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We consider large random matrices with a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent away from the spectral edges, generalizing the recent result of Ajanki et al. [‘Stability of the matrix Dyson equation and random matrices with correlations’, Probab. Theory Related Fields173(1–2) (2019), 293–373] to allow slow correlation decay and arbitrary expectation. The main novel tool is a systematic diagrammatic control of a multivariate cumulant expansion.
We generalize the Cohen–Lenstra heuristics over function fields to étale group schemes
$G$
(with the classical case of abelian groups corresponding to constant group schemes). By using the results of Ellenberg–Venkatesh–Westerland, we make progress towards the proof of these heuristics. Moreover, by keeping track of the image of the Weil-pairing as an element of
$\wedge ^{2}G(1)$
, we formulate more refined heuristics which nicely explain the deviation from the usual Cohen–Lenstra heuristics for abelian
$\ell$
-groups in cases where
$\ell \mid q-1$
; the nature of this failure was suggested already in the works of Malle, Garton, Ellenberg–Venkatesh–Westerland, and others. On the purely large random matrix side, we provide a natural model which has the correct moments, and we conjecture that these moments uniquely determine a limiting probability measure.
We consider a class of sample covariance matrices of the form Q = TXX*T*, where X = (xij) is an M×N rectangular matrix consisting of independent and identically distributed entries, and T is a deterministic matrix such that T*T is diagonal. Assuming that M is comparable to N, we prove that the distribution of the components of the right singular vectors close to the edge singular values agrees with that of Gaussian ensembles provided the first two moments of xij coincide with the Gaussian random variables. For the right singular vectors associated with the bulk singular values, the same conclusion holds if the first four moments of xij match those of the Gaussian random variables. Similar results hold for the left singular vectors if we further assume that T is diagonal.
We consider the spectrum of additive, polynomially vanishing random perturbations of deterministic matrices, as follows. Let
$M_{N}$
be a deterministic
$N\times N$
matrix, and let
$G_{N}$
be a complex Ginibre matrix. We consider the matrix
${\mathcal{M}}_{N}=M_{N}+N^{-\unicode[STIX]{x1D6FE}}G_{N}$
, where
$\unicode[STIX]{x1D6FE}>1/2$
. With
$L_{N}$
the empirical measure of eigenvalues of
${\mathcal{M}}_{N}$
, we provide a general deterministic equivalence theorem that ties
$L_{N}$
to the singular values of
$z-M_{N}$
, with
$z\in \mathbb{C}$
. We then compute the limit of
$L_{N}$
when
$M_{N}$
is an upper-triangular Toeplitz matrix of finite symbol: if
$M_{N}=\sum _{i=0}^{\mathfrak{d}}a_{i}J^{i}$
where
$\mathfrak{d}$
is fixed,
$a_{i}\in \mathbb{C}$
are deterministic scalars and
$J$
is the nilpotent matrix
$J(i,j)=\mathbf{1}_{j=i+1}$
, then
$L_{N}$
converges, as
$N\rightarrow \infty$
, to the law of
$\sum _{i=0}^{\mathfrak{d}}a_{i}U^{i}$
where
$U$
is a uniform random variable on the unit circle in the complex plane. We also consider the case of slowly varying diagonals (twisted Toeplitz matrices), and, when
$\mathfrak{d}=1$
, also of independent and identically distributed entries on the diagonals in
$M_{N}$
.
We show that the partial transposes of complex Wishart random matrices are asymptotically free. We also investigate regimes where the number of blocks is fixed but the size of the blocks increases. This gives an example where the partial transpose produces freeness at the operator level. Finally, we investigate the case of real Wishart matrices.
