http about jstor org terms

universal distance ratios for 2d saws series results the mean square distance of a monomer to the origin and the mean square distance of the centre of mass to the origin are studied by exact enumeration on the square and triangular lattices the numerical results are inconsistent with the theoretical prediction of cardy and saleur 1989 by more than fifteen times the numerical uncertainty in the authors calculation

spectra of sparse non hermitian random matrices an analytical solution we present the exact analytical expression for the spectrum of a sparse non hermitian random matrix ensemble generalizing two standard results in random matrix theory this analytical expression constitutes a non hermitian version of the kesten mckay measure as well as a sparse realization of girko s elliptic law our exact result opens new perspectives in the study of several physical problems modelled on sparse random graphs which are locally treelike in this context we show analytically that the convergence rate of a transport process on a very sparse graph depends in a nonmonotonic way upon the degree of symmetry of the graph edges

disentangling giant component and finite cluster contributions in sparse random matrix spectra we describe a method for disentangling giant component and finite cluster contributions to sparse random matrix spectra using sparse symmetric random matrices defined on erd ifmmode mbox h o else h o fi s r enyi graphs as an example and test bed our methods apply to sparse matrices defined in terms of arbitrary graphs in the configuration model class as long as they have finite mean degree

freezing and collapse of flexible polymers on regular lattices in three dimensions we analyze the crystallization and collapse transition of a simple model for flexible polymer chains on simple cubic and face centered cubic lattices by means of sophisticated chain growth methods in contrast to the bond fluctuation polymer model in certain parameter ranges where these two conformational transitions were found to merge in the thermodynamic limit we conclude from our results that the two transitions remain well separated in the limit of infinite chain lengths the reason for this qualitatively distinct behavior is presumably due to the ultrashort attractive interaction range in the lattice models considered here

properties of networks with partially structured and partially random connectivity networks studied in many disciplines including neuroscience and mathematical biology have connectivity that may be stochastic about some underlying mean connectivity represented by a non normal matrix furthermore the stochasticity may not be independent and identically distributed iid across elements of the connectivity matrix more generally the problem of understanding the behavior of stochastic matrices with nontrivial mean structure and correlations arises in many settings we address this by characterizing large random n ifmmode times else texttimes fi n matrices of the form a m ljr where m phantom rule 0 16em 0ex l and r are arbitrary deterministic matrices and j is a random matrix of zero mean iid elements m can be non normal and l and r allow correlations that have separable dependence on row and column indices we first provide a general formula for the eigenvalue density of a for a non normal the eigenvalues do not suffice to specify the dynamics induced by a so we also provide general formulas for the transient evolution of the magnitude of activity and frequency power spectrum in an n dimensional linear dynamical system with a coupling matrix given by a these quantities can also be thought of as characterizing the stability and the magnitude of the linear response of a nonlinear network to small perturbations about a fixed point we derive these formulas and work them out analytically for some examples of m phantom rule 0 16em 0ex l and r motivated by neurobiological models we also argue that the persistence as n ensuremath rightarrow ensuremath infty of a finite number of randomly distributed outlying eigenvalues outside the support of the eigenvalue density of a as previously observed arises in regions of the complex plane ensuremath omega where there are nonzero singular values of l ensuremath 1 z mathbf 1 ensuremath m r ensuremath 1 for z ensuremath in ensuremath omega that vanish as n ensuremath rightarrow ensuremath infty when such singular values do not exist and l and r are equal to the identity there is a correspondence in the normalized frobenius norm but not in the operator norm between the support of the spectrum of a for j of norm ensuremath sigma and the ensuremath sigma pseudospectrum of m

universal amplitude ratios for three dimensional self avoiding walks we have calculated exactly the number the mean square end to end distance the mean square radius of gyration and the mean square distance of a monomer from the origin for n step self avoiding walks on the simple cubic diamond body centred cubic and face centred cubic lattices respectively up to 20 30 16 and 13 steps by a computer two universal amplitude ratios are estimated

