Pub Date : 2021-03-04DOI: 10.1142/s2010326322500307
Alicja Dembczak-Kołodziejczyk, A. Lytova
Given [Formula: see text], we study two classes of large random matrices of the form [Formula: see text] where for every [Formula: see text], [Formula: see text] are iid copies of a random variable [Formula: see text], [Formula: see text], [Formula: see text] are two (not necessarily independent) sets of independent random vectors having different covariance matrices and generating well concentrated bilinear forms. We consider two main asymptotic regimes as [Formula: see text]: a standard one, where [Formula: see text], and a slightly modified one, where [Formula: see text] and [Formula: see text] while [Formula: see text] for some [Formula: see text]. Assuming that vectors [Formula: see text] and [Formula: see text] are normalized and isotropic “in average”, we prove the convergence in probability of the empirical spectral distributions of [Formula: see text] and [Formula: see text] to a version of the Marchenko–Pastur law and the so-called effective medium spectral distribution, correspondingly. In particular, choosing normalized Rademacher random variables as [Formula: see text], in the modified regime one can get a shifted semicircle and semicircle laws. We also apply our results to the certain classes of matrices having block structures, which were studied in [G. M. Cicuta, J. Krausser, R. Milkus and A. Zaccone, Unifying model for random matrix theory in arbitrary space dimensions, Phys. Rev. E 97(3) (2018) 032113, MR3789138; M. Pernici and G. M. Cicuta, Proof of a conjecture on the infinite dimension limit of a unifying model for random matrix theory, J. Stat. Phys. 175(2) (2019) 384–401, MR3968860].
给定[公式:见文]，我们研究两类形式为[公式:见文]的大型随机矩阵，其中每个[公式:见文]，[公式:见文]是一个随机变量[公式:见文]，[公式:见文]，[公式:见文]的iid副本，是两个(不一定是独立的)独立随机向量集合，具有不同的协方差矩阵，并产生很集中的双线性形式。我们考虑两种主要的渐近机制[公式:见文]:一个标准的，其中[公式:见文]，和一个稍微修改的，其中[公式:见文]和[公式:见文]，而[公式:见文]的一些[公式:见文]。假设向量[公式:见文]和[公式:见文]是归一化且“平均”各向同性的，我们相应地证明了[公式:见文]和[公式:见文]的经验光谱分布在概率上收敛于Marchenko-Pastur定律的一个版本和所谓的有效介质光谱分布。特别地，选取归一化Rademacher随机变量为[公式:见文]，在修正的制度下可以得到移位的半圆定律和半圆定律。我们还将我们的结果应用于[G]中研究的具有块结构的某类矩阵。M. Cicuta, J. Krausser, R. Milkus和A. Zaccone，任意空间维度随机矩阵理论的统一模型，物理学报。Rev. E 97(3) (2018) 032113, MR3789138;M. Pernici和G. M. Cicuta，随机矩阵理论统一模型的无限维极限猜想的证明[j].物理学报，17 (2)(2019)384-401,MR3968860。
{"title":"On the empirical spectral distribution for certain models related to sample covariance matrices with different correlations","authors":"Alicja Dembczak-Kołodziejczyk, A. Lytova","doi":"10.1142/s2010326322500307","DOIUrl":"https://doi.org/10.1142/s2010326322500307","url":null,"abstract":"Given [Formula: see text], we study two classes of large random matrices of the form [Formula: see text] where for every [Formula: see text], [Formula: see text] are iid copies of a random variable [Formula: see text], [Formula: see text], [Formula: see text] are two (not necessarily independent) sets of independent random vectors having different covariance matrices and generating well concentrated bilinear forms. We consider two main asymptotic regimes as [Formula: see text]: a standard one, where [Formula: see text], and a slightly modified one, where [Formula: see text] and [Formula: see text] while [Formula: see text] for some [Formula: see text]. Assuming that vectors [Formula: see text] and [Formula: see text] are normalized and isotropic “in average”, we prove the convergence in probability of the empirical spectral distributions of [Formula: see text] and [Formula: see text] to a version of the Marchenko–Pastur law and the so-called effective medium spectral distribution, correspondingly. In particular, choosing normalized Rademacher random variables as [Formula: see text], in the modified regime one can get a shifted semicircle and semicircle laws. We also apply our results to the certain classes of matrices having block structures, which were studied in [G. M. Cicuta, J. Krausser, R. Milkus and A. Zaccone, Unifying model for random matrix theory in arbitrary space dimensions, Phys. Rev. E 97(3) (2018) 032113, MR3789138; M. Pernici and G. M. Cicuta, Proof of a conjecture on the infinite dimension limit of a unifying model for random matrix theory, J. Stat. Phys. 175(2) (2019) 384–401, MR3968860].","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83652553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-03DOI: 10.1142/S201032632250023X
Giovanni Barbarino, V. Noferini
We study the empirical spectral distribution (ESD) for complex [Formula: see text] matrix polynomials of degree [Formula: see text] under relatively mild assumptions on the underlying distributions, thus highlighting universality phenomena. In particular, we assume that the entries of each matrix coefficient of the matrix polynomial have mean zero and finite variance, potentially allowing for distinct distributions for entries of distinct coefficients. We derive the almost sure limit of the ESD in two distinct scenarios: (1) [Formula: see text] with [Formula: see text] constant and (2) [Formula: see text] with [Formula: see text] bounded by [Formula: see text] for some [Formula: see text]; the second result additionally requires that the underlying distributions are continuous and uniformly bounded. Our results are universal in the sense that they depend on the choice of the variances and possibly on [Formula: see text] (if it is kept constant), but not on the underlying distributions. The results can be specialized to specific models by fixing the variances, thus obtaining matrix polynomial analogues of results known for special classes of scalar polynomials, such as Kac, Weyl, elliptic and hyperbolic polynomials.
