bulk universality for generalized wigner matrices consider n n hermitian or symmetric random matrices h where the distribution of the i j matrix element is given by a probability measure ij with a subexponential decay let sigma ij 2 be the variance for the probability measure ij with the normalization property that sum i sigma 2 ij 1 for all j under essentially the only condition that c le n sigma ij 2 le c 1 for some constant c 0 we prove that in the limit n the eigenvalue spacing statistics of h in the bulk of the spectrum coincide with those of the gaussian unitary or orthogonal ensemble gue or goe we also show that for band matrices with bandwidth m the local semicircle law holds to the energy scale m 1

convergence of the largest singular value of a polynomial in independent wigner matrices for polynomials in independent wigner matrices we prove convergence of the largest singular value to the operator norm of the corresponding polynomial in free semicircular variables under fourth moment hypotheses we actually prove a more general result of the form no eigenvalues outside the support of the limiting eigenvalue distribution we build on ideas of haagerup schultz thorbjornsen on the one hand and bai silverstein on the other we refine the linearization trick so as to preserve self adjointness and we develop a secondary trick bearing on the calculation of correction terms instead of poincare type inequalities we use a variety of matrix identities and lp estimates the schwinger dyson equation controls much of the analysis

local kesten mckay law for random regular graphs we study the adjacency matrices of random d regular graphs with large but fixed degree d in the bulk of the spectrum 2 sqrt d 1 varepsilon 2 sqrt d 1 varepsilon down to the optimal spectral scale we prove that the green s functions can be approximated by those of certain infinite tree like few cycles graphs that depend only on the local structure of the original graphs this result implies that the kesten mckay law holds for the spectral density down to the smallest scale and the complete delocalization of bulk eigenvectors our method is based on estimating the green s function of the adjacency matrices and a resampling of the boundary edges of large balls in the graphs

structured noncommutative multidimensional linear systems we introduce a class of multidimensional linear systems with evolution along a free semigroup the transfer function for such a system is a formal power series in noncommuting indeterminates standard system theoretic properties the operations of cascade parallel connection and inversion controllability observability kalman decomposition state space similarity theorem minimal state space realizations hankel operators realization theory are developed for this class of systems we also draw out the connections with the much earlier studied theory of rational and recognizable formal power series applications include linear fractional models for classical discrete time systems with structured time varying uncertainty dimensionless formulas in robust control multiscale systems and automata theory and the theory of formal languages

freely independent random variables with non atomic distributions 2015 american mathematical society we examine the distributions of non commutative polynomials of non atomic freely independent random variables in particular we obtain an analogue of the strong atiyah conjecture for free groups thus proving that the measure of each atom of any n n matricial polynomial of nonatomic freely independent random variables is an integer multiple of n 1 in addition we show that the cauchy transform of the distribution of any matricial polynomial of freely independent semicircular variables is algebraic and thus the polynomial has a distribution that is real analytic except at a finite number of points

spontaneous symmetry breaking of a hyperbolic sigma model in three dimensions non linear sigma models that arise from the supersymmetric approach to disordered electron systems contain a non compact bosonic sector we study the model with target space h2 the two hyperboloid with isometry group su 1 1 and prove that in three dimensions moments of the fields are finite in the thermodynamic limit thus the non compact symmetry su 1 1 is spontaneously broken the bound on moments is compatible with the presence of extended states

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

minimum sample size for developing a multivariable prediction model part ii binary and time to event outcomes when designing a study to develop a new prediction model with binary or time to event outcomes researchers should ensure their sample size is adequate in terms of the number of participants n and outcome events e relative to the number of predictor parameters p considered for inclusion we propose that the minimum values of n and e and subsequently the minimum number of events per predictor parameter epp should be calculated to meet the following three criteria i small optimism in predictor effect estimates as defined by a global shrinkage factor of 0 9 ii small absolute difference of 0 05 in the model s apparent and adjusted nagelkerke s r2 and iii precise estimation of the overall risk in the population criteria i and ii aim to reduce overfitting conditional on a chosen p and require prespecification of the model s anticipated cox snell r2 which we show can be obtained from previous studies the values of n and e that meet all three criteria provides the minimum sample size required for model development upon application of our approach a new diagnostic model for chagas disease requires an epp of at least 4 8 and a new prognostic model for recurrent venous thromboembolism requires an epp of at least 23 this reinforces why rules of thumb eg 10 epp should be avoided researchers might additionally ensure the sample size gives precise estimates of key predictor effects this is especially important when key categorical predictors have few events in some categories as this may substantially increase the numbers required

