Matthew Kwan, Lisa Sauermann
Let $M_{n}$ denote a random symmetric $n\times n$ matrix, whose entries on and above the diagonal are i.i.d. Rademacher random variables (taking values $\pm 1$ with probability $1/2$ each). Resolving a conjecture of Vu, we prove that the permanent of $M_{n}$ has magnitude $n^{n/2+o(n)}$ with probability $1-o(1)$. Our result can also be extended to more general models of random matrices.
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A t by n random matrix A is formed by sampling n independent random column vectors, each containing t components. The random Gram matrix of size n, G_n, contains the dot products between all pairs of column vectors in the randomly generated matrix A; that is, G_n = transpose(A) A. The matrix G_n has characteristic roots coinciding with the singular values of A. Furthermore, the sequences det(G_i) and per(G_i) (for i = 0, 1, ..., n) are factors that comprise the expected coefficients of the characteristic and permanental polynomials of G_n. We prove theorems that relate the generating functions…
Kernels of $α$-permanental processes of the form \[ v(x,y)=u(x,y)+f(y),\qquad x,y\in S, \] in which $u(x,y)$ is symmetric, and $f$ is an excessive function for the Borel right process with potential densities $u(x,y)$, are considered. Conditions are given that determine whether $\{v(x,y);x,y\in S\}$ is symmetrizable or asymptotically symmetrizable.
We prove a stability version of a general result that bounds the permanent of a matrix in terms of its operator norm. More specifically, suppose $A$ is an $n \times n$ matrix over $\mathbb{C}$ (resp. $\mathbb{R}$), and let $\mathcal{P}$ denote the set of $n \times n$ matrices over $\mathbb{C}$ (resp. $\mathbb{R}$) that can be written as a permutation matrix times a unitary diagonal matrix. Then it is known that the permanent of $A$ satisfies $|\text{perm}(A)| \leq \Vert A \Vert_{2} ^n$ with equality iff $A/ \Vert A \Vert_{2} \in \mathcal{P}$ (where $\Vert A \Vert_2$ is the operator $2$-norm of…
We show that the permanent of an $n \times n$ matrix with iid Bernoulli entries $\pm 1$ is of magnitude $n^{({1/2}+o(1))n}$ with probability $1-o(1)$. In particular, it is almost surely non-zero.
We show that the absolute value of the determinant of a matrix with random independent (but not necessarily iid) entries is strongly concentrated around its mean. As an application, we show that the Godsil-Gutman and Barvinok estimators for the permanent of a strictly positive matrix give sub-exponential approximation ratios with high probability.
An Exponential Lower Bound for the Permanent of Random Bernoulli Matrix
该研究针对元素为独立均匀符号项的$n$阶随机伯努利矩阵，证明其行列式积存在指数尺度下界，通过逐行暴露跟踪子式总平方行列式积的方法，证实了陶哲轩与武提出的相关猜想。
We introduce a new notion of the determinant, called symmetrized determinant, for a square matrix with the entries in an associative algebra $Å$. The monomial expansion of the symmetrized determinant is obtained from the standard expansion of the commutative determinant by averaging the products of entries of the matrix in all possible orders. We show that for any fixed finite-dimensional associative algebra $Å$, the symmetrized determinant of an $n\times n$ matrix with the entries in $Å$ can be computed in polynomial in $n$ time (the degree of the polynomial is linear in the dimension of $Å$)…
The paper addresses the calculation of correlation functions of permanental polynomials of matrices with random entries. By exploiting a convenient contour integral representation of the matrix permanent some explicit results are provided for several random matrix ensembles. When compared with the corresponding formulae for characteristic polynomials, our results show both striking similarities and interesting differences. Based on these findings, we conjecture the asymptotic forms of the density of permanental roots in the complex plane for Gaussian ensembles as well as for the Circular Unita…
Permanental rank versus determinantal rank of random matrices over finite fields
研究随机矩阵在有限域上永久性秩与行列式秩的区别，发现永久性秩概率与零列概率相似，而行列式秩概率则不同。
Let A_n=(a_{ij})_{i,j=1}^n be an n\times n positive matrix with entries in [a,b], 0<a\le b. Let X_n=(\sqrta_{ij}x_{ij})_{i,j=1}^n be a random matrix, where {x_{ij}} are i.i.d. N(0,1) random variables. We show that for large n, \det (X_n^TX_n) concentrates sharply at the permanent of A_n, in the sense that n^{-1}\log (\det(X_n^TX_n)/perA_n)\to_{n\to\infty}0 in probability.