{"title":"The limit empirical spectral distribution of complex matrix polynomials","authors":"Giovanni Barbarino, V. Noferini","doi":"10.1142/S201032632250023X","DOIUrl":"https://doi.org/10.1142/S201032632250023X","url":null,"abstract":"We study the empirical spectral distribution (ESD) for complex [Formula: see text] matrix polynomials of degree [Formula: see text] under relatively mild assumptions on the underlying distributions, thus highlighting universality phenomena. In particular, we assume that the entries of each matrix coefficient of the matrix polynomial have mean zero and finite variance, potentially allowing for distinct distributions for entries of distinct coefficients. We derive the almost sure limit of the ESD in two distinct scenarios: (1) [Formula: see text] with [Formula: see text] constant and (2) [Formula: see text] with [Formula: see text] bounded by [Formula: see text] for some [Formula: see text]; the second result additionally requires that the underlying distributions are continuous and uniformly bounded. Our results are universal in the sense that they depend on the choice of the variances and possibly on [Formula: see text] (if it is kept constant), but not on the underlying distributions. The results can be specialized to specific models by fixing the variances, thus obtaining matrix polynomial analogues of results known for special classes of scalar polynomials, such as Kac, Weyl, elliptic and hyperbolic polynomials.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":"309 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77629061","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-02DOI: 10.1142/s2010326322500125
M. Fukuda, Takahiro Hasebe, Shinya Sato
Additivity violation of minimum output entropy, which shows non-classical properties in quantum communication, had been proved in most cases for random quantum channels defined by Haar-distributed unitary matrices. In this paper, we investigate random completely positive maps made of Gaussian Unitary Ensembles and Ginibre Ensembles regarding this matter. Using semi-circular systems and circular systems of free probability, we not only show the multiplicativity violation of maximum output norms in the asymptotic regimes but also prove the additivity violation via Haagerup inequality for a new class of random quantum channels constructed by rectifying the above completely positive maps based on strong convergence.
{"title":"Additivity violation of quantum channels via strong convergence to semi-circular and circular elements","authors":"M. Fukuda, Takahiro Hasebe, Shinya Sato","doi":"10.1142/s2010326322500125","DOIUrl":"https://doi.org/10.1142/s2010326322500125","url":null,"abstract":"Additivity violation of minimum output entropy, which shows non-classical properties in quantum communication, had been proved in most cases for random quantum channels defined by Haar-distributed unitary matrices. In this paper, we investigate random completely positive maps made of Gaussian Unitary Ensembles and Ginibre Ensembles regarding this matter. Using semi-circular systems and circular systems of free probability, we not only show the multiplicativity violation of maximum output norms in the asymptotic regimes but also prove the additivity violation via Haagerup inequality for a new class of random quantum channels constructed by rectifying the above completely positive maps based on strong convergence.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87380629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-23DOI: 10.1142/s2010326322500332
J. W. Silverstein
For each n, let Un be Haar distributed on the group of n × n unitary matrices. Let xn,1, . . . ,xn,m denote orthogonal nonrandom unit vectors in C n and let un,k = (uk, . . . , u n k) ∗ = U∗ nxn,k, k = 1, . . . , m. Define the following functions on [0,1]: X k,k n (t) = √ n ∑[nt] i=1(|uk|− 1 n ),X ′ n (t) = √ 2n ∑[nt] i=1 ū i ku i k′ , k < k ′. Then it is proven thatX n ,RXk,k ′ n , IXk,k′ n , considered as random processes in D[0, 1], converge weakly, as n → ∞, to m independent copies of Brownian bridge. The same result holds for the m(m + 1)/2 processes in the real case, where On is real orthogonal Haar distributed and xn,i ∈ R, with √ n in X n and √ 2n in X ′ n replaced with √ n 2 and √ n, respectively. This latter result will be shown to hold for the matrix of eigenvectors of Mn = (1/s)VnV T n where Vn is n × s consisting of the entries of {vij}, i, j = 1, 2, . . . , i.i.d. standardized and symmetrically distributed, with each xn,i = {±1/ √ n, . . . ,±1/√n}, and n/s→ y > 0 as n→ ∞. This result extends the result in J.W. Silverstein Ann. Probab. 