eigenvector localization for random band matrices with power law band width it is shown that certain ensembles of random matrices with entries that vanish outside a band around the diagonal satisfy a localization condition on the resolvent which guarantees that eigenvectors have strong overlap with a vanishing fraction of standard basis vectors provided the band width w raised to a power remains smaller than the matrix size n for a gaussian band ensemble with matrix elements given by i i d centered gaussians within a band of width w the estimate 8 holds

a new application of random matrices ext c _ mathrm red f_2 is not a group in the process of developing the theory of free probability and free entropy voiculescu introduced in 1991 a random matrix model for a free semicircular system since then random matrices have played a key role in von neumann algebra theory cf v8 v9 the main result of this paper is the following extension of voiculescui s random matrix result let x n 1 x n r be a system of r stochastically independent n i n gaussian self adjoint random matrices as in voiculescui s random matrix paper v4 and let x1 xr be a semi circular system in a c probability space then for every polynomial p in r noncommuting variables lim niaeia p x n 1 o x n r o p x1 xr for almost all o in the underlying probability space we use the result to show that the ext invariant for the reduced c algebra of the free group on 2 generators is not a group but only a semi group this problem has been open since anderson in 1978 found the first example of a c algebra a for which ext a is not a group

delocalization for a class of random block band matrices we consider n times n hermitian random matrices h consisting of blocks of size m ge n 6 7 the matrix elements are i i d within the blocks close to a gaussian in the four moment matching sense but their distribution varies from block to block to form a block band structure with an essential band width m we show that the entries of the green s function g z h z 1 satisfy the local semicircle law with spectral parameter z e mathbf i eta down to the real axis for any eta gg n 1 using a combination of the supersymmetry method inspired by shcherbina j stat phys 155 3 466 499 2014 and the green s function comparison strategy previous estimates were valid only for eta gg m 1 the new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized

no eigenvalues outside the support of the limiting spectral distribution of large dimensional sample covariance matrices let b n 1 n t n 1 2 x n x n t n 1 2 where x n is n times n with i i d complex standardized entries having finite fourth moment and t n 1 2 is a hermitian square root of the nonnegative definite hermitian matrix t n it is known that as n to infty if n n converges to a positive number and the empirical distribution of the eigenvalues of t n converges to a proper probability distribution then the empirical distribution of the eigenvalues of b n converges a s to a nonrandom limit in this paper we prove that under certain conditions on the eigenvalues of t n for any closed interval outside the support of the limit with probability 1 there will be no eigenvalues in this interval for all n sufficiently large

extended states in the anderson model on the bethe lattice abstract we prove that the anderson hamiltonianh von the bethe lattice has extended states for small disorder more precisely given any closed intervalicontained in the interior of the spectrum of the laplacian on the bethe lattice we prove that for small disorderh has purely absolutely continuous spectrum iniwith probability one i e ac h i iand pp h i sc h i with probability one and its integrated density of states is continuously differentiable on the intervali

minimum sample size for developing a multivariable prediction model part i continuous outcomes in the medical literature hundreds of prediction models are being developed to predict health outcomes in individuals for continuous outcomes typically a linear regression model is developed to predict an individual s outcome value conditional on values of multiple predictors covariates to improve model development and reduce the potential for overfitting a suitable sample size is required in terms of the number of subjects n relative to the number of predictor parameters p for potential inclusion we propose that the minimum value of n should meet the following four key criteria i small optimism in predictor effect estimates as defined by a global shrinkage factor of 0 9 ii small absolute difference of 0 05 in the apparent and adjusted r2 iii precise estimation a margin of error 10 of the true value of the model s residual standard deviation and similarly iv precise estimation of the mean predicted outcome value model intercept the criteria require prespecification of the user s chosen p and the model s anticipated r2 as informed by previous studies the value of n that meets all four criteria provides the minimum sample size required for model development in an applied example a new model to predict lung function in african american women using 25 predictor parameters requires at least 918 subjects to meet all criteria corresponding to at least 36 7 subjects per predictor parameter even larger sample sizes may be needed to additionally ensure precise estimates of key predictor effects especially when important categorical predictors have low prevalence in certain categories