self avoiding walks on the simple cubic lattice we have substantially extended the series for the number of self avoiding walks and the mean square end to end distance on the simple cubic lattice our analysis of the series gives refined estimates for the critical point and critical exponents our estimates of the exponents and are in good agreement with recent high precision monte carlo estimates and also with recent renormalization group estimates critical amplitude estimates are also given a new improved rigorous upper bound for the connective constant 4 7114 is obtained

universal amplitude combinations for self avoiding walks polygons and trails the authors give exact relations for a number of amplitude combinations that occur in the study of self avoiding walks polygons and lattice trails in particular they elucidate the lattice dependent factors which occur in those combinations which are otherwise universal show how these are modified for oriented lattices and give new results for amplitude ratios involving even moments of the area of polygons they also survey numerical results for a wide range of amplitudes on a number of oriented and regular lattices and provide some new ones

spectral theory of sparse non hermitian random matrices sparse non hermitian random matrices arise in the study of disordereda 13 physical systems with asymmetric local interactions and have applications ranginga 13 from neural networks to ecosystem dynamics the spectral characteristics of thesea 13 matrices provide crucial information on system stability and susceptibility however a 13 their study is greatly complicated by the twin challenges of a lack of symmetry anda 13 a sparse interaction structure in this review we provide a concise and systematica 13 introduction to the main tools and results in this field we show how the spectraa 13 of sparse non hermitian matrices can be computed via an analogy with infinitea 13 dimensional operators obeying certain recursion relations with reference to threea 13 illustrative examples adjacency matrices of regular oriented graphs adjacencya 13 matrices of oriented erd s renyi graphs and adjacency matrices of weighted orienteda 13 erd s renyi graphs we demonstrate the use of these methods to obtain both analytica 13 and numerical results for the spectrum the spectral distribution the location of outliera 13 eigenvalues and the statistical properties of eigenvectors a 13

critical exponents hyperscaling and universal amplitude ratios for two and three dimensional self avoiding walks we make a high precision monte carlo study of two and three dimensional self avoiding walks saws of length up to 80 000 steps using the pivot algorithm and the karp luby algorithm we study the critical exponents v and 2 triangle sub 4 gamma as well as several universal amplitude ratios in particular we make an extremely sensitive test of the hyperscaling relation dv 2 triangle sub 4 gamma in two dimensions we confirm the predicted exponent v 3 4 and the hyperscaling relation we estimate the universal ratios 0 14026 0 00007 0 43961 0 00034 and psi 0 66296 0 00043 68 confidence limits in three dimensions we estimate v 0 5877 0 0006 with a correction to scaling exponent triangle sub 1 0 56 0 03 subjective 68 confidence limits this value for v agrees excellently with the field theoretic renormalization group prediction but there is some discrepancy for triangle 1 earlier monte carlo estimates of v which were approximately 0 592 are now seen to be biased by corrections to scaling we estimate the universal ratios 0 1599 0 0002 and psi 0 2471 0 0003 since psi 0 hyperscaling holds the approach to psi is from above contrary to the prediction of the two parameter renormalization group theory we critically reexamine this theory and explain where the error lies in an appendix we prove rigorously modulo some standard scalingmore assumptions the hyperscaling relation dv 2 triangle sub 4 gamma for two dimensional saws less

comment on the universality class of self avoiding walks on a two layer square lattice abstract we have calculated exactly the mean square radius of gyration and the mean square distance of a monomer from the origin for n step self avoiding walks on a two layer square lattice up to 24 steps we estimated two universal amplitude ratios and the results provide additional evidence that self avoiding walks on a two layer lattice belongs to the same universality class as the self avoiding walk on a planar lattice

index statistical properties of sparse random graphs using the replica method we develop an analytical approach to compute the characteristic function for the probability p n k that a large n n adjacency matrix of sparse random graphs has k eigenvalues below a threshold the method allows to determine in principle all moments of p n k from which the typical sample to sample fluctuations can be fully characterized for random graph models with localized eigenvectors we show that the index variance scales linearly with n 1 for 0 with a model dependent prefactor that can be exactly calculated explicit results are discussed for erdos renyi and regular random graphs both exhibiting a prefactor with a nonmonotonic behavior as a function of these results contrast with rotationally invariant random matrices where the index variance scales only as lnn with an universal prefactor that is independent of numerical diagonalization results confirm the exactness of our approach and in addition strongly support the gaussian nature of the index fluctuations