Let
$n\geq 1$
be an integer and
$f$
be an arithmetical function. Let
$S=\{x_{1},\ldots ,x_{n}\}$
be a set of
$n$
distinct positive integers with the property that
$d\in S$
if
$x\in S$
and
$d|x$
. Then
$\min (S)=1$
. Let
$(f(S))=(f(\gcd (x_{i},x_{j})))$
and
$(f[S])=(f(\text{lcm}(x_{i},x_{j})))$
denote the
$n\times n$
matrices whose
$(i,j)$
-entries are
$f$
evaluated at the greatest common divisor of
$x_{i}$
and
$x_{j}$
and the least common multiple of
$x_{i}$
and
$x_{j}$
, respectively. In 1875, Smith [‘On the value of a certain arithmetical determinant’, Proc. Lond. Math. Soc.7 (1875–76), 208–212] showed that
$\det (f(S))=\prod _{l=1}^{n}(f\ast \unicode[STIX]{x1D707})(x_{l})$
, where
$f\ast \unicode[STIX]{x1D707}$
is the Dirichlet convolution of
$f$
and the Möbius function
$\unicode[STIX]{x1D707}$
. Bourque and Ligh [‘Matrices associated with classes of multiplicative functions’, Linear Algebra Appl.216 (1995), 267–275] computed the determinant
$\det (f[S])$
if
$f$
is multiplicative and, Hong, Hu and Lin [‘On a certain arithmetical determinant’, Acta Math. Hungar.150 (2016), 372–382] gave formulae for the determinants
$\det (f(S\setminus \{1\}))$
and
$\det (f[S\setminus \{1\}])$
. In this paper, we evaluate the determinant
$\det (f(S\setminus \{x_{t}\}))$
for any integer
$t$
with
$1\leq t\leq n$
and also the determinant
$\det (f[S\setminus \{x_{t}\}])$
if
$f$
is multiplicative.
Let [An,k]n,k⩾0 be an infinite lower triangular array satisfying the recurrence
for n ⩾ 1 and k ⩾ 0, where A0,0 = 1, A0,k = Ak,–1 = 0 for k > 0. We present some criteria for the log-concavity of rows and strong q-log-convexity of generating functions of rows. Our results can be applied to many well-known triangular arrays, such as the Pascal triangle, the Stirling triangle of the second kind, the Bell triangle, the large Schröder triangle, the Motzkin triangle, and the Catalan triangles of Aigner and Shapiro, in a unified approach. In addition, we prove that the binomial transformation not only preserves the strong q-log-convexity property, but also preserves the strong q-log-concavity property. Finally, we demonstrate that the strong q-log-convexity property is preserved by the Stirling transformation and Whitney transformation of the second kind, which extends some known results for the strong q-log-convexity property.
In this paper, we consider two innovative structured matrices, CUPL-Toeplitz matrix and CUPL-Hankel matrix. The inverses of CUPL-Toeplitz and CUPL-Hankel matrices can be expressed by the Gohberg-Heinig type formulas, and the stability of the inverse matrices is verified in terms of 1-, ∞- and 2-norms, respectively. In addition, two algorithms for the inverses of CUPL-Toeplitz and CUPL-Hankel matrices are given and examples are provided to verify the feasibility of these algorithms.
This paper presents a parallel algorithm for finding the smallest eigenvalue of a family of Hankel matrices that are ill-conditioned. Such matrices arise in random matrix theory and require the use of extremely high precision arithmetic. Surprisingly, we find that a group of commonly-used approaches that are designed for high efficiency are actually less efficient than a direct approach for this class of matrices. We then develop a parallel implementation of the algorithm that takes into account the unusually high cost of individual arithmetic operations. Our approach combines message passing and shared memory, achieving near-perfect scalability and high tolerance for network latency. We are thus able to find solutions for much larger matrices than previously possible, with the potential for extending this work to systems with greater levels of parallelism. The contributions of this work are in three areas: determination that a direct algorithm based on the secant method is more effective when extreme fixed-point precision is required than are the algorithms more typically used in parallel floating-point computations; the particular mix of optimizations required for extreme precision large matrix operations on a modern multi-core cluster, and the numerical results themselves.
We study systems of
$n$
points in the Euclidean space of dimension
$d\geqslant 1$
interacting via a Riesz kernel
$|x|^{-s}$
and confined by an external potential, in the regime where
$d-2\leqslant s<d$
. We also treat the case of logarithmic interactions in dimensions 1 and 2. Our study includes and retrieves all cases previously studied in Sandier and Serfaty [2D Coulomb gases and the renormalized energy, Ann. Probab. (to appear); 1D log gases and the renormalized energy: crystallization at vanishing temperature (2013)] and Rougerie and Serfaty [Higher dimensional Coulomb gases and renormalized energy functionals, Comm. Pure Appl. Math. (to appear)]. Our approach is based on the Caffarelli–Silvestre extension formula, which allows one to view the Riesz kernel as the kernel of an (inhomogeneous) local operator in the extended space
$\mathbb{R}^{d+1}$
.
