scholarly journals $S$-constrained random matrices

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Sylvain Gravier ◽  
Bernard Ycart
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Explicit Construction ◽  
Superimposed Codes ◽  
International Audience ◽  
Asymptotic Probability

International audience Let $S$ be a set of $d$-dimensional row vectors with entries in a $q$-ary alphabet. A matrix $M$ with entries in the same $q$-ary alphabet is $S$-constrained if every set of $d$ columns of $M$ contains as a submatrix a copy of the vectors in $S$, up to permutation. For a given set $S$ of $d$-dimensional vectors, we compute the asymptotic probability for a random matrix $M$ to be $S$-constrained, as the numbers of rows and columns both tend to infinity. If $n$ is the number of columns and $m=m_n$ the number of rows, then the threshold is at $m_n= \alpha_d \log (n)$, where $\alpha_d$ only depends on the dimension $d$ of vectors and not on the particular set $S$. Applications to superimposed codes, shattering classes of functions, and Sidon families of sets are proposed. For $d=2$, an explicit construction of a $S$-constrained matrix is given.

VICTORIA transform, RESPECT and REFORM methods for the proof of the G-Elliptic Law under G-Lindeberg condition and twice stochastic condition for the variances and covariances of the entries of some random matrices

2020 ◽  
Vol 28 (2) ◽  
pp. 131-162
Author(s):  
Vyacheslav L. Girko
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Lindeberg Condition

AbstractThe G-Elliptic law under the G-Lindeberg condition for the independent pairs of the entries of a random matrix is proven.

scholarly journals On the limiting spectral density of random matrices filled with stochastic processes

2019 ◽  
Vol 27 (2) ◽  
pp. 89-105 ◽  
Author(s):  
Matthias Löwe ◽  
Kristina Schubert
Keyword(s):  
Stochastic Process ◽  
Spectral Density ◽  
Random Matrices ◽  
Markov Processes ◽  
Random Matrix ◽  
Matrix Theory ◽  
Gibbs Measures ◽  
State Spaces ◽  
The Matrix ◽  
Finite State

Abstract We discuss the limiting spectral density of real symmetric random matrices. In contrast to standard random matrix theory, the upper diagonal entries are not assumed to be independent, but we will fill them with the entries of a stochastic process. Under assumptions on this process which are satisfied, e.g., by stationary Markov chains on finite sets, by stationary Gibbs measures on finite state spaces, or by Gaussian Markov processes, we show that the limiting spectral distribution depends on the way the matrix is filled with the stochastic process. If the filling is in a certain way compatible with the symmetry condition on the matrix, the limiting law of the empirical eigenvalue distribution is the well-known semi-circle law. For other fillings we show that the semi-circle law cannot be the limiting spectral density.

scholarly journals The Expected Norm of Random Matrices

2000 ◽  
Vol 9 (2) ◽  
pp. 149-166 ◽  
Author(s):  
YOAV SEGINER
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Operator Norm ◽  
Euclidean Norm ◽  
Constant Independent ◽  
Multiplicative Constant ◽  
The Matrix

We compare the Euclidean operator norm of a random matrix with the Euclidean norm of its rows and columns. In the first part of this paper, we show that if A is a random matrix with i.i.d. zero mean entries, then E∥A∥h [les ] Kh (E maxi ∥ai[bull ] ∥h + E maxj ∥aj[bull ] ∥h), where K is a constant which does not depend on the dimensions or distribution of A (h, however, does depend on the dimensions). In the second part we drop the assumption that the entries of A are i.i.d. We therefore consider the Euclidean operator norm of a random matrix, A, obtained from a (non-random) matrix by randomizing the signs of the matrix's entries. We show that in this case, the best inequality possible (up to a multiplicative constant) is E∥A∥h [les ] (c log1/4 min {m, n})h (E maxi ∥ai[bull ] ∥h + E maxj ∥aj[bull ] ∥h) (m, n the dimensions of the matrix and c a constant independent of m, n).

scholarly journals FLUCTUATIONS OF MATRIX ENTRIES OF ANALYTIC FUNCTIONS OF NON-HERMITIAN RANDOM MATRICES

2012 ◽  
Vol 01 (03) ◽  
pp. 1250008
Author(s):  
SEAN O'ROURKE
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Analytic Functions ◽  
Random Variables

Consider an n × n non-Hermitian random matrix Mn whose entries are independent real random variables. Under suitable conditions on the entries, we study the fluctuations of the entries of f(Mn) as n tends to infinity, where f is analytic on an appropriate domain. This extends the results in [19, 20, 23] from symmetric random matrices to the non-Hermitian case.