statistical properties of eigenvectors in non hermitian gaussian random matrix ensembles statistical properties of eigenvectors in non hermitian random matrix ensembles are discussed with an emphasis on correlations between left and right eigenvectors two approaches are described one is an exact calculation for ginibre s ensemble in which each matrix element is an independent identically distributed gaussian complex random variable the other is a simpler calculation using n 1 as an expansion parameter where n is the rank of the random matrix this is applied to girko s ensemble consequences of eigenvector correlations which may be of physical importance in applications are also discussed it is shown that eigenvalues are much more sensitive to perturbations than in the corresponding hermitian random matrix ensembles it is also shown that in problems with time evolution governed by a non hermitian random matrix transients are controlled by eigenvector correlations

spacing ratio characterization of the spectra of directed random networks previous literature on random matrix and network science has traditionally employed measures derived from nearest neighbor level spacing distributions to characterize the eigenvalue statistics of random matrices this approach however depends crucially on eigenvalue unfolding procedures which in many situations represent a major hindrance due to constraints in the calculation especially in the case of complex spectra here we study the spectra of directed networks using the recently introduced ratios between nearest and next to nearest eigenvalue spacing thus circumventing the shortcomings imposed by spectral unfolding specifically we characterize the eigenvalue statistics of directed erd ifmmode mbox h o else h o fi s r enyi er random networks by means of two adjacency matrix representations namely 1 weighted non hermitian random matrices and 2 a transformation on non hermitian adjacency matrices which produces weighted hermitian matrices for both representations we find that the distribution of spacing ratios becomes universal for a fixed average degree in accordance with undirected random networks furthermore by calculating the average spacing ratio as a function of the average degree we show that the spectral statistics of directed er random networks undergoes a transition from poisson to ginibre statistics for model 1 and from poisson to gaussian unitary ensemble statistics for model 2 eigenvector delocalization effects of directed networks are also discussed

on the critical behaviour of self avoiding walks for pt i see ibid vol 22 p 1839 1989 for the self avoiding walk problem the coefficients of the chain generating function and of the generating function for the sum of square end to end distances have been extended to 20 terms for the triangular lattice to 27 terms for the diamond lattice to 21 terms for the simple cubic lattice and to 16 terms for the bcc lattice precise estimates of the critical points are obtained and for the exponents we find that gamma 1 161 or 0 002 and nu 0 592 or 0 003 encompasses all the three dimensional lattice data

random incidence matrices moments of the spectral density we study numerically and analytically the spectrum of incidence matrices of random labeled graphs on n vertices any pair of vertices is connected by an edge with probability p we give two algorithms to compute the moments of the eigenvalue distribution as explicit polynomials in n and p for large n and fixed p the spectrum contains a large eigenvalue at np and a semicircle of small eigenvalues for large n and fixed average connectivity pn dilute or sparse random matrices limit we show that the spectrum always contains a discrete component an anomaly in the spectrum near eigenvalue 0 for connectivity close to e is observed we develop recursion relations to compute the moments as explicit polynomials in pn their growth is slow enough so that they determine the spectrum the extension of our methods to the laplacian matrix is given in appendix

eigenvalue spectra of random matrices for neural networks the dynamics of neural networks is influenced strongly by the spectrum of eigenvalues of the matrix describing their synaptic connectivity in large networks elements of the synaptic connectivity matrix can be chosen randomly from appropriate distributions making results from random matrix theory highly relevant unfortunately classic results on the eigenvalue spectra of random matrices do not apply to synaptic connectivity matrices because of the constraint that individual neurons are either excitatory or inhibitory therefore we compute eigenvalue spectra of large random matrices with excitatory and inhibitory columns drawn from distributions with different means and equal or different variances

localization transition in symmetric random matrices we study the behavior of the inverse participation ratio and the localization transition in infinitely large random matrices through the cavity method results are shown for two ensembles of random matrices laplacian matrices on sparse random graphs and fully connected levy matrices we derive a critical line separating localized from extended states in the case of levy matrices comparison between theoretical results and diagonalization of finite random matrices is shown