simulating bosonic baths with error bars we derive rigorous truncation error bounds for the spin boson model and its generalizations to arbitrary quantum systems interacting with bosonic baths for the numerical simulation of such baths the truncation of both the number of modes and the local hilbert space dimensions is necessary we derive superexponential lieb robinson type bounds on the error when restricting the bath to finitely many modes and show how the error introduced by truncating the local hilbert spaces may be efficiently monitored numerically in this way we give error bounds for approximating the infinite system by a finite dimensional one as a consequence numerical simulations such as the time evolving density with orthogonal polynomials algorithm tedopa now allow for the fully certified treatment of the system environment interaction

a law of large numbers for finite range dependent random matrices we consider random hermitian matrices in which distant above diagonal entries are independent but nearby entries may be correlated we find the limit of the empirical distribution of eigenvalues by combinatorial methods we also prove that the limit has an algebraic stieltjes transform by an argument based on dimension theory of noetherian local rings 2008 wiley periodicals inc

absence of algebraic relations and of zero divisors under the assumption of full non microstates free entropy dimension abstract we show that in a tracial and finitely generated w probability space existence of conjugate variables excludes algebraic relations for the generators moreover under the assumption of maximal non microstates free entropy dimension we prove that there are no zero divisors in the sense that the product of any non commutative polynomial in the generators with any element from the von neumann algebra is zero if and only if at least one of those factors is zero in particular this shows that in this case the distribution of any non constant self adjoint non commutative polynomial in the generators does not have atoms questions on the absence of atoms for polynomials in non commuting random variables or for polynomials in random matrices have been an open problem for quite a while we solve this general problem by showing that maximality of free entropy dimension excludes atoms

superbosonization formula and its application to random matrix theory starting from gaussian random matrix models we derive a new supermatrix field theory model in contrast to the conventional non linear sigma models the new model is applicable for any range of correlations of the elements of the random matrices we clarify the domain of integration for the supermatrices and give a demonstration of how the model works by calculating the density of states for an ensemble of almost diagonal matrices it is also shown how one can reduce the supermatrix model to the conventional sigma model

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

noncommutative convexity arises from linear matrix inequalities this paper concerns polynomials in g noncommutative variables x x1 xg inverses of such polynomials and more generally noncommutative rational expressions with real coefficients which are formally symmetric and analytic near 0 the focus is on rational expressions r r x which are matrix convex near 0 i e those rational expressions r for which there is an 0 such that if x x1 xg is a g tuple of n n symmetric matrices satisfying in x12 xg2 is positive definite and y is also then the symmetric matrix tr x 1 t r y r tx 1 t y is positive semidefinite for all numbers t 0 t 1 this article gives a complete classification of matrix convex rational expressions see theorem 3 3 by representing such r in terms of a symmetric linear pencil l x id jajxj 0d 100 1 r 0 in the noncommuting variables xj where aj are symmetric d d matrices namely for a real number r is a schur complement of the linear pencil l moreover given a matrix convex r the set consisting of g tuples x of n n symmetric matrices 0 1 x r x iis negative definite has component containing 0 which is the same as the negativity set 0 2 x l x is negative definite for l conditions like l x is negative definite are known as linear matrix inequalities lmis in the engineering literature and arguably the main advance in linear systems theory in the 1990s was the introduction of lmi techniques in this language what we have shown in 0 1 vs 0 2 is that the set of solutions to a convex matrix inequality with noncommutative unknowns is the same as the set of solutions to some lmi in many engineering systems problems convexity would have all of the advantages of lmis indeed convexity guarantees that solutions are global and convexity bodes well for reliability of the numerics since lmis have a structure which is seemingly much more rigid than convexity there is the continual hope that a convexity based theory will be more far reaching than lmis but will it there are two natural situations one where the unknowns are scalars and one where the unknowns are matrices appearing in formulas which respect matrix multiplication these latter problems mathematically yield expressions with noncommutative unknowns and they arise in engineering systems problems which are dimensionless in the sense that they scale automatically with dimension as do most of the classics of control theory that is the case we study here and the result stated above suggests the surprising conclusion that for dimensionless systems problems convexity offers no greater generality than lmis indeed the result proves this for a class of model problems furthermore we show that existing algorithms together with algorithms described here construct the lmis above which are equivalent to the matrix inequalities based on the given matrix convex rational function r in a very different direction we prove that a symmetric polynomial p in g noncommutative symmetric variables has a symmetric determinantal representation namely there are symmetric matrices a0 ag in srd d with a0 invertible such that 0 3 detp x det a0 la x for each x a g tuple of symmetric n n matrices of course taking n 1 implies immediately that a commuting variables polynomial p on rg has a symmetric determinantal representation for g 2 much stronger commutative results can be obtained using tools of algebraic geometry but these do not seem to generalize to the higher dimensional case on the other hand a nonsymmetric commutative determinantal representation for any g is due to valiant universality of determinant in algebraic complexity theory our determinantal representation theorem is a bi product of the theory of systems realizations of noncommutative rational functions and can be read independently of much of the rest of the paper while the notion of noncommutative rational functions is standard the equivalence relation we use on rational expressions in our construction based on evaluating rational expressions on matrices is new and gives a new approach to noncommutative rational functions

