probability of all eigenvalues real for products of standard gaussian matrices with xi independent n n standard gaussian random matrices the probability that all eigenvalues are real for the matrix product pm xmxm 1 x1 is expressed in terms of an n 2 n 2 n even and n 1 2 n 1 2 n odd determinant the entries of the determinant are certain meijer g functions in the case m 2 high precision computation indicates that the entries are rational multiples of 2 with the denominator a power of 2 and that to leading order in n decays as we are able to show that for general m and large n with an explicit bm an analytic demonstration that as m is given

supersymmetric quantum mechanics with levy disorder in one dimension we consider the schrodinger equation with a random potential of the form open image in new window where w is a levy noise we focus on the problem of computing the so called complex lyapunov exponent open image in new window where n is the integrated density of states of the system and is the lyapunov exponent in the case where the levy process is non decreasing we show that the calculation of reduces to a stieltjes moment problem we ascertain the low energy behaviour of the density of states in some generality and relate it to the distributional properties of the levy process we review the known solvable cases where can be expressed in terms of special functions and discover a new one

exactly soluble random field ising models in one dimension the authors solve exactly the one dimensional random field ising model for two classes of magnetic field distributions symmetric exponential model i and non symmetric exponential model ii for both models expressions for the free energy at all finite temperatures are presented the low temperature region is examined in more detail they obtain the zero temperature energy and entropy in closed form it is shown that the free energy of both models has an expansion in integer powers of temperature model i has a non vanishing zero point entropy for all values of the parameters as soon as randomness is diluted in model ii the zero point entropy is zero except for a discrete sequence of values of one parameter in some cases the zero temperature magnetisation is positive whereas the average magnetic field is negative the magnetisation may also change sign as a function of temperature

products of random matrices and generalised quantum point scatterers to every product of 2 2 matrices there corresponds a one dimensional schrodinger equation whose potential consists of generalised point scatterers products of random matrices are obtained by making these interactions and their positions random we exhibit a simple one dimensional quantum model corresponding to the most general product of matrices in sl 2 we use this correspondence to find new examples of products of random matrices for which the invariant measure can be expressed in simple analytical terms

a fast segmentation algorithm for piecewise polynomial numeric function generators we give an efficient algorithm for partitioning the domain of a numeric function f into segments the function f is realized as a polynomial in each segment and a lookup table stores the coefficients of the polynomial such an algorithm is an essential part of the design of lookup table methods ercepovac et al 2000 5 lee et al 2003 7 nagayama et al 2007 12 paul et al 2007 6 and sasao et al 2004 8 for realizing numeric functions such as sin px ln x and ln x our algorithm requires many fewer steps than a previous algorithm given in frenzen et al 2010 10 and makes tractable the design of numeric function generators based on table lookup for high accuracy applications we show that an estimate of segment width based on local derivatives greatly reduces the search needed to determine the exact segment width we apply the new algorithm to a suite of 15 numeric functions and show that the estimates are sufficiently accurate to produce a minimum or near minimum number of computational steps

random lie group actions on compact manifolds a perturbative analysis a random lie group action on a compact manifold generates a discrete time markov process the main object of this paper is the evaluation of associated birkhoff sums in a regime of weak but sufficiently effective coupling of the randomness this effectiveness is expressed in terms of random lie algebra elements and replaces the transience or furstenberg s irreducibility hypothesis in related problems the birkhoff sum of any given smooth function then turns out to be equal to its integral w r t a unique smooth measure on the manifold up to errors of the order of the coupling constant applications to the theory of products of random matrices and a model of a disordered quantum wire are presented

fluctuations of the product of random matrices and generalized lyapunov exponent i present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products varpi n m nm n 1 ldots m 1 where m i s are i i d following tutubalin theor probab appl 10 1 15 27 1965 the calculation of the generating function is reduced to finding the largest eigenvalue of a certain transfer operator associated with a family of representations of the group the formalism is illustrated by considering products of random matrices from the group mathrm sl 2 mathbb r where explicit calculations are possible for concreteness i study in detail transfer matrix products for the one dimensional schrodinger equation where the random potential is a levy noise derivative of a levy process in this case i obtain a general formula for the variance of ln varpi n and for the variance of ln psi x where psi x is the wavefunction in terms of a single integral involving the fourier transform of the invariant density of the matrix product finally i discuss the continuum limit of random matrix products matrices close to the identity in particular i investigate a simple case where the spectral problem providing the generalized lyapunov exponent can be solved exactly