universal sum and product rules for random matrices the spectral density of random matrices is studied through a quaternionic generalization of the green s function which precisely describes the mean spectral density of a given matrix under a particular type of random perturbation exact and universal expressions are found in the high dimension limit for the quaternionic green s functions of random matrices with independent entries when summed or multiplied with deterministic matrices from these the limiting spectral density can be accurately predicted

sparse random matrices the eigenvalue spectrum revisited we revisit the derivation of the density of states of sparse random matrices we derive a recursion relation that allows one to compute the spectrum of the matrix of incidence for finite trees that determines completely the low concentration limit using the iterative scheme introduced by biroli and monasson 1999 j phys a math gen 32 l255 we find an approximate expression for the density of states expected to hold exactly in the opposite limit of large but finite concentration the combination of the two methods yields a very simple geometric interpretation of the tails of the spectrum we test the analytic results with numerical simulations and we suggest an indirect numerical method to explore the tails of the spectrum

self avoiding walks in two to five dimensions exact enumerations and series study the method of concatenation the addition of precomputed shorter chains to the ends of a centrally generated longer chain has permitted the extension of the exact series for cn the number of distinct configurations for self avoiding walks of length n the authors report on the leading exponent y and xc the reciprocal of the connectivity constant for the 2d honeycomb lattice 42 terms 1 3437 0 541 1968 the 2d square lattice 30 terms 1 3436 0 379 0520 the 3d simple cubic lattice 23 terms 1 161 932 0 213 4987 the 4d hypercubic 18 terms y approximately 1 0 147 60 and the 5d hypercubic lattice 13 terms y or 1 025 0 113 05 in addition they have also evaluated the leading correction terms honeycomb delta approximately 1 square delta approximately 0 85 simple cubic delta approximately 1 0 and the 4d hypercubic logarithmic correction with delta approximately 0 25

self avoiding walks on self similar structures finite versus infinite ramification self avoiding walks saws generated by exact enumeration techniques are studied on the sierpinski carpet d 2 and on the sierpinski sponge d 3 also called sierpinski square lattices a detailed comparison of the results for saws on these infinitely ramified fractals to saws on finitely ramified sierpinski gaskets sierpinski triangular lattices on regular lattices and on the incipient percolation cluster is done providing insight into the behaviour of saws on ordered and disordered structures the saws on sierpinski square lattices are found to display a kind of intermediate behaviour sharing aspects of both saws on ordered and on fractal structures as a consequence a des cloizeaux relation does not seem to hold for this structure as opposed to its validity for saws on regular lattices on sierpinski triangular lattices and on the incipient percolation cluster

eigenvalue statistics of the real ginibre ensemble the real ginibre ensemble consists of random n ifmmode times else texttimes fi n matrices formed from independent and identically distributed standard gaussian entries by using the method of skew orthogonal polynomials the general n point correlations for the real eigenvalues and for the complex eigenvalues are given as n ifmmode times else texttimes fi n pfaffians with explicit entries a computationally tractable formula for the cumulative probability density of the largest real eigenvalue is presented this is relevant to may s stability analysis of biological webs

spectra of sparse regular graphs with loops we derive exact equations that determine the spectra of undirected and directed sparsely connected regular graphs containing loops of arbitrary lengths the implications of our results for the structural and dynamical properties of network models are discussed by showing how loops influence the size of the spectral gap and the propensity for synchronization analytical formulas for the spectrum are obtained for specific lengths of the loops

non normal amplification in random balanced neuronal networks in dynamical models of cortical networks the recurrent connectivity can amplify the input given to the network in two distinct ways one is induced by the presence of near critical eigenvalues in the connectivity matrix w producing large but slow activity fluctuations along the corresponding eigenvectors dynamical slowing the other relies on w not being normal which allows the network activity to make large but fast excursions along specific directions here we investigate the trade off between non normal amplification and dynamical slowing in the spontaneous activity of large random neuronal networks composed of excitatory and inhibitory neurons we use a schur decomposition of w to separate the two amplification mechanisms assuming linear stochastic dynamics we derive an exact expression for the expected amount of purely non normal amplification we find that amplification is very limited if dynamical slowing must be kept weak we conclude that to achieve strong transient amplification with little slowing the connectivity must be structured we show that unidirectional connections between neurons of the same type together with reciprocal connections between neurons of different types allow for amplification already in the fast dynamical regime finally our results also shed light on the differences between balanced networks in which inhibition exactly cancels excitation and those where inhibition dominates