on certain free product factors via an extended matrix model abstract a random matrix model for freeness is extended and used to investigate free products of free group factors with matrix algebras and with the hyperfinite ii 1 factor the latter is shown to be isomorphic to a free group factor having one additional generator

dynamical error bounds for continuum discretisation via gauss quadrature rules a lieb robinson bound approach instances of discrete quantum systems coupled to a continuum of oscillators are ubiquitous in physics often the continua are approximated by a discrete set of modes we derive error bounds on expectation values of system observables that have been time evolved under such discretised hamiltonians these bounds take on the form of a function of time and the number of discrete modes where the discrete modes are chosen according to gauss quadrature rules the derivation makes use of tools from the field of lieb robinson bounds and the theory of orthonormal polynomials

open systems with error bounds spin boson model with spectral density variations in the study of open quantum systems one of the most common ways to describe environmental effects on the reduced dynamics is through the spectral density however in many models this object cannot be computed from first principles and needs to be inferred on phenomenological grounds or fitted to experimental data consequently some uncertainty regarding its form and parameters is unavoidable this in turn calls into question the accuracy of any theoretical predictions based on a given spectral density here we focus on the spin boson model as a prototypical open quantum system find two error bounds on predicted expectation values in terms of the spectral density variation considered and state a sufficient condition for the strongest one to apply we further demonstrate an application of our result by bounding the error brought about by the approximations involved in the hierarchical equations of motion resolution method for spin boson dynamics

universality for general wigner type matrices we consider the local eigenvalue distribution of large self adjoint n times n random matrices mathbf h mathbf h with centered independent entries in contrast to previous works the matrix of variances s i j mathbbm e h i j 2 is not assumed to be stochastic hence the density of states is not the wigner semicircle law its possible shapes are described in the companion paper ajanki et al in quadratic vector equations on the complex upper half plane arxiv 1506 05095 we show that as n grows the resolvent mathbf g z mathbf h z 1 converges to a diagonal matrix mathrm diag mathbf m z where mathbf m z m 1 z dots m n z solves the vector equation 1 m i z z sum j s i j m j z that has been analyzed in ajanki et al quadratic vector equations on the complex upper half plane arxiv 1506 05095 we prove a local law down to the smallest spectral resolution scale and bulk universality for both real symmetric and complex hermitian symmetry classes

local semicircle law for random regular graphs we consider random d regular graphs on n vertices with degree d at least log n 4 we prove that the green s function of the adjacency matrix and the stieltjes transform of its empirical spectral measure are well approximated by wigner s semicircle law down to the optimal scale given by the typical eigenvalue spacing up to a logarithmic correction aside from well known consequences for the local eigenvalue distribution this result implies the complete isotropic delocalization of all eigenvectors and a probabilistic version of quantum unique ergodicity 2017 wiley periodicals inc