18 1174-1194. These results are applied to the detection problem in sampling random vectors mostly made of noise and detecting whether the sample includes a nonrandom vector. The matrix Bn = θvnv ∗ n + Sn is studied where Sn is Hermitian or symmetric and nonnegative definite with either its matrix of eigenvectors being Haar distributed, or Sn = Mn, θ > 0 nonrandom, and vn is a nonrandom unit vector. Results are derived on the distributional behavior of the inner product of vectors orthogonal to vn with the eigenvector associated with the largest eigenvalue of Bn.
为每n,让联合国成为Haar集团》按on n×n unitary matrices。我们走xn 1…第xn,m, don ' t be orthogonal unrandom单位vectors在C, u n k)∗= u∗nxn, k, k = 1,。。,《跟踪functions on [0.1 m .定义:X k, k n (t) =√n∑(nt) i = 1 (| uk | n−1),X′n (t) =√2n∑(nt) i = 1 kū我我k′,< k′。然后是proven thatX n, k′n, IXk RXk k′n,美国认为随机processes in D[0, 1],美国converge虚弱地n→∞,到公元独立报copies of Brownian大桥。不变论点珍藏》(m + 1) / 2 processes in The real凯斯,真正在哪里是orthogonal Haar按和n, i∈R,在X√n n和√2n在X′n replaced 2√n和√n, respectively。这个后期圣徒论点将展示拥抱eigenvectors矩阵》为Mn = (1 / n) VnV T s哪里Vn是n×s consisting of之。{vij}, i, j = 1, 2,。。, i . i . d . standardized和symmetrically按,每一起,i ={±1 /√n个,。。,±1 /√n的和美国n / s > 0 y→n→∞。这是西尔弗斯坦·安的最新提议。181174 -1194号提案。这些结果大多应用于随机标本的发现问题，基本上是制造噪音，如果样本包括一个不随机的向量，就会发现。矩阵Bn =θvnv∗n + Sn是studied Sn在哪里Hermitian或symmetric和nonnegative当然不管它的矩阵eigenvectors身为Haar按,或Sn = Mn,θa > 0 nonrandom,和vn是nonrandom单位向量。结果导致了从内部生产到vn的正规性行为的传播，与Bn的最高等级关系有关。
{"title":"Weak Convergence of a Collection of Random Functions Defined by the Eigenvectors of Large Dimensional Random Matrices","authors":"J. W. Silverstein","doi":"10.1142/s2010326322500332","DOIUrl":"https://doi.org/10.1142/s2010326322500332","url":null,"abstract":"For each n, let Un be Haar distributed on the group of n × n unitary matrices. Let xn,1, . . . ,xn,m denote orthogonal nonrandom unit vectors in C n and let un,k = (uk, . . . , u n k) ∗ = U∗ nxn,k, k = 1, . . . , m. Define the following functions on [0,1]: X k,k n (t) = √ n ∑[nt] i=1(|uk|− 1 n ),X ′ n (t) = √ 2n ∑[nt] i=1 ū i ku i k′ , k < k ′. Then it is proven thatX n ,RXk,k ′ n , IXk,k′ n , considered as random processes in D[0, 1], converge weakly, as n → ∞, to m independent copies of Brownian bridge. The same result holds for the m(m + 1)/2 processes in the real case, where On is real orthogonal Haar distributed and xn,i ∈ R, with √ n in X n and √ 2n in X ′ n replaced with √ n 2 and √ n, respectively. This latter result will be shown to hold for the matrix of eigenvectors of Mn = (1/s)VnV T n where Vn is n × s consisting of the entries of {vij}, i, j = 1, 2, . . . , i.i.d. standardized and symmetrically distributed, with each xn,i = {±1/ √ n, . . . ,±1/√n}, and n/s→ y > 0 as n→ ∞. This result extends the result in J.W. Silverstein Ann. Probab. 18 1174-1194. These results are applied to the detection problem in sampling random vectors mostly made of noise and detecting whether the sample includes a nonrandom vector. The matrix Bn = θvnv ∗ n + Sn is studied where Sn is Hermitian or symmetric and nonnegative definite with either its matrix of eigenvectors being Haar distributed, or Sn = Mn, θ > 0 nonrandom, and vn is a nonrandom unit vector. Results are derived on the distributional behavior of the inner product of vectors orthogonal to vn with the eigenvector associated with the largest eigenvalue of Bn.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":"451 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79705275","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-17DOI: 10.1142/S2010326321500350
A. Aguirre, A. Soshnikov, Joshua Sumpter
We study the limiting distribution of a pair counting statistics of the form [Formula: see text] for the circular [Formula: see text]-ensemble (C[Formula: see text]E) of random matrices for sufficiently smooth test function [Formula: see text] and [Formula: see text] For [Formula: see text] and [Formula: see text] our results are inspired by a classical result of Montgomery on pair correlation of zeros of Riemann zeta function.