chebyshev polynomial expansion of the localization length of hermitian and non hermitian random chains we study chebyshev polynomial expansion of the inverse localization length of hermitian and non hermitian random chains as a function of energy for hermitian models the expansion produces this energy dependent function numerically in one run of the algorithm this is in strong contrast to the standard transfer matrix method which produces the inverse localization length for a fixed energy in each run for non hermitian models as in the transfer matrix method our algorithm computes the inverse localization length for a fixed complex energy we also find a formula of the chebyshev polynomial expansion of the density of states of non hermitian models as explained in detail our algorithm for non hermitian models may be the only available efficient algorithm for finding the density of states of models with interactions

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

universal hypotrochoidic law for random matrices with cyclic correlations the celebrated elliptic law describes the distribution of eigenvalues of random matrices with correlations between off diagonal pairs of elements having applications to a wide range of physical and biological systems here we investigate the generalization of this law to random matrices exhibiting higher order cyclic correlations between k tuples of matrix entries we show that the eigenvalue spectrum in this ensemble is bounded by a hypotrochoid curve with k fold rotational symmetry this hypotrochoid law applies to full matrices as well as sparse ones and thereby holds with remarkable universality we further extend our analysis to matrices and graphs with competing cycle motifs which are described more generally by polytrochoid spectral boundaries

universal distance ratios for two dimensional polymers by using a quantitative version of the c theorem in conformal theories the authors determine some universal geometrical features of two dimensional critical systems with emphasis on the ratios of mean square distances for polymers

non hermitian localization in biological networks we explore the spectra and localization properties of the n site banded one dimensional non hermitian random matrices that arise naturally in sparse neural networks approximately equal numbers of random excitatory and inhibitory connections lead to spatially localized eigenfunctions and an intricate eigenvalue spectrum in the complex plane that controls the spontaneous activity and induced response a finite fraction of the eigenvalues condense onto the real or imaginary axes for large n the spectrum has remarkable symmetries not only with respect to reflections across the real and imaginary axes but also with respect to 90 rotations with an unusual anisotropic divergence in the localization length near the origin when chains with periodic boundary conditions become directed with a systematic directional bias superimposed on the randomness a hole centered on the origin opens up in the density of states in the complex plane all states are extended on the rim of this hole while the localized eigenvalues outside the hole are unchanged the bias dependent shape of this hole tracks the bias independent contours of constant localization length we treat the large n limit by a combination of direct numerical diagonalization and using transfer matrices an approach that allows us to exploit an electrostatic analogy connecting the charges embodied in the eigenvalue distribution with the contours of constant localization length we show that similar results are obtained for more realistic neural networks that obey dale s law each site is purely excitatory or inhibitory and conclude with perturbation theory results that describe the limit of large directional bias when all states are extended related problems arise in random ecological networks and in chains of artificial cells with randomly coupled gene expression patterns

random matrices universality of esds and the circular law given an n x n complex matrix a let mu a x y 1 n vertical bar 1 i n re lambda i x im lambda i y vertical bar be the empirical spectral distribution esd of its eigenvalues lambda i is an element of c i l n we consider the limiting distribution both in probability and in the almost sure convergence sense of the normalized esd mu 1 root n an of a random matrix a n a ij 1 i j n where the random variables a ij e a ij are i i d copies of a fixed random variable x with unit variance we prove a universality principle for such ensembles namely that the limit distribution in question is independent of the actual choice of x in particular in order to compute this distribution one can assume that x is real or complex gaussian as a related result we show how laws for this esd follow from laws for the singular value distribution of 1 root n a n zi for complex z as a corollary we establish the circular law conjecture both almost surely and in probability which asserts that mu 1 root n an converges to the uniform measure on the unit disc when the a ij have zero mean

universal distance ratios for 2d saws monte carlo and exact series results the mean square distance of a monomer to the origin and the mean square distance of the centre of mass to the origin are studied by monte carlo and exact series methods on the square and lattice the numerical results give strong evidence of a discrepancy with the theoretical prediction of cardy and saleur 1989 based on conformal invariance theory

