Teaching

Winter 2024: STAT/MTHE 353 Slot 12, MATH 892, Slot 5

Research

Seminar on Free Probability and Random Matrices
Research Group in Free Probability

Research interests

Free probability theory, Random Matrices, C*-algebras, W*-algebras, Tilings, K-theory, C*-dynamical systems.

Preprint version of my book with Roland Speicher Free Probability and Random Matrices published by Springer in 2017. List of typos.

James A. Mingo's articles on arXiv

[1]   Infinitesimal freeness for orthogonally invariant random matrices
Guillaume Cébron, James A Mingo

Comments: 50 pages, corrected typos, added a few points of clarification, and added some more figures
Subjects: Probability (math.PR); Probability (math.PR),Operator Algebras (math.OA),46L54, 60B20

We introduce a new kind of free independence, called real infinitesimal freeness. We show that independent orthogonally invariant with infinitesimal laws are asymptotically real infinitesimally free. We introduce new cumulants, called real infinitesimal cumulants and show that real infinitesimal freeness is equivalent to vanishing of mixed cumulants. We prove the formula for cumulants with products as entries.

[2]  Asymptotic limit of cumulants and higher order free cumulants of complex Wigner matrices
James A. Mingo, Daniel Munoz George

Comments: 68 pages, 7 Figures
Subjects: Probability (math.PR); Probability (math.PR),60B20 (Primary) 46L54, 15B52 (Secondary)

We compute the fluctuation moments $\alpha_{m_1,\dots,m_r}$ of a Complex Wigner Matrix $X_N$ given by the limit $\lim_{N\rightarrow\infty}N^{r-2}k_r(Tr(X_N^{m_1}),\dots,Tr(X_N^{m_r}))$. We prove the limit exists and characterize the leading order via planar graphs that result to be trees. We prove these graphs can be counted by the set of non-crossing partitioned permutations which permit us to express the moments $\alpha_{m_1,\dots,m_r}$ in terms of simpler quantities $\kappa_{m_1,\dots,m_r}$ known as the higher order cumulants. As for lower order dimensions ($r \leq 3$) we observe that while the moments have a more elaborated expression the cumulants are simpler.

[3]  On partial transposes of unitarily invariant random matrices
James A. Mingo, Mihai Popa

Comments: 18 pages, small typos corrected, to appear in Int. J. Math
Subjects: Probability (math.PR); Probability (math.PR),Combinatorics (math.CO),46L54, 15A52, 05A99

We compute the limit distribution of partial transposes (when both the number and the size of blocks tends to infinity) for a large class of ensembles of unitarily invariant random matrices. Furthermore, it is shown the asymptotic freeness relation between the ensembles of random matrices, their transposes and their left and right partial transposes.

[4]  Infinitesimal Operators and the Distribution of Anticommutators and Commutators
James A. Mingo, Pei-Lun Tseng

Comments: 26 pages, we have corrected some typos, updated the references and added a comment on the connection to the Markov-Krein transform
Subjects: Operator Algebras (math.OA); Operator Algebras (math.OA),46L54

In an infinitesimal probability space we consider operators which are infinitesimally free and one of which is infinitesimal, in that all its moments vanish. Many previously analysed random matrix models are captured by this framework. We show that there is a simple way of finding non-commutative distributions involving infinitesimal operators and apply this to the commutator and anticommutator. We show the joint infinitesimal distribution of an operator and an infinitesimal idempotent gives us the Boolean cumulants of the given operator. We also show that Boolean cumulants can be expressed as infinitesimal moments thus giving matrix models which exhibit asymptotic Boolean independence and monotone independence. Finally we demonstrate a connection to the Markov-Krein transform.

[5]  Second Order Cumulants: second order even elements and R-diagonal elements
Octavio Arizmendi, James A. Mingo

Comments: 55 pp, we corrected a few typos and updated the references, to appear in Annales de l'Institut Henri Poincar\'e, D,
Subjects: Operator Algebras (math.OA); Operator Algebras (math.OA),Probability (math.PR),46L54 (15B52 46L53)

We introduce $R$-diagonal and even operators of second order. We give a formula for the second order free cumulants of the square $x^2$ of a second order even element in terms of the second order free cumulants of $x$. Similar formulas are proved for the second order free cumulants of $aa^*$, when $a$ is a second order $R$-diagonal operator. We also show that if $r$ is second order $R$-diagonal and $b$ is second order free from $r$, then $rb$ is also second order $R$-diagonal. We present a large number of examples, in particular the limit distribution of products of Ginibre matrices. We prove the conjectured formula of Dartois and Forrester for the fluctuations moments of the product of two independent complex Wishart matrices and generalize it to any number of factors.