{"title":"Pair dependent linear statistics for CβE","authors":"A. Aguirre, A. Soshnikov, Joshua Sumpter","doi":"10.1142/S2010326321500350","DOIUrl":"https://doi.org/10.1142/S2010326321500350","url":null,"abstract":"We study the limiting distribution of a pair counting statistics of the form [Formula: see text] for the circular [Formula: see text]-ensemble (C[Formula: see text]E) of random matrices for sufficiently smooth test function [Formula: see text] and [Formula: see text] For [Formula: see text] and [Formula: see text] our results are inspired by a classical result of Montgomery on pair correlation of zeros of Riemann zeta function.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":"96 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78068055","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-03DOI: 10.1142/S2010326322500186
Phanuel Mariano, Hugo Panzo
We prove a central limit theorem (CLT) for the product of a class of random singular matrices related to a random Hill’s equation studied by Adams–Bloch–Lagarias. The CLT features an explicit formula for the variance in terms of the distribution of the matrix entries and this allows for exact calculation in some examples. Our proof relies on a novel connection to the theory of [Formula: see text]-dependent sequences which also leads to an interesting and precise nondegeneracy condition.
{"title":"CLT with explicit variance for products of random singular matrices related to Hill’s equation","authors":"Phanuel Mariano, Hugo Panzo","doi":"10.1142/S2010326322500186","DOIUrl":"https://doi.org/10.1142/S2010326322500186","url":null,"abstract":"We prove a central limit theorem (CLT) for the product of a class of random singular matrices related to a random Hill’s equation studied by Adams–Bloch–Lagarias. The CLT features an explicit formula for the variance in terms of the distribution of the matrix entries and this allows for exact calculation in some examples. Our proof relies on a novel connection to the theory of [Formula: see text]-dependent sequences which also leads to an interesting and precise nondegeneracy condition.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":"99 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77612023","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-02DOI: 10.1142/s2010326322500228
E. Blackstone, C. Charlier, J. Lenells
We consider the probability that no points lie on [Formula: see text] large intervals in the bulk of the Airy point process. We make a conjecture for all the terms in the asymptotics up to and including the oscillations of order [Formula: see text], and we prove this conjecture for [Formula: see text].
{"title":"Gap probabilities in the bulk of the Airy process","authors":"E. Blackstone, C. Charlier, J. Lenells","doi":"10.1142/s2010326322500228","DOIUrl":"https://doi.org/10.1142/s2010326322500228","url":null,"abstract":"We consider the probability that no points lie on [Formula: see text] large intervals in the bulk of the Airy point process. We make a conjecture for all the terms in the asymptotics up to and including the oscillations of order [Formula: see text], and we prove this conjecture for [Formula: see text].","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":"496 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80022352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-27DOI: 10.1142/S2010326321500301
A. Bose, Koushik Saha, Priyanka Sen
Patterned random matrices such as the reverse circulant, the symmetric circulant, the Toeplitz and the Hankel matrices and their almost sure limiting spectral distribution (LSD), have attracted much attention. Under the assumption that the entries are taken from an i.i.d. sequence with finite variance, the LSD are tied together by a common thread — the [Formula: see text]th moment of the limit equals a weighted sum over different types of pair-partitions of the set [Formula: see text] and are universal. Some results are also known for the sparse case. In this paper, we generalize these results by relaxing significantly the i.i.d. assumption. For our models, the limits are defined via a larger class of partitions and are also not universal. Several existing and new results for patterned matrices, their band and sparse versions, as well as for matrices with continuous and discrete variance profile follow as special cases.