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

on statistics of bi orthogonal eigenvectors in real and complex ginibre ensembles combining partial schur decomposition with supersymmetry we suggest a method of studying the joint probability density jpd of an eigenvalue and the associated non orthogonality overlap factor also known as the eigenvalue condition number of the left and right eigenvectors for non selfadjoint gaussian random matrices of size n times n first we derive the general finite n expression for the jpd of a real eigenvalue lambda and the associated non orthogonality factor in the real ginibre ensemble and then analyze its bulk and edge scaling limits the ensuing distribution is maximally heavy tailed so that all integer moments beyond normalization are divergent a similar calculation for a complex eigenvalue z and the associated non orthogonality factor in the complex ginibre ensemble is presented as well and yields a distribution with the finite first moment its bulk scaling limit yields a distribution whose first moment reproduces the well known result of chalker and mehlig phys rev lett 81 16 3367 3370 1998 and we provide the edge scaling distribution for this case as well our method involves evaluating the ensemble average of products and ratios of integer and half integer powers of characteristic polynomials for ginibre matrices which we perform in the framework of a supersymmetry approach our paper complements recent studies by bourgade and dubach the distribution of overlaps between eigenvectors of ginibre matrices 2018 arxiv 1801 01219

giant strongly connected component of directed networks we describe how to calculate the sizes of all giant connected components of a directed graph including the strongly connected one in particular the world wide web is a directed network the results are obtained for graphs with statistically uncorrelated vertices and an arbitrary joint in and out degree distribution p k i k o we show that if p k i k o does not factorize the relative size of the giant strongly connected component deviates from the product of the relative sizes of the giant in and out components the calculations of the relative sizes of all the giant components are demonstrated using the simplest examples we explain that the giant strongly connected component may be less resilient to random damage than the giant weakly connected one

a novel method of calculating amplitude ratios by series expansions the author introduces a new method of calculating critical amplitude ratios using series which is both simple and powerful this method which gives estimates for the amplitude ratios that are neither biased by the values of the critical points nor by the critical exponents is applied to several models it is shown that this method produces results where no reliable estimates from series expansion exist in particular one finds 0 025 or 0 001 for at b2 for the 3d ising model and 220 or 10 for c c for the two dimensional percolation model in agreement with and with more accuracy than values obtained by other methods

random regularization of brown spectral measure abstract we generalize a recent result of haagerup namely we show that a convolution with a standard gaussian random matrix regularizes the behavior of fuglede kadison determinant and brown spectral distribution measure in this way it is possible to establish a connection between the limit eigenvalues distributions of a wide class of random matrices and the brown measure of the corresponding limits

cavity approach to the spectral density of non hermitian sparse matrices the spectral densities of ensembles of non hermitian sparse random matrices are analyzed using the cavity method we present a set of equations from which the spectral density of a given ensemble can be efficiently and exactly calculated within this approach the generalized girko s law is recovered easily we compare our results with direct diagonalisation for a number of random matrix ensembles finding excellent agreement

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

integrable structure of ginibre s ensemble of real random matrices and a pfaffian integration theorem in the recent publication e kanzieper and g akemann in phys rev lett 95 230201 2005 an exact solution was reported for the probability pn k to find exactly k real eigenvalues in the spectrum of an n n real asymmetric matrix drawn at random from ginibre s orthogonal ensemble ginoe in the present paper we offer a detailed derivation of the above result by concentrating on the proof of the pfaffian integration theorem the key ingredient of our analysis of the statistics of real eigenvalues in the ginoe we also initiate a study of the correlations of complex eigenvalues and derive a formula for the joint probability density function of all complex eigenvalues of a ginoe matrix restricted to have exactly k real eigenvalues in the particular case of k 0 all correlation functions of complex eigenvalues are determined