Inequalities for rank-two permanents and finite free convolutions
该研究强化了Bang的矩阵永久式不等式，将其适用范围扩展至秩不超过2的实矩阵，并利用该强化结果推导得出多项式有限自由卷积的新不等式。
Permanents of matrix ensembles: computation, distribution, and geometry
研究通过GPU加速计算，分析不同矩阵族永久性的分布特性与几何行为，发现高斯矩阵永久性遵循α-稳定分布，而海耳正交矩阵永久性具有正超额峰度，且反集中性在高斯族中更稳健。
Exponential anticoncentration of the permanent
该论文证明随机矩阵的永久式具有指数反集中性，显著改进先前界，并应用于解决 Ingram 和 Razborov 关于±1矩阵永久式取值数量的问题。
Bobkov (J. Theoret. Probab. 18(2) (2005) 399-412) investigated an approximate de Finetti representation for probability measures, on product measurable spaces, which are symmetric under permutations of coordinates. One of the main results of that paper was an explicit approximation bound for permanents of complex rectangular matrices, which was shown by a somewhat complicated induction argument. In this paper, we indicate how to avoid the induction argument using an (asymptotic) expansion. Our approach makes it possible to give new explicit higher order approximation bounds for such permanents…
Permanents of random matrices over finite fields
研究有限域上随机矩阵永久的渐近分布，发现其比行列式更均匀。
An $\al$-permanental process $\{X_{ t},t\in T \}$ is a stochastic process determined by a kernel $K=\{K(s,t),s,t\in T \}$, with the property that for all $t_{1},\ldots,t_{n}\in T $, $ |I+K( t_{1},\ldots,t_{n}) S|^{- \al} $ is the Laplace transform of $(X_{t_{1}},\ldots,X_{t_{n}})$, where $ K( t_{1},\ldots,t_{n})$ denotes the matrix $\{K(t_{i}, t_{j})\}_{i,j=1}^{n}$ and $S$ is the diagonal matrix with entries $s_{1},\ldots,s_{n} $. $ (X_{t_{1}},\ldots,X_{t_{n}})$ is called a permanental vector. Under the condition that $K$ is the potential density of a transient Markov process, $(X_{t_{1}},\ldo…
Approximating the Permanent of a Random Matrix with Polynomially Small Mean: Zeros and Universality
研究在矩阵条目略微偏离零时近似随机矩阵永久性的算法，通过零点区域分析揭示复杂性理论中的核心问题与贡献。
We study the asymptotic behavior of permanents of $n \times n$ random matrices $A$ with positive entries. We assume that $A$ has either i.i.d. entries or is a symmetric matrix with the i.i.d. upper triangle. Under the assumption that elements have power law decaying tails, we prove a strong law of large numbers for $\log \perm A$. We calculate the values of the limit $\lim_{n \to \infty}\frac{\log \perm A}{n \log n}$ in terms of the exponent of the power law distribution decay, and observe a first order phase transition in the limit as the mean becomes infinite. The methods extend to a wide cl…
Permanents of random matrices with independent and identically distributed (i.i.d.) entries have extensively studied in literature and convergence and concentration properties are known under varying assumptions on the distributions. In this paper we consider constrained~\(n \times n\) random matrices where each row has a deterministic number~\(r = r(n)\) zero entries and the rest of the entries are independent random variables that are positive almost surely. The positions of the zeros within each row are also random and we establish sufficient conditions for the existence of a weak law of la…
Anticoncentration of the Permanent in Ginibre Ensembles
研究吉尼贝尔系综中归一化行序积和式的反集中性，通过比较拉普拉斯变换阶的平方高斯积和式与平方行列式来证明，给出其径向密度相关条件及上界，特别解决了\(\mathbb{K}=\mathbb{C}\)时的积和式反集中猜想。
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