superbosonization of invariant random matrix ensembles superbosonization is a new variant of the method of commuting and anti commuting variables as used in studying random matrix models of disordered and chaotic quantum systems we here give a concise mathematical exposition of the key formulas of superbosonization conceived by analogy with the bosonization technique for dirac fermions the new method differs from the traditional one in that the superbosonization field is dual to the usual hubbard stratonovich field the present paper addresses invariant random matrix ensembles with symmetry group un on or uspn giving precise definitions and conditions of validity in each case the method is illustrated at the example of wegner s n orbital model superbosonization promises to become a powerful tool for investigating the universality of spectral correlation functions for a broad class of random matrix ensembles of non gaussian and or non invariant type

a simple method of sample size calculation for linear and logistic regression a sample size calculation for logistic regression involves complicated formulae this paper suggests use of sample size formulae for comparing means or for comparing proportions in order to calculate the required sample size for a simple logistic regression model one can then adjust the required sample size for a multiple logistic regression model by a variance inflation factor this method requires no assumption of low response probability in the logistic model as in a previous publication one can similarly calculate the sample size for linear regression models this paper also compares the accuracy of some existing sample size software for logistic regression with computer power simulations an example illustrates the methods

localization at large disorder and at extreme energies an elementary derivation the work presents a short proof of localization under the conditions of either strong disorder 0 or extreme energies for a wide class of self adjoint operators with random matrix elements acting inl 2 spaces a prototypical example is the discrete schrodinger operatorh u 0 x v x onz d d 1 withu 0 x a specified background potential and v x generated as random variables the general results apply to operators with replaced by a non local self adjoint operatort whose matrix elements satisfy y t x y s const uniformly inx for somes 1 localization means here that within a specified energy range the spectrum ofh is of the pure point type or equivalently the wave functions do not spread indefinitely under the unitary time evolution generated byh the effect is produced by strong disorder in either the potential or in the off diagonal matrix elementst x y under rapid decay oft x y the corresponding eigenfunctions are also proven to decay exponentially the method is based on resolvent techniques the central technical ideas include the use of low moments of the resolvent kernel i e g e x y s withs small enough 1 to avoid the divergence caused by the distribution s cauchy tails and an effective use of the simple form of the dependence ofg e x y on the individual matrix elements ofh in elucidating the implications of the fundamental equation h e g e x x 0 x x0 this approach simplifies previous derivations of localization results avoiding the small denominator difficulties which have been hitherto encountered in the subject it also yields some new results which include localization under the following sets of conditions i potentials with an inhomogeneous non random partu 0 x ii the bethe lattice iii operators with very slow decay in the off diagonal terms t x y 1 x y d e and iv localization produced by disordered boundary conditions

computing norms of free operators with matrix coefficients let g 1 g 2 g n be the generators of the free group f n and denote by its left regular representation a formula for norms of operators of the form a i g i where a i are complex matrices or operators is derived similarly to the formula of akemann and ostrand for the scalar case this formula can be used to compute norms of arbitrary finitely supported convolution operators on the free group essentially the same technique yields a formula for the norms of sums of matrix valued creation and annihilation operators on full fock space finally some results about the numerical evaluation of the formula are proved

an estimate for the average spectral measure of random band matrices for a class of random band matrices of band width w we prove regularity of the average spectral measure at scales w 0 99 and find its asymptotics at these scales

the norm of polynomials in large random and deterministic matrices let bf x n x 1 n ldots x p n be a family of n n independent normalized random matrices from the gaussian unitary ensemble we state sufficient conditions on matrices bf y n y 1 n ldots y q n possibly random but independent of xn for which the operator norm of p bf x n bf y n bf y n converges almost surely for all polynomials p limits are described by operator norms of objects from free probability theory taking advantage of the choice of the matrices yn and of the polynomials p we get for a large class of matrices the no eigenvalues outside a neighborhood of the limiting spectrum phenomena we give examples of diagonal matrices yn for which the convergence holds convergence of the operator norm is shown to hold for block matrices even with rectangular gaussian blocks a situation including non white wishart matrices and some matrices encountered in mimo systems

