Bruno Schapira
The aim of this note is to describe the Poisson boundary of the group of invertible triangular matrices with coefficients in a number field. It generalizes to any dimension and to any number field a result of Brofferio concerning the Poisson boundary of random rational affinities.
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Universal constant order fluctuations for the cokernels of block triangular matrices
该研究证明随机分块下三角矩阵余核的Sylow p-子群具有与特定矩阵乘积一致的常数阶涨落，且相关矩阵乘积定理在更弱的因子数量假设下依然成立。
Multivariate stochastic recurrence equations (SREs) are investigated when coefficients are triangular matrices. If coefficient matrices of SREs have all strictly positive elements, the Kesten's classical result yields solutions with regularly varying tails such that the tail indices of solutions are the same through coordinates. This framework is too restrictive for applications. In order to widen the applicability of the SREs, we study SREs with triangular matrix coefficients and prove that they have regularly varying solutions which may exhibit coordinate-wisely different tail exponents. We …
We prove weak convergence of triangular arrays to the compound Poisson limit using Tikhomirov's method. The result is applied to statistical estimation of the threshold parameter in autoregressive models.
We study solution X of the stochastic equation X = AX +B, where A is a random matrix and B,X are random vectors, the law of (A,B) is given and X is independent of (A,B). The equation is meant in law, the matrix A is 2x2 upper triangular, A_{11}=A_{22}>0, A_{12} is real. A sharp asymptotics of the tail of X =(X _1,X_2) is obtained. We show that under "so called" Kesten-Goldie conditions P (X_2>t)\sim t^{-a} and P (X_1>t )\sim t^{-a}(\log t)^b, where b =a or a\2.
We study a natural random walk over the upper triangular matrices, with entries in the field $\Z_2$, generated by steps which add row $i+1$ to row $i$. We show that the mixing time of the lazy random walk is $O(n^2)$ which is optimal up to constants. Our proof makes key use of the linear structure of the group and extends to walks on the upper triangular matrices over the fields $\Z_q$ for $q$ prime.
We study the eigenvalue problem for some special class of anti-triangular matrices. Though the eigenvalue problem is quite classical, as far as we know, almost nothing is known about properties of eigenvalues for anti-triangular matrices. In this paper, we show that there is a nice class of anti-triangular matrices whose eigenvalues are given explicitly by their elements. Moreover, this class contains several interesting subclasses which we characterize in terms of probability measures. We also discuss the application of our main theorem to the study of interacting particle systems, which are …
We prove the existence of the limiting spectral distribution (LSD) of symmetric triangular patterned matrices and also establish the joint convergence of sequences of such matrices. For the particular case of the symmetric triangular Wigner matrix, we derive expression for the moments of the LSD using properties of Catalan words. The problem of deriving explicit formulae for the moments of the LSD does not seem to be easy to solve for other patterned matrices. The LSD of the non-symmetric triangular Wigner matrix also does not seem to be easy to establish.
We analyze the macroscale behavior of random lower (and therefore upper) triangular matrices with entries drawn iid from a distribution with nonzero mean and finite variance. We show that such a matrix behaves like a probabilistic version of a Riemann sum and therefore in the limit behaves like the Volterra operator. Specifically, we analyze certain SOT-like and WOT-like modes of convergence for random lower triangular matrices to a scaled Volterra operator. We close with a brief discussion of moments.
We show that the empirical distribution of the eigenvalues of the sample covariance matrix of certain random vectors (not necessarily independent entries) with bounded marginal $L^{4}$ norms converges weakly to a compound free Poisson distribution.
The fluctuations of the mod p rank of triangular matrices
本文研究了元素为独立同分布整数的随机下三角矩阵，证明了其余核的 Sylow p-子群具有与已知矩阵乘积相同的常数阶波动，并以此描述了该矩阵在有限域 F_p 上秩的极限波动。
We present a Hilbert space approach to the limit joint *-distributions of complex independent Gaussian random matrices. For that purpose, we use a suitably defined family of creation and annihilation operators living in some direct integral of Hilbert spaces. These operators are decomposed in terms of continuous circular systems of operators acting between the fibers of the considered Hilbert space direct integral. In the case of square matrices with i.i.d. entries, we obtain the circular operators of Voiculescu, whereas in the case of upper-triangular matrices with i.i.d. entries, we obtain t…
Consider the random walk on the $n \times n$ upper triangular matrices with ones on the diagonal and elements over $\mathbb{F}_p$ where we pick a row at random and either add it or subtract it from the row directly above it. The main result of this paper is to prove that the dependency of the mixing time on $p$ is $p^2$. This is proven by combining super-character theory and comparison theory arguments.
We study the singular values of certain triangular random matrices. When their elements are i.i.d. standard complex Gaussian random variables, the squares of the singular values form a biorthogonal ensemble, and with an appropriate change in the distribution of the diagonal elements, they give the biorthogonal Laguerre ensemble. For triangular Wigner matrices, we give alternative proofs for the convergence of the empirical distribution of the appropriately scaled squares of the singular eigenvalues to a distribution with support $[0, e]$, as well as for the almost sure convergence of the resca…
The random walk on upper triangular matrices over $\mathbb{Z}/m \mathbb{Z}$
本文研究了 $\mathbb{Z}/m \mathbb{Z}$ 上 $n \times n$ 上三角矩阵的随机游走，通过加减均匀随机行生成步骤，并证明其混合时间为 $O(m^2n \log n+ n^2 m^{o(1)})$，回答了 Stong 等人的问题。
We study bivariate stochastic recurrence equations with triangular matrix coefficients and we characterize the tail behavior of their stationary solutions ${\bf W} =(W_1,W_2)$. Recently it has been observed that $W_1,W_2$ may exhibit regularly varying tails with different indices, which is in contrast to well-known Kesten-type results. However, only partial results have been derived. Under typical "Kesten-Goldie" and "Grey" conditions, we completely characterize tail behavior of $W_1,W_2$. The tail asymptotics we obtain has not been observed in previous settings of stochastic recurrence equati…
Persistence diagrams of random triangular matrices over finite fields
研究有限域上随机下三角矩阵行张成子空间的演化，利用持续图刻画其性质，给出分布显式公式、寿命大数定律及持续Betti数波动描述。
Toeplitz matrices from permutation displacements and the triangular kernel
本文研究了通过缩放核a_k=f(k/n)生成的Toeplitz矩阵，证明其特征值均值收敛于f的加权积分，并引入置换位移计数的组合构造，最终分析了与置换位移相关的极限谱结构。
We study large Wigner random matrices in the case when the marginal distributions of matrix entries have heavy tails. We prove that the largest eigenvalues of such matrices have Poisson statistics.
Quantum inverse scattering for the 20-vertex model up to Dynkin automorphism: 3D Poisson structure, triangular height functions, weak integrability
本文基于量子反散射方法研究20顶点模型，利用更高维的L算子实现类似可积性的概念，探讨3D泊松结构和三角高度函数对相关性的影响。
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