A Block Compressive Sensing Based Scalable Encryption Framework for Protecting Significant Image Regions

2016 ◽  
Vol 26 (11) ◽  
pp. 1650191 ◽  
Author(s):  
Yushu Zhang ◽  
Jiantao Zhou ◽  
Fei Chen ◽  
Leo Yu Zhang ◽  
Di Xiao ◽  
...  
Keyword(s):  
Compressive Sensing ◽  
Random Matrices ◽  
Random Matrix ◽  
Chaotic Maps ◽  
Sampling Rate ◽  
Equal Size ◽  
Edge Detector ◽  
Scalable Encryption ◽  
Significant Block ◽  
Block Encryption

The existing Block Compressive Sensing (BCS) based image ciphers adopted the same sampling rate for all the blocks, which may lead to the desirable result that after subsampling, significant blocks lose some more-useful information while insignificant blocks still retain some less-useful information. Motivated by this observation, we propose a scalable encryption framework (SEF) based on BCS together with a Sobel Edge Detector and Cascade Chaotic Maps. Our work is firstly dedicated to the design of two new fusion techniques, chaos-based structurally random matrices and chaos-based random convolution and subsampling. The basic idea is to divide an image into some blocks with an equal size and then diagnose their respective significance with the help of the Sobel Edge Detector. For significant block encryption, chaos-based structurally random matrix is applied to significant blocks whereas chaos-based random convolution and subsampling are responsible for the remaining insignificant ones. In comparison with the BCS based image ciphers, the SEF takes lightweight subsampling and severe sensitivity encryption for the significant blocks and severe subsampling and lightweight robustness encryption for the insignificant ones in parallel, thus better protecting significant image regions.

scholarly journals VC-dimensions of random function classes

2008 ◽  
Vol Vol. 10 no. 1 (Combinatorics) ◽  
Author(s):  
Bernard Ycart ◽  
Joel Ratsaby
Keyword(s):  
Random Function ◽  
Classical Result ◽  
High Probability ◽  
Necessary Condition ◽  
Vc Dimension ◽  
Uniform Sampling ◽  
Function Classes ◽  
International Audience ◽  
Asymptotic Probability ◽  
Binary Functions

Combinatorics International audience For any class of binary functions on [n]={1, ..., n} a classical result by Sauer states a sufficient condition for its VC-dimension to be at least d: its cardinality should be at least O(nd-1). A necessary condition is that its cardinality be at least 2d (which is O(1) with respect to n). How does the size of a 'typical' class of VC-dimension d compare to these two extreme thresholds ? To answer this, we consider classes generated randomly by two methods, repeated biased coin flips on the n-dimensional hypercube or uniform sampling over the space of all possible classes of cardinality k on [n]. As it turns out, the typical behavior of such classes is much more similar to the necessary condition; the cardinality k need only be larger than a threshold of 2d for its VC-dimension to be at least d with high probability. If its expected size is greater than a threshold of O(&log;n) (which is still significantly smaller than the sufficient size of O(nd-1)) then it shatters every set of size d with high probability. The behavior in the neighborhood of these thresholds is described by the asymptotic probability distribution of the VC-dimension and of the largest d such that all sets of size d are shattered.

scholarly journals Identifiability of parametric random matrix models

2019 ◽  
Vol 22 (03) ◽  
pp. 1950018
Author(s):  
Tomohiro Hayase
Keyword(s):  
Probability Theory ◽  
Matrix Models ◽  
Random Matrices ◽  
Random Matrix ◽  
Free Probability ◽  
Parameter Identifiability ◽  
Free Probability Theory ◽  
Noise Type ◽  
Random Matrix Models ◽  
Spectral Distributions

We investigate parameter identifiability of spectral distributions of random matrices. In particular, we treat compound Wishart type and signal-plus-noise type. We show that each model is identifiable up to some kind of rotation of parameter space. Our method is based on free probability theory.

scholarly journals Permutations Without Long Decreasing Subsequences and Random Matrices