[6]  Third order moments of complex Wigner matrices
Daniel Munoz George, James A. Mingo

Comments: 64 pages, 14 figures
Subjects: Probability (math.PR); Probability (math.PR),Combinatorics (math.CO),46L54
 

We compute the third order moments of a complex Wigner matrix. We provide a formula for the third order moments $\alpha_{m_1,m_2,m_3}$ in terms of quotient graphs $T_{m_1,m_2,m_3}^{\pi}$ where $\pi$ is the Kreweras complement of a non-crossing pairing on the annulus. We prove that these graphs can be counted using the set of partitioned permutations, this permits us to write the third order moments in terms of the high order free cumulants which have a simple expression.

[7]  The Asymptotic Infinitesimal Distribution of a Real Wishart Random Matrix
James A. Mingo, Josue Vazquez-Becerra

Comments: 36 pages, updated the references, added some comments, main results unchanged
Subjects: Probability (math.PR); Probability (math.PR),Combinatorics (math.CO),Operator Algebras (math.OA),60B20, 15B52, 46L54
DOI: 10.1063/5.0147470
 

Let $X_N$ be a $N \times N$ real Wishart random matrix with aspect ratio $M/N$. The limit eigenvalue distribution of $X_N$ is the Marchenko-Pastur law with parameter $c = \lim_N M/N$. The limit moments $\{m_n\}_n$ are given by $m_n = \sum_{\pi} c^{\#(\pi)}$ where the sum runs over $NC(n)$. Let $m_n'$ be the limit of $N( \mathrm{E}(\mathrm {tr}(X_N^n)) - m_n)$. These are the asymptotic infinitesimal moments of a real Wishart matrix. We show that $m'_n$ can be written as a sum over planar diagrams with two terms, $\sum_{\pi} c'(\#(\pi) -1) c^{\#(\pi)-1}$, and $\sum_{\pi \in S_{NC}^\delta(n,-n)} c^{\#(\pi)/2}$, where $S_{NC}^\delta(n,-n)$ is a set of non-crossing annular permutations satisfying a symmetry condition. Moreover we present a recursion formula for the second term which is related to one for higher order freeness.

[8]  On the Analytic Structure of Second-Order Non-Commutative Probability Spaces and Functions of Bounded Fr\'echet Variation
Mario Diaz, James A. Mingo

Comments: 28 pages
Subjects: Probability (math.PR); Probability (math.PR),Operator Algebras (math.OA),60B20, 15B52, 46L54

In this paper we propose a new approach to the central limit theorem (CLT), based on functions of bounded F\'echet variation for the continuously differentiable linear statistics of random matrix ensembles which relies on: a weaker form of a large deviation principle for the operator norm; a Poincar\'{e}-type inequality for the linear statistics; and the existence of a second-order limit distribution. This approach frames into a single setting many known random matrix ensembles and, as a consequence, classical central limit theorems for linear statistics are recovered and new ones are established, e.g., the CLT for the continuously differentiable linear statistics of block Gaussian matrices. In addition, our main results contribute to the understanding of the analytical structure of second-order non-commutative probability spaces. On the one hand, they pinpoint the source of the unbounded nature of the bilinear functional associated to these spaces; on the other hand, they lead to a general archetype for the integral representation of the second-order Cauchy transform, $G_2$. Furthermore, we establish that the covariance of resolvents converges to this transform and that the limiting covariance of analytic linear statistics can be expressed as a contour integral in $G_2$.

[9]  Freely Independent Coin Tosses, Standard Young Tableaux, and the Kesten--McKay Law
Iris Stephanie Arenas Longoria, James A. Mingo

Comments: 18 pages, we have added a connection to standard Young tableaux [reflected in the title], and made a few small corrections
Subjects: Probability (math.PR); Probability (math.PR),Combinatorics (math.CO),Operator Algebras (math.OA),60G50, 05A15, 46L54, 60B20
In this article, we shall start with a closed walk on a regular tree of degree $d$. These walks are described by the Kesten-McKay law which arises as the asymptotic distribution of a random $d$-regular graph on $n$ vertices. We will show that the moments of the Kesten-McKay law are given by counting standard Young tableaux with at most 2 rows, and how some properties of the walk make sense even when $d$ is not an integer. We will use free probability to instruct us how to build an explicit model in random matrix theory.

[10]  On the Partial Transpose of a Haar Unitary Matrix
James A. Mingo, Mihai Popa, Kamil Szpojankowski

Comments: 23 pages
Subjects: Operator Algebras (math.OA); Operator Algebras (math.OA),Probability (math.PR),Primary: 46L54, Secondary: 60B20
We consider the effect of a partial transpose on the limit $*$-distribution of a Haar distributed random unitary matrix. If we fix, $b$, the number of blocks, we show that the partial transpose can be decomposed into a sum of $b$ matrices which are asymptotically free and identically distributed. We then consider the joint effect of different block decompositions and show that under some mild assumptions we also get asymptotic freeness.

[ Showing 10 of 31 total entries, additional 21 entries available at arXiv.org ]