{"title":"Some patterned matrices with independent entries","authors":"A. Bose, Koushik Saha, Priyanka Sen","doi":"10.1142/S2010326321500301","DOIUrl":"https://doi.org/10.1142/S2010326321500301","url":null,"abstract":"Patterned random matrices such as the reverse circulant, the symmetric circulant, the Toeplitz and the Hankel matrices and their almost sure limiting spectral distribution (LSD), have attracted much attention. Under the assumption that the entries are taken from an i.i.d. sequence with finite variance, the LSD are tied together by a common thread — the [Formula: see text]th moment of the limit equals a weighted sum over different types of pair-partitions of the set [Formula: see text] and are universal. Some results are also known for the sparse case. In this paper, we generalize these results by relaxing significantly the i.i.d. assumption. For our models, the limits are defined via a larger class of partitions and are also not universal. Several existing and new results for patterned matrices, their band and sparse versions, as well as for matrices with continuous and discrete variance profile follow as special cases.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":"35 1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78067209","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-23DOI: 10.1142/s2010326322500204
P. Forrester
The eigenvalue probability density function (PDF) for the Gaussian unitary ensemble has a well-known analogy with the Boltzmann factor for a classical log-gas with pair potential [Formula: see text], confined by a one-body harmonic potential. A generalization is to replace the pair potential by [Formula: see text]. The resulting PDF first appeared in the statistical physics literature in relation to non-intersecting Brownian walkers, equally spaced at time [Formula: see text], and subsequently in the study of quantum many-body systems of the Calogero–Sutherland type, and also in Chern–Simons field theory. It is an example of a determinantal point process with correlation kernel based on the Stieltjes–Wigert polynomials. We take up the problem of determining the moments of this ensemble, and find an exact expression in terms of a particular little [Formula: see text]-Jacobi polynomial. From their large [Formula: see text] form, the global density can be computed. Previous work has evaluated the edge scaling limit of the correlation kernel in terms of the Ramanujan ([Formula: see text]-Airy) function. We show how in a particular [Formula: see text] scaling limit, this reduces to the Airy kernel.
{"title":"Global and local scaling limits for the β = 2 Stieltjes–Wigert random matrix ensemble","authors":"P. Forrester","doi":"10.1142/s2010326322500204","DOIUrl":"https://doi.org/10.1142/s2010326322500204","url":null,"abstract":"The eigenvalue probability density function (PDF) for the Gaussian unitary ensemble has a well-known analogy with the Boltzmann factor for a classical log-gas with pair potential [Formula: see text], confined by a one-body harmonic potential. A generalization is to replace the pair potential by [Formula: see text]. The resulting PDF first appeared in the statistical physics literature in relation to non-intersecting Brownian walkers, equally spaced at time [Formula: see text], and subsequently in the study of quantum many-body systems of the Calogero–Sutherland type, and also in Chern–Simons field theory. It is an example of a determinantal point process with correlation kernel based on the Stieltjes–Wigert polynomials. We take up the problem of determining the moments of this ensemble, and find an exact expression in terms of a particular little [Formula: see text]-Jacobi polynomial. From their large [Formula: see text] form, the global density can be computed. Previous work has evaluated the edge scaling limit of the correlation kernel in terms of the Ramanujan ([Formula: see text]-Airy) function. We show how in a particular [Formula: see text] scaling limit, this reduces to the Airy kernel.","PeriodicalId":54329,"journal":{"name":"Random Matrices-Theory and Applications","volume":"181 ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72428028","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-20DOI: 10.1142/s2010326322500460
L. Pastur, V. Slavin
We study the distribution of singular values of product of random matrices pertinent to the analysis of deep neural networks. The matrices resemble the product of the sample covariance matrices, however, an important difference is that the population covariance matrices assumed to be non-random or random but independent of the random data matrix in statistics and random matrix theory are now certain functions of random data matrices (synaptic weight matrices in the deep neural network terminology). The problem has been treated in recent work [25, 13] by using the techniques of free probability theory. Since, however, free probability theory deals with population covariance matrices which are independent of the data matrices, its applicability has to be justified. The justification has been given in [22] for Gaussian data matrices with independent entries, a standard analytical model of free probability, by using a version of the techniques of random matrix theory. In this paper we use another, more streamlined, version of the techniques of random matrix theory to generalize the results of [22] to the case where the entries of the synaptic weight matrices are just independent identically distributed random variables with zero mean and finite fourth moment. This, in particular, extends the property of the so-called macroscopic universality on the considered random matrices.
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