Random Discrete Matrices

关于酉不变随机矩阵的部分转置

On partial transposes of unitarily invariant random matrices

本文研究酉不变随机矩阵的部分转置在块数和大小趋于无穷时的极限分布，并证明其与转置及左右部分转置间的渐近自由关系。

A countdown process, with application to the rank of random matrices over $\mathbb F_q(n)$

Motivated by the work of Fulman and Goldstein, comparing the distribution of the corank of random matrices in $\mathbb F_q[n]$ with the limit distribution as $n \to \infty$, we define a countdown process, driven by independent geometric random variables related to random integer partitions. Analysis of this process leads to sharper bounds on the total variation distance.

Permutation Matrices, Their Discrete Derivatives and Extremal Properties

For a permutation $π$, and the corresponding permutation matrix, we introduce the notion of {\em discrete derivative}, obtained by taking differences of successive entries in $π$. We characterize the possible derivatives of permutations, and consider questions for permutations with certain properties satisfied by the derivative. For instance, we consider permutations with distinct derivatives, and the relationship to so-called Costas arrays.

Random Permutations, Random Sudoku Matrices and Randomized Algorithms

Some randomized algorithms, used to obtain a random $n^2 \times n^2$ Sudoku matrix, where $n$ is a natural number, is reviewed in this study. Below is described the set $Π_n$ of all $(2n) \times n$ matrices, consisting of elements of the set $\mathbb{Z}_n =\{ 1,2,\ldots ,n\}$, such that every row is a permutation. It is proved that such matrices would be particularly useful in developing efficient algorithms in generating Sudoku matrices. An algorithm to obtain random $Π_n$ matrices is presented in this paper. The algorithms are evaluated according to two criteria - probability evaluation, and…

Enumerating traceless matrices over compact discrete valuation rings

We enumerate traceless square matrices over finite quotients of compact discrete valuation rings by their image sizes. We express the associated rational generating functions in terms of statistics on symmetric and hyperoctahedral groups, viz. Coxeter groups of types A and B, respectively. These rational functions may also be interpreted as local representation zeta functions associated to the members of an infinite family of finitely generated class-2-nilpotent groups. As a byproduct of our work, we obtain descriptions of the numbers of traceless square matrices over a finite field of fixed r…

Discrete Tomography: Reconstruction under periodicity constraints

This paper studies the problem of reconstructing binary matrices that are only accessible through few evaluations of their discrete X-rays. Such question is prominently motivated by the demand in material science for developing a tool for the reconstruction of crystalline structures from their images obtained by high-resolution transmission electron microscopy. Various approaches have been suggested for solving the general problem of reconstructing binary matrices that are given by their discrete X-rays in a number of directions, but more work have to be done to handle the ill-posedness of the…

On the singularity probability of discrete random matrices

Let $M_n$ be an $n$ by $n$ random matrix where each entry is +1 or -1 independently with probability 1/2. Our main result implies that the probability that $M_n$ is singular is at most $(1/\sqrt{2} + o(1))^n$, improving on the previous best upper bound of $(3/4 + o(1))^n$ proven by Tao and Vu in arXiv:math/0501313v2. This paper follows a similar approach to the Tao and Vu result, including using a variant of their structure theorem. We also extend this type of exponential upper bound on the probability that a random matrix is singular to a large class of discrete random matrices taking values …

离散周期矩阵和离散全纯积分在黎曼球面分支覆盖中的收敛性

Convergence of discrete period matrices and discrete holomorphic integrals for ramified coverings of the Riemann sphere

研究离散全纯函数在黎曼球面分支覆盖中的收敛性，通过三角剖分和立体投影定义离散调和函数，并证明离散周期矩阵收敛于连续情形，同时给出误差估计和离散全纯积分的收敛结果。

Shuffling matrices, Kronecker product and Discrete Fourier Transform

We define and investigate a family of permutations matrices, called shuffling matrices, acting on a set of $N=n_1\cdots n_m$ elements, where $m\geq 2$ and $n_i\geq 2$ for any $i=1,\ldots, m$. These elements are identified with the vertices of the $m$-th level of a rooted tree with branch indices $(n_1,\ldots, n_m)$. Each of such matrices is induced by a permutation of $Sym(m)$ and it turns out that, in the case in which one considers the cyclic permutation $(1\ \ldots\ m)$, the corresponding permutation is the classical perfect shuffle. We give a combinatorial interpretation of these permutati…

Further results on discrete unitary invariance

In arXiv:1607.06679, Marcus proved that certain functions of multiple matrices, when summed over the symmetries of the cube, decompose into functions of the original matrices. In this note, we generalize the results from the Marcus paper to a larger class of functions of multiple matrices. We also answer a problem posed in the Marcus paper.

Toric networks, geometric $R$-matrices and generalized discrete Toda lattices

We use the combinatorics of toric networks and the double affine geometric $R$-matrix to define a three-parameter family of generalizations of the discrete Toda lattice. We construct the integrals of motion and a spectral map for this system. The family of commuting time evolutions arising from the action of the $R$-matrix is explicitly linearized on the Jacobian of the spectral curve. The solution to the initial value problem is constructed using Riemann theta functions.

离散周期矩阵的收敛性

The convergence of discrete period matrices

本文研究紧多面体曲面作为黎曼曲面及其离散化，证明了离散 Dirichlet 能量与离散周期矩阵的收敛性，并通过调整锥奇点附近的四边形剖分将收敛速率提升为最大边长的线性函数。

周期矩阵与离散黎曼面上的同调近树

Period matrices and homological quasi-trees on discrete Riemann surfaces

本文研究离散周期矩阵与同调近树的关系，推导了Teichmüller空间上的Weil-Petersson势的组合类比，并探讨了其在圆平面网络中的作用。

Surjectivity of near square random matrices

We show that a nearly square iid random integral matrix is surjective over the integral lattice with very high probability. This answers a question by Koplewitz. Our result extends to sparse matrices as well as to matrices of dependent entries.

Asymptotic free independence and entry permutations for Gaussian random matrices

The paper presents conditions on entry permutations that induce asymptotic freeness when acting on Gaussian random matrices. The class of permutations described includes the matrix transpose, as well as entry permutations relevant in Quantum Information Theory and Quantum Physics.