incorporating geostatistical constraints in nonlinear inversion problems 1 in this paper we present a method of incorporating semivariogram constraints into nonlinear inversion problems that is we describe a method of sampling the space of inverse solutions that honor a specified semivariogram or set of semivariograms and also explain a set of state data the approach can be considered a method of conditional simulation where model conditioning is based upon state data as opposed to parameter data the difference between this approach and other simulation approaches is that the simulation is posed as an optimization problem with the joint objective of matching the semivariograms and honoring the state data this approach requires computing the sensitivities of the semivariograms with respect to the distributed parameter we derive these sensitivities and find that they are efficient to compute and store making the method tenable for large models we demonstrate the method with one synthetic and one field example using radar velocity tomography where radar velocity is related through a petrophysical transform to saturated porosity we address biasing issues and demonstrate ensemble generation and the resulting resolution and uncertainty analysis using ensemble statistics we also demonstrate how the method can be applied to existing deterministic inversion codes with the field example

finite size corrections to the spectrum of regular random graphs an analytical solution we develop a thorough analytical study of the o 1 n correction to the spectrum of regular random graphs with n nodes the finite size fluctuations of the resolvent are given in terms of a weighted series over the contributions coming from loops of all possible lengths from which we obtain the isolated eigenvalue as well as an analytical expression for the o 1 n correction to the continuous part of the spectrum the comparison between this analytical formula and direct diagonalization results exhibits an excellent agreement confirming the correctness of our expression

eigenvalue outliers of non hermitian random matrices with a local tree structure spectra of sparse non hermitian random matrices determine the dynamics of complex processes on graphs eigenvalue outliers in the spectrum are of particular interest since they determine the stationary state and the stability of dynamical processes we present a general and exact theory for the eigenvalue outliers of random matrices with a local tree structure for adjacency and laplacian matrices of oriented random graphs we derive analytical expressions for the eigenvalue outliers the first moments of the distribution of eigenvector elements associated with an outlier the support of the spectral density and the spectral gap we show that these spectral observables obey universal expressions which hold for a broad class of oriented random matrices

first eigenvalue eigenvector in sparse random symmetric matrices influences of degree fluctuation the properties of the first largest eigenvalue and its eigenvector first eigenvector are investigated for large sparse random symmetric matrices that are characterized by bimodal degree distributions in principle one should be able to accurately calculate them by solving a functional equation concerning auxiliary fields which come out in an analysis based on replica cavity methods however the difficulty in analytically solving this equation makes an accurate calculation infeasible in practice to overcome this problem we develop approximation schemes on the basis of two exceptionally solvable examples the schemes are reasonably consistent with numerical experiments when the statistical bias of positive matrix entries is sufficiently large and they qualitatively explain why considerably large finite size effects of the first eigenvalue can be observed when the bias is relatively small

eigenvalue distribution of large weighted random graphs we study eigenvalue distribution of the adjacency matrix a n p of weighted random graphs n p we assume that the graphs have n vertices and the average number of edges attached to one vertex is p to each edge of the graph eij we assign a weight given by a random variable aij with zero mathematical expectation and all moments finite in the first part of the paper we consider the moments of normalized eigenvalue counting function n p of a n p assuming all moments of a finite we obtain recurrent relations that determine the moments of the limiting measure p limn n p the method developed is applied to the laplace operator closely related with a n p using the recurrent relations we analyze the form of p for the both of random matrix families in the second part of the paper we consider the resolvents g a z of a n p and of n p and study the functions fn a z u 1 n k 1n exp ugkk a z in the limit n we derive closed equations that uniquely determine the limiting functio

eigenvalues of non hermitian random matrices and brown measure of non normal operators hermitian reduction and linearization method abstract we study the brown measure of certain non hermitian operators arising from voiculescu s free probability theory usually those operators appear as the limit in moments of certain ensembles of non hermitian random matrices and the brown measure gives then a canonical candidate for the limit eigenvalue distribution of the random matrices a prominent class for our operators is given by polynomials in free variables other explicit examples include r diagonal elements and elliptic elements for which the brown measure was already known and a new class of triangular elliptic elements our method for the calculation of the brown measure is based on a rigorous mathematical treatment of the hermitian reduction method as considered in the physical literature combined with subordination and the linearization trick