lyapunov exponents one dimensional anderson localization and products of random matrices the concept of lyapunov exponent has long occupied a central place in the theory of anderson localisation its interest in this particular context is that it provides a reasonable measure of the localisation length the lyapunov exponent also features prominently in the theory of products of random matrices pioneered by furstenberg after a brief historical survey we describe some recent work that exploits the close connections between these topics we review the known solvable cases of disordered quantum mechanics involving random point scatterers and discuss a new solvable case finally we point out some limitations of the lyapunov exponent as a means of studying localisation properties

the lyapunov exponent of products of random 2 2 matrices close to the identity we study products of arbitrary random real 2 2 matrices that are close to the identity matrix using the iwasawa decomposition of sl 2 we identify a continuum regime where the mean values and the covariances of the three iwasawa parameters are simultaneously small in this regime the lyapunov exponent of the product is shown to assume a scaling form in the general case the corresponding scaling function is expressed in terms of gauss hypergeometric function a number of particular cases are also considered where the scaling function of the lyapunov exponent involves other special functions airy bessel whittaker elliptic the general solution thus obtained allows us among other things to recover in a unified framework many results known previously from exactly solvable models of one dimensional disordered systems

explicit invariant measures for products of random matrices we construct explicit invariant measures for a family of infinite products of random independent identically distributed elements of sl 2 c the matrices in the product are such that one entry is gamma distributed along a ray in the complex plane when the ray is the positive real axis the products are those associated with a continued fraction studied by letac seshadri z wahr verw geb 62 1983 485 489 who showed that the distribution of the continued fraction is a generalised inverse gaussian we extend this result by finding the distribution for an arbitrary ray in the complex right half plane and thus compute the corresponding lyapunov exponent explicitly when the ray lies on the imaginary axis the matrices in the infinite product coincide with the transfer matrices associated with a one dimensional discrete schrodinger operator with a random gamma distributed potential hence the explicit knowledge of the lyapunov exponent may be used to estimate the exponential rate of localisation of the eigenstates

single parameter scaling in one dimensional localization revisited the variance of the lyapunov exponent is calculated exactly in the one dimensional anderson model with random site energies distributed according to the cauchy distribution we find a new significant scaling parameter in the system and derive an exact analytical criterion for single parameter scaling which differs from the commonly used condition of phase randomization the results obtained are applied to the kronig penney model with the potential in the form of periodically positioned ensuremath delta functions with random strength

the distribution of the reflection phase of disordered conductors the reflection coefficients of one dimensional disordered conductors is asymptotically a pure phase when the sample length is much larger than the localisation length the authors study the stationary distribution of this reflection phase for tight binding chains with diagonal disorder i e random site potentials they derive the dyson integral equation obeyed by this distribution and relate it to the lyapunov exponent and the integrated density of states this general approach allows them to consider then more specifically two types of disorder in the case of a binary alloy where the potentials take two values the distribution of the reflection phase is shown to have generically an infinity of power law singularities related to the occurrence of periodic patterns of potentials this phenomenon produces divergences in the distribution when the strength of disorder exceeds some energy dependent threshold they also obtain an exact expression for the distribution of the reflection phase for a symmetric exponential distribution of potentials this non trivial solvable model allows them to examine analytically various properties of the phase distribution such as the anomalies which occur at weak disorder

weak disorder expansion of liapunov exponents in a degenerate case it is shown how the weak disorder expansion of the liapunov exponents of a product of random matrices can be derived when the unperturbed matrices have two degenerate eigenvalues the general expression of the liapunov exponents at the lowest nontrivial order in disorder is given