10.37236/929 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Piotr Šniady
Keyword(s):  
Random Matrices ◽  
Young Diagram ◽  
Random Matrix ◽  
Random Permutation ◽  
Fixed Number ◽  
Plancherel Measure ◽  
Young Diagrams ◽  
Gaussian Unitary Ensemble ◽  
Longest Increasing Subsequence ◽  
Unitary Ensemble

We study the shape of the Young diagram $\lambda$ associated via the Robinson–Schensted–Knuth algorithm to a random permutation in $S_n$ such that the length of the longest decreasing subsequence is not bigger than a fixed number $d$; in other words we study the restriction of the Plancherel measure to Young diagrams with at most $d$ rows. We prove that in the limit $n\to\infty$ the rows of $\lambda$ behave like the eigenvalues of a certain random matrix (namely the traceless Gaussian Unitary Ensemble random matrix) with $d$ rows and columns. In particular, the length of the longest increasing subsequence of such a random permutation behaves asymptotically like the largest eigenvalue of the corresponding random matrix.

scholarly journals Indecomposable permutations with a given number of cycles

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Robert Cori ◽  
Claire Mathieu
Keyword(s):  
Fixed Point ◽  
Error Term ◽  
Nous Obtenons ◽  
Threshold Phenomenon ◽  
Arbitrary Genus ◽  
International Audience ◽  
Asymptotic Probability ◽  
Number Of Cycles ◽  
Pour Cent ◽  
Orientable Surfaces

International audience A permutation $a_1a_2 \ldots a_n$ is $\textit{indecomposable}$ if there does not exist $p \lt n$ such that $a_1a_2 \ldots a_p$ is a permutation of $\{ 1,2, \ldots ,p\}$. We compute the asymptotic probability that a permutation of $\mathbb{S}_n$ with $m$ cycles is indecomposable as $n$ goes to infinity with $m/n$ fixed. The error term is $O(\frac{\log(n-m)}{ n-m})$. The asymptotic probability is monotone in $m/n$, and there is no threshold phenomenon: it degrades gracefully from $1$ to $0$. When $n=2m$, a slight majority ($51.1 \ldots$ percent) of the permutations are indecomposable. We also consider indecomposable fixed point free involutions which are in bijection with maps of arbitrary genus on orientable surfaces, for these involutions with $m$ left-to-right maxima we obtain a lower bound for the probability of being indecomposable. Une permutation $a_1a_2 \ldots a_n$ est $\textit{indécomposable}$, s’il n’existe pas de $p \lt n$ tel que $a_1a_2 \ldots a_p$ est une permutation de $\{ 1,2, \ldots ,p\}$. Nous calculons la probabilité pour qu’une permutation de $\mathbb{S}_n$ ayant $m$ cycles soit indécomposable et plus particulièrement son comportement asymptotique lorsque $n$ tend vers l’infini et que $m=n$ est fixé. Cette valeur décroît régulièrement de $1$ à $0$ lorsque $m=n$ croît, et il n’y a pas de phénomène de seuil. Lorsque $n=2m$, une faible majorité ($51.1 \ldots$ pour cent) des permutations sont indécomposables. Nous considérons aussi les involutions sans point fixe indécomposables qui sont en bijection avec les cartes de genre quelconque plongées dans une surface orientable, pour ces involutions ayant $m$ maxima partiels (ou records) nous obtenons une borne inférieure pour leur probabilité d’êtres indécomposables.

scholarly journals UNIVERSAL CORRELATIONS IN RANDOM MATRICES: QUANTUM CHAOS, THE 1/r2 INTEGRABLE MODEL, AND QUANTUM GRAVITY

1996 ◽  
Vol 11 (15) ◽  
pp. 1201-1219 ◽  
Author(s):  
SANJAY JAIN
Keyword(s):  
Quantum Gravity ◽  
Random Matrices ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Correlation Functions ◽  
Quantum Chaos ◽  
Matrix Theory ◽  
Mathematical Formulation ◽  
Integrable Model ◽  
Moser System

Random matrix theory (RMT) provides a common mathematical formulation of distinct physical questions in three different areas: quantum chaos, the 1-D integrable model with the 1/r2 interaction (the Calogero-Sutherland-Moser system) and 2-D quantum gravity. We review the connection of RMT with these areas. We also discuss the method of loop equations for determining correlation functions in RMT, and smoothed global eigenvalue correlators in the two-matrix model for Gaussian orthogonal, unitary and symplectic ensembles.

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