Octavio Arizmendi, Adrian Celestino
We provide a generalization and new proofs of the formulas of Collins, Hasebe and Sakuma for the spectrum of polynomials in cyclic monotone elements. This is applied to random matrices with discrete spectrum.
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Matrix models for cyclic monotone and monotone independences
本文证明同一随机矩阵模型在适当状态下同时作为循环单调独立和单调独立的模型，从而为两种独立概念提供统一框架。
Cyclic independence: Boolean and monotone
本文引入了循环-单调独立性的修正版本及其自然类比循环-布尔独立性，研究了卷积公式、极限定理与无穷可分分布分类，并应用于图的迭代星积与梳积邻接矩阵的特征值。
We show that there is no cyclically monotone stationary matching of two independent Poisson processes in dimension $d=2$. The proof combines the harmonic approximation result from \cite{GHO} with local asymptotics for the two-dimensional matching problem for which we give a new self-contained proof using martingale arguments.
We extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations. We introduce the concept of "second order freeness" and derive the global fluctuations of Gaussian and Wishart random matrices by a general limit theorem for second order freeness. By introducing cyclic Fock space, we also give an operator algebraic model for the fluctuations of our random matrices in terms of the usual creation, annihilation, and preservation operators. We show that orthogonal families of Gaussian and Wishart random matrices are asymptoticall…
We introduce the concept of ``R-cyclic family'' of matrices with entries in a non-commutative probability space; the definition consists in asking that only the ``cyclic'' non-crossing cumulants of the entries of the matrices are allowed to be non-zero. Let A_{1}, ..., A_{s} be an R-cyclic family of d \times d matrices over a non-commutative probability space. We prove a convolution-type formula for the explicit computation of the joint distribution of A_{1}, ..., A_{s} (considered in M_{d} (\A) with the natural state), in terms of the joint distribution (considered in the original space) of t…
In the setting of optimal transport with $N\ge 2$ marginals, a necessary condition for transport plans to be optimal is that they are $c$-cyclically monotone. For $N=2$ there exist several proofs that in very general settings $c$-cyclical monotoncity is also sufficient for optimality, while for $N\ge 3$ this is only known under strong conditions on $c$. Here we give a counterexample which shows that $c$-cylclical monotonicity is in general not sufficient for optimality if $N\ge 3$. Comparison with the $N=2$ case shows how the main proof strategies valid for the case $N=2$ might fail for $N\ge …
The practically important classes of equal-input and of monotone Markov matrices are revisited, with special focus on embeddability, infinite divisibility, and mutual relations. Several uniqueness results for the classic Markov embedding problem are obtained in the process. To achieve our results, we need to employ various algebraic and geometric tools, including commutativity, permutation invariance and convexity. Of particular relevance in several demarcation results are Markov matrices that are idempotents.
Voiculescu's freeness emerges in computing the asymptotic of spectra of polynomials on $N\times N$ random matrices with eigenspaces in generic positions: they are randomly rotated with a uniform unitary random matrix $U_N$. In this article we elaborate on the previous point by proposing a random matrix model, which we name the Vortex model, where $U_N$ has the law of a uniform unitary random matrix conditioned to leave invariant one deterministic vector $v_N$. In the limit $N \to +\infty$, we show that $N\times N$ matrices randomly rotated by the matrix $U_N$ are asymptotically conditionally f…
Generalised eigenvector expansion of infinite Toeplitz matrices with absolutely/completely monotone entries
研究在序列完全单调条件下无穷Toeplitz矩阵的谱理论，推导广义特征向量表达式并证明广义特征向量展开，即使矩阵非正规，表达式仅涉及实特征值和特征向量。
We identify a relationship between a certain family of random walks on Euclidean lattices and difference matrices over cyclic groups. We then use the techniques of Fourier analysis to estimate the return probabilities of these random walks, which in turn yields the asymptotic number of difference matrices over cyclic groups as the number of columns increases.
The celebrated elliptic law describes the distribution of eigenvalues of random matrices with correlations between off-diagonal pairs of elements, having applications to a wide range of physical and biological systems. Here, we investigate the generalization of this law to random matrices exhibiting higher-order cyclic correlations between $k$-tuples of matrix entries. We show that the eigenvalue spectrum in this ensemble is bounded by a hypotrochoid curve with $k$-fold rotational symmetry. This hypotrochoid law applies to full matrices as well as sparse ones, and thereby holds with remarkable…
Addition to "Structured random matrices and cyclic cumulants: A free probability approach"
本文精化随机矩阵系综定义，扩展第四公理处理不相交循环累积量，并证明非线性变换下的稳定性定理仍成立。
Structured random matrices and cyclic cumulants: A free probability approach
本文提出一类由四个基本性质刻画的大型结构化随机矩阵，证明其在矩阵值及逐点非线性运算下稳定，并基于极值化问题给出计算子块谱的高效方法，通过组合与代数证明验证其与自由概率的联系。
Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics
本研究将球变换与Pólya系综框架拓展至酉矩阵乘积研究，定义了循环Pólya频率函数并明确其与循环Pólya系综的关联，给出对应权重唯一性结论，推导了固定维度下特征值的行列式点过程，还概述了局部谱统计研究的待解问题。
The Distribution of Polynomials in Monotone Independent Elements
本文研究单调独立与无穷小单调独立元素中多项式的 $*$-分布，显式给出 $αab+βba$ 的分布，并应用于随机矩阵多项式的偏迹极限分布。
We give formulae for the cumulants of complex Wishart (LUE) and inverse Wishart matrices (inverse LUE). Their large-$N$ expansions are generating functions of double (strictly and weakly) monotone Hurwitz numbers which count constrained factorisations in the symmetric group. The two expansions can be compared and combined with a duality relation proved in [F. D. Cunden, F. Mezzadri, N. O'Connell and N. J. Simm, arXiv:1805.08760] to obtain: i) a combinatorial proof of the reflection formula between moments of LUE and inverse LUE at genus zero and, ii) a new functional relation between the gener…
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