semianalytical solution of the random product problem of matrices and discrete time random evolution we consider the product of a large number of matrices chosen randomly with some correlation as a or b we introduce an exact functional equation whose solution gives the limiting distribution and the lyapunov exponent for the product of large number of matrices thus we suggest new semianalytical method to solve the random product problem of correlated matrices the method can be easily generalized to high order matrices as we deal with a larger systems of functional equations we have provided a known numerical algorithm for the solution of the matrix product problem we have also applied our method for a solution of discrete time random evolution our functional equation method provides exact transition point

universal distribution of lyapunov exponents for products of ginibre matrices starting from exact analytical results on singular values and complex eigenvalues of products of independent gaussian complex random n n matrices also called the ginibre ensemble we rederive the lyapunov exponents for an infinite product we show that for a large number t of product matrices the distribution of each lyapunov exponent is normal and we compute its t dependent variance as well as corrections in a large t expansion originally lyapunov exponents are defined for the singular values of the product matrix that represents a linear time evolution surprisingly a similar construction for the moduli of the complex eigenvalues yields the very same exponents and normal distributions to leading order we discuss a general mechanism for 2 2 matrices showing why the singular values and the radii of complex eigenvalues collapse onto the same value in the large t limit we thereby rederive newman s triangular law which has a simple interpretation as the radial density of complex eigenvalues in the circular law and study the commutativity of the two limits and on the global scale and on the local scale we show as a mathematical byproduct that a particular asymptotic expansion of a meijer g function with large index leads to a gaussian

fluctuations of random matrix products and 1d dirac equation with random mass we study the fluctuations of certain random matrix products pi n m n cdots m 2m 1 of mathrm sl 2 mathbb r describing localisation properties of the one dimensional dirac equation with random mass in the continuum limit i e when matrices m n s are close to the identity matrix we obtain convenient integral representations for the variance gamma 2 lim n rightarrow infty mathrm var ln pi n n the case studied exhibits a saturation of the variance at low energy varepsilon along with a vanishing lyapunov exponent gamma 1 lim n rightarrow infty ln pi n n leading to the behaviour gamma 2 gamma 1 sim ln 1 varepsilon rightarrow infty as varepsilon rightarrow 0 our continuum description sheds new light on the kappus wegner band center anomaly

exact solutions for ising chains in a random field the authors present an exact solution for the one dimensional ising model in a random field the distribution of field strengths is site independent symmetric and has a three peak structure the free energy is obtained for all finite temperatures the low temperature behaviour is studied in detail they find an expansion for the free energy in integer powers of temperature the ground state energy and the zero point entropy are calculated explicitly the specific heat is linear for low temperatures in contradiction with mean field theory the origin of an additional exponential parameter is discussed

calculus on arithmetic surfaces in a2 and a3 arakelov introduces an intersection calculus for arithmetic surfaces that is for stable models of curves over a number field in this paper we intend to show that his intersection product has a lot of useful properties more precisely we show that the following properties from the theory of algebraic surfaces have an analogue in our situation

global approximation in dimension two this paper is concerned with artin approximation and related results for normed rings in section 2 we prove a normed version of tougeron s lemma to and in section 3 we prove a normed version of elkik s theorems el theorems 2 and 2bis then in section 4 we prove an artin approximation theorem for normed rings having dimension at most 2 this has analytic applications example 4 7 and applications for arithmetic surfaces example 4 5 c it also applies to henselizations of curves and surfaces at closed subvarieties in this latter case the proof can be simplified and we treat that case separately in section 1 section 4 also proves approximation for closed complex polydiscs in arbitrary dimensions in the isolated singularity case in addition it shows that the ring of algebraic holomorphic functions on such a polydisc is factorial and indeed all finitely generated projective modules are free