spectral properties of polynomials in independent wigner and deterministic matrices abstract on the one hand we prove that almost surely for large dimension there is no eigenvalue of a hermitian polynomial in independent wigner and deterministic matrices in any interval lying at some distance from the supports of a sequence of deterministic probability measures which is computed with the tools of free probability on the other hand we establish the strong asymptotic freeness of independent wigner matrices and any family of deterministic matrices with strong limiting distribution

isotropic self consistent equations for mean field random matrices we present a simple and versatile method for deriving an isotropic local laws for general random matrices constructed from independent random variables our method is applicable to mean field random matrices where all independent variables have comparable variances it is entirely insensitive to the expectation of the matrix in this paper we focus on the probabilistic part of the proof the derivation of the self consistent equations as a concrete application we settle in complete generality the local law for wigner matrices with arbitrary expectation

local laws for polynomials of wigner matrices abstract we consider general self adjoint polynomials in several independent random matrices whose entries are centered and have the same variance we show that under certain conditions the local law holds up to the optimal scale i e the eigenvalue density on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory we prove that these conditions hold for general homogeneous polynomials of degree two and for symmetrized products of independent matrices with i i d entries thus establishing the optimal bulk local law for these classes of ensembles in particular we generalize a similar result of anderson for anticommutator for more general polynomials our conditions are effectively checkable numerically

free entropy dimension and regularity of non commutative polynomials abstract we show that the spectral measure of any non constant non commutative polynomial evaluated at a non commutative n tuple cannot have atoms if the free entropy dimension of that n tuple is n see also work of mai speicher and weber under stronger assumptions on the n tuple we prove that the spectral measure of any non constant non commutative polynomial function is not singular and measures of intervals surrounding any point may not decay slower than polynomially as a function of the interval s length

predictive value of statistical models a review is given of different ways of estimating the error rate of a prediction rule based on a statistical model a distinction is drawn between apparent optimum and actual error rates moreover it is shown how cross validation can be used to obtain an adjusted predictor with smaller error rate a detailed discussion is given for ordinary least squares logistic regression and cox regression in survival analysis finally the splitsample approach is discussed and demonstrated on two data sets

stability of the matrix dyson equation and random matrices with correlations we consider real symmetric or complex hermitian random matrices with correlated entries we prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum the correlations have fast decay but are otherwise of general form the key novelty is the detailed stability analysis of the corresponding matrix valued dyson equation whose solution is the deterministic limit of the resolvent

a random matrix approach to the lack of projections in cred f2 abstract in 1982 pimsner and voiculescu computed the k 0 and k 1 groups of the reduced group c algebra c red f k of the free group f k on k generators and settled thereby a long standing conjecture c red f k has no projections except for the trivial projections 0 and 1 later simpler proofs of this conjecture were found by methods from k theory or from non commutative differential geometry in this paper we provide a new proof of the fact that c red f k is projectionless the new proof is based on random matrices and is obtained by a refinement of the methods recently used by the first and the third named author to show that the semigroup ext c red f k is not a group for k 2 by the same type of methods we also obtain that two phenomena proved by bai and silverstein for certain classes of random matrices no eigenvalues outside a small neighbourhood of the support of the limiting distribution and exact separation of eigenvalues by gaps in the limiting distribution also hold for arbitrary non commutative selfadjoint polynomials of independent gue goe or gse random matrices with matrix coefficients

applications of realizations aka linearizations to free probability abstract we show how the combination of new linearization ideas in free probability theory with the powerful realization machinery developed over the last 50 years in fields including systems engineering and automata theory allows solving the problem of determining the eigenvalue distribution or even the brown measure in the non selfadjoint case of noncommutative rational functions of random matrices when their size tends to infinity along the way we extend evaluations of noncommutative rational expressions from matrices to stably finite algebras e g type ii 1 von neumann algebras with a precise control of the domains of the rational expressions the paper provides sufficient background information with the intention that it should be accessible both to functional analysts and to algebraists