As
$n\rightarrow \infty$
, we exhibit a next to leading order term in
$n^{1+s/d}$
in the asymptotic expansion of the total energy of the system, where the constant term in factor of
$n^{1+s/d}$
depends on the microscopic arrangement of the points and is expressed in terms of a ‘renormalized energy’. This new object is expected to penalize the disorder of an infinite set of points in whole space, and to be minimized by Bravais lattice (or crystalline) configurations. We give applications to the statistical mechanics in the case where temperature is added to the system, and identify an expected ‘crystallization regime’. We also obtain a result of separation of the points for minimizers of the energy.
For a positive integer
$n\geq 2$
, let
$M_{n}$
be the set of
$n\times n$
complex matrices and
$H_{n}$
the set of Hermitian matrices in
$M_{n}$
. We characterize injective linear maps
${\it\phi}:H_{m_{1}\cdots m_{l}}\rightarrow H_{n}$
satisfying
for all
$A_{k}\in H_{m_{k}}$
,
$k=1,\dots ,l$
, where
$l,m_{1},\dots ,m_{l}\geq 2$
are positive integers. The necessity of the injectivity assumption is shown. Moreover, the connection of the problem to quantum information science is mentioned.
We propose a discrete state-space model for storage of urban stormwater in two connected dams using an optimal pump-to-fill policy to transfer water from the capture dam to the holding dam. We assume stochastic supply to the capture dam and independent stochastic demand from the holding dam. We find new analytic formulae to calculate steady-state probabilities for the contents of each dam and thereby enable operators to better understand system behaviour. We illustrate our methods by considering some particular examples and discuss extension of our analysis to a series of three connected dams.
When a discrete-time homogenous Markov chain is observed at time intervals that correspond to its time unit, then the transition probabilities of the chain can be estimated using known maximum likelihood estimators. In this paper we consider a situation when a Markov chain is observed on time intervals with length equal to twice the time unit of the Markov chain. The issue then arises of characterizing probability matrices whose square root(s) are also probability matrices. This characterization is referred to in the literature as the embedding problem for discrete time Markov chains. The probability matrix which has probability root(s) is called embeddable.
In this paper for two-state Markov chains, necessary and sufficient conditions for embeddability are formulated and the probability square roots of the transition matrix are presented in analytic form. In finding conditions for the existence of probability square roots for (k x k) transition matrices, properties of row-normalized matrices are examined. Besides the existence of probability square roots, the uniqueness of these solutions is discussed: In the case of nonuniqueness, a procedure is introduced to identify a transition matrix that takes into account the specificity of the concrete context. In the case of nonexistence of a probability root, the concept of an approximate probability root is introduced as a solution of an optimization problem related to approximate nonnegative matrix factorization.
The critical paths of a max-plus linear system with noise are random variables. In this paper we introduce the edge criticalities which measure how often the critical paths traverse each edge in the precedence graph. We also present the parallel path approximation, a novel method for approximating these new statistics as well as the previously studied max-plus exponent. We show that, for low amplitude noise, the critical paths spend most of their time traversing the deterministic maximally weighted cycle and that, as the noise amplitude is increased, the critical paths become more random and their distribution over the edges in the precedence graph approaches a highly uniform measure of maximal entropy.
We prove that a continuous map
$\phi $
defined on the set of all
$n\times n$
Hermitian matrices preserving order in both directions is up to a translation a congruence transformation or a congruence transformation composed with the transposition.
We prove an upper bound on sums of squares of minors of
$\{+1, -1\}$
-matrices. The bound is sharp for Hadamard matrices, a result due to de Launey and Levin [‘
$(1,-1)$
-matrices with near-extremal properties’, SIAM J. Discrete Math.23(2009), 1422–1440], but our proof is simpler. We give several corollaries relevant to minors of Hadamard matrices.
Continuous-time discrete-state random Markov chains generated by a random linear differential equation with a random tridiagonal matrix are shown to have a random attractor consisting of singleton subsets, essentially a random path, in the simplex of probability vectors. The proof uses comparison theorems for Carathéodory random differential equations and the fact that the linear cocycle generated by the Markov chain is a uniformly contractive mapping of the positive cone into itself with respect to the Hilbert projective metric. It does not involve probabilistic properties of the sample path and is thus equally valid in the nonautonomous deterministic context of Markov chains with, say, periodically varying transition probabilities, in which case the attractor is a periodic path.
We investigate the number of symmetric matrices of nonnegative integers with zero diagonal such that each row sum is the same. Equivalently, these are zero-diagonal symmetric contingency tables with uniform margins, or loop-free regular multigraphs. We determine the asymptotic value of this number as the size of the matrix tends to infinity, provided the row sum is large enough. We conjecture that one form of our answer is valid for all row sums. An example appears in Figure 1.
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