exact electronic spectra and inverse localization lengths in one dimensional random systems i random alloy liquid metal and liquid alloy analytic continuations into the complex energy plane of dyson schmidt type of equations for the calculation of the density of states are constructed for a random alloy model a liquid metal and for a liquid alloy in all these models the characteristic function follows from the solution of this equation its imaginary part yields the accumulated density of states and its real part is a measure for the inverse of the localization length of the eigenfunctions the equations have been solved exactly for some distributions of the random variables in the random alloy case the strengths of the delta potentials have an exponential distribution they may also have finite exponentially distributed values with probability 0 p 1 and be infinite with probability q 1 p in the liquid metal the liquid particles are assumed to behave like hard rods this implies an exponential distribution of the distances between the particles the common electronic potential may be arbitrary but is assumed to vanish outside the rods in the one dimensional liquid alloy there is apart from positional randomness of the liquid particles a distribution of the strengths of the electronic delta potentials for cauchy distributions an argument of lloyd is extended to obtain the characteristic function from the one in the model with equal strengths for the case of a liquid of point particles a three parameter class of distributions of the strengths is shown to yield a solution in the form of known functions of the equation mentioned above for several cases numerical calculations of the density of states and the inverse localization length of the eigenfunctions are presented and discussed new results are found exponential decay of the density of states near special energies in the random alloy and liquid metal divergence of the density of states at certain energies with non classical exponent 13 in the random alloy if the average of the potential strengths vanishes exponentially small broadening of the bound state levels for low concentrations of the liquid particles peak in the localization length at the bound state energies which becomes exponentially narrow for low concentrations different exponent in the decay of the inverse localization length at large energies for delta potentials and square well potentials further an expression for the grand potential is given involving a sum over the characteristic function at certain points and divergence of the zero point energy is found for cauchy distributions of the delta potential strengths

exact solutions for spectra and green s functions in random one dimensional systems for chains of harmonic oscillators with random masses a set of equations is derived which determine the spatial fourier components of the average one particle green s function these equations are valid for complex values of the frequency a relation between the spectral density and functions introduced by schmidt is discussed exact solutions for this green s function and the less complicated characteristics function the analytic continuation into the complex frequency plane of the accumulated spectral density and the inverse localization length of the eigenfunctions are derived for exponential distributions of the masses for some cases the characteristic function is calculated numerically for gamma distributions the equations are cast in the form of ordinary higher order differential equations these have been solved numerically for determining the characteristic function for arbitrary mass distributions a cumulant expansion and a peculiar symmetry of the green s function are discussed the method is also applied to chains where the spring constants and or the masses have random values also for these systems exact solutions are discussed for exponential distributions e g of both masses and spring constants the characteristic function is expressed in bessel functions the relation with certain random relaxation models is shown finally x y hamiltonians with random exchange constants and or magnetic fields or equivalently tight binding electron models with diagonal and or off diagonal disorder are considered here the green s function does not depend on the wave number if the distribution of exchange constants is symmetric around the origin new solutions for the characteristic function and green s function are derived for a number of cases including exponentially distributed magnetic fields and power law distributed exchange constants

weak commutation relations and eigenvalue statistics for products of rectangular random matrices we study the joint probability density of the eigenvalues of a product of rectangular real complex or quaternion random matrices in a unified way the random matrices are distributed according to arbitrary probability densities whose only restriction is the invariance under left and right multiplication by orthogonal unitary or unitary symplectic matrices respectively we show that a product of rectangular matrices is statistically equivalent to a product of square matrices hereby we prove a weak commutation relation of the random matrices at finite matrix sizes which previously has been discussed for infinite matrix size moreover we derive the joint probability densities of the eigenvalues to illustrate our results we apply them to a product of random matrices drawn from ginibre ensembles and jacobi ensembles as well as a mixed version thereof for these weights we show that the product of complex random matrices yields a determinantal point process while the real and quaternion matrix ensembles correspond to pfaffian point processes our results are visualized by numerical simulations furthermore we present an application to a transport on a closed disordered chain coupled to a particle bath

diffusion in a one dimensional random medium and hyperbolic brownian motion classical diffusion in a random medium involves an exponential functional of brownian motion this functional also appears in the study of brownian diffusion on a riemann surface of constant negative curvature we analyse in detail this relationship and study various distributions using stochastic calculus and functional integration

correlation function of random field ising chains is it lorentzian or not the two point connected correlation function or wavevector dependent susceptibility of the ferromagnetic ising chain in a random field is calculated exactly at any temperature for a two parameter family of diluted symmetric exponential distributions of the magnetic fields thermodynamic properties of this model have been derived in a previous work by the authors besides the correlation function itself the solution provides exact results for the usual susceptibility the correlation length and the edwards anderson parameter the low temperature regime is examined in full detail the authors obtain in closed form the limit value at zero temperature of various quantities and the first correction to this limit which behaves linearly with temperature the correlation length is discontinuous at zero temperature they also derive the scaling form of the correlation function for small p where the dilution p is the probability for a spin to be subjected to a non zero random field even in this limit the correlation function is more complicated than the simple lorentzian predicted e g by mean field theory

low temperature thermodynamics of random field ising chains exact results the authors consider the random field ferromagnetic ising chain for a class of continuous symmetric probability distributions of the random magnetic fields of a diluted power times exponential type this class of distribution is exactly solvable in the sense that the disorder can be integrated explicitly at any temperature a detailed analysis of the low temperature thermodynamics is presented exact expressions are obtained for the ground state energy the zero temperature entropy and the amplitude of the specific heat which vanishes linearly at low temperature the diluted symmetric binary distribution where the magnetic fields can only assume the values or hb or zero can be viewed as a limiting case of the class of exactly solvable distributions the low temperature physics of this discrete model is investigated in detail including the exponential fall off of the specific heat

exact evaluations of some meijer g functions and probability of all eigenvalues real for the product of two gaussian matrices we provide a proof to a recent conjecture by forrester 2014 j phys a math theor 47 065202 regarding the algebraic and arithmetic structure of meijer g functions which appear in the expression for probability of all eigenvalues real for the product of two real gaussian matrices in the process we come across several interesting identities involving meijer g functions

breaking supersymmetry in a one dimensional random hamiltonian the one dimensional supersymmetric random hamiltonian where x is a gaussian white noise of zero mean and variance g presents particular spectral and localization properties at low energy a dyson singularity in the integrated density of states idos n e 1 ln2e and a delocalization transition related to the behavior of the lyapunov exponent inverse localization length vanishing like e 1 ln e as e 0 we study how this picture is affected by breaking supersymmetry with a scalar random potential h hsusy v x where v x is a gaussian white noise of variance in the limit g3 a fraction of states n 0 g ln2 g3 migrate to the negative spectrum and the lyapunov exponent reaches a finite value 0 g ln g3 at e 0 the exponential lifshits tail of the idos for e is studied in detail and is shown to involve a competition between the two noises x and v x whatever the larger is this analysis relies on analytic results for n e and e obtained by two different methods a stochastic method and the replica method the problem of extreme value statistics of eigenvalues is also considered distribution of the nth excited state energy the results are analyzed in the context of classical diffusion in a random force field in the presence of random annihilation creation local rates

convergent arithmetic power series this paper considers rings of power series on discs of radius r 0 with coefficients lying in a subring a of the rational numbers one such ring denoted here by ar t is an arithmetic version of the disc algebra studied in analysis a slightly smaller ring ar t consists of the holomorphic functions on a neighborhood of the closed disc of radius r with coefficients lying in a the rings z1 t and z1 t are in fact equal to z t and so the families zr ti 0 r i and zr t i 0 r 1 can be regarded as interpolating between z t and z t these convergent power series rings may thus be of use in a deformation theoretic context the maximal spectra of ar t and ar t are described in section 1 and the latter ring is shown to be a noetherian regular unique factorization domain the ring zr t meanwhile is shown to be complete under the uniform norm section 2 shows that every finitely generated projective module over ar t is free the proof uses a patching lemma for free modules this section also contains a more general module patching result which can be viewed as an arithmetic analog of grothendieck s existence theorem from this point of view z is analogous to the polynomial ring k xl over a field k z t and z t correspond to k x t and k x t and zr t corresponds to k t x whose spectrum is a tubular neighborhood of the affine k line section 3 considers the subring ar kitih of ar t consisting of algebraic power series every such power series is shown to lie in an etale neighborhood of the disc it is also shown that the ring z t h of algebraic power series over z on the open unit disc consists entirely of rational functions given an algebraic structure over z t h obtained for example by