Central Limit Theorems for the Brownian motion on large unitary groups

In this paper, we are concerned with the large N limit of linear combinations of the entries of a Brownian motion on the group of N by N unitary matrices. We prove that the process of such a linear combination converges to a Gaussian one. Various scales of time and various initial distribution are concerned, giving rise to various limit processes, related to the geometric construction of the unitary Brownian motion. As an application, we propose a quite short proof of the asymptotic Gaussian feature of the linear combinations of the entries of Haar distributed random unitary matrices, a result already proved by Diaconis et al.Comment: 14 page

Rectangular R-transform as the limit of rectangular spherical integrals

In this paper, we connect rectangular free probability theory and spherical integrals. In this way, we prove the analogue, for rectangular or square non-Hermitian matrices, of a result that Guionnet and Maida proved for Hermitian matrices in 2005. More specifically, we study the limit, as $n,m$ tend to infinity, of the logarithm (divided by $n$) of the expectation of $\exp[\sqrt{nm}\theta X_n]$, where $X_n$ is the real part of an entry of $U_n M_n V_m$, $\theta$ is a real number, $M_n$ is a certain $n\times m$ deterministic matrix and $U_n, V_m$ are independent Haar-distributed orthogonal or unitary matrices with respective sizes $n\times n$, $m\times m$. We prove that when the singular law of $M_n$ converges to a probability measure $\mu$, for $\theta$ small enough, this limit actually exists and can be expressed with the rectangular R-transform of $\mu$. This gives an interpretation of this transform, which linearizes the rectangular free convolution, as the limit of a sequence of log-Laplace transforms.Comment: 17 page

On a surprising relation between rectangular and square free convolutions

Debbah and Ryan have recently proved a result about the limit empirical singular distribution of the sum of two rectangular random matrices whose dimensions tend to infinity. In this paper, we reformulate it in terms of the rectangular free convolution introduced in a previous paper and then we give a new, shorter, proof of this result under weaker hypothesis: we do not suppose the \pro measure in question in this result to be compactly supported anymore. At last, we discuss the inclusion of this result in the family of relations between rectangular and square random matrices.Comment: 8 page

Exponential bounds for the support convergence in the Single Ring Theorem

We consider an $n$ by $n$ matrix of the form $A=UTV$, with $U, V$ some independent Haar-distributed unitary matrices and $T$ a deterministic matrix. We prove that for $k\sim n^{1/6}$ and $b^2:=\frac{1}{n}\operatorname{Tr}(|T|^2)$, as $n$ tends to infinity, we have $$\mathbb{E} \operatorname{Tr} (A^{k}(A^{k})^*) \ \lesssim \ b^{2k}\qquad \textrm{and} \qquad\mathbb{E}[|\operatorname{Tr} (A^{k})|^2] \ \lesssim \ b^{2k}.$$ This gives a simple proof (with slightly weakened hypothesis) of the convergence of the support in the Single Ring Theorem, improves the available error bound for this convergence from $n^{-\alpha}$ to $e^{-cn^{1/6}}$ and proves that the rate of this convergence is at most $n^{-1/6}\log n$.Comment: 15 pages, 1 figure. Minor typos corrected, references added. To appear in J. Funct. Ana

Cycles of free words in several independent random permutations with restricted cycle lengths

In this text, we consider random permutations which can be written as free words in several independent random permutations: firstly, we fix a non trivial word $w$ in letters $g_1,g_1^{-1},..., g_k,g_k^{-1}$, secondly, for all $n$, we introduce a $k$-tuple $s_1(n),..., s_k(n)$ of independent random permutations of $\{1,..., n\}$, and the random permutation $\sigma_n$ we are going to consider is the one obtained by replacing each letter $g_i$ in $w$ by $s_i(n)$. For example, for $w=g_1g_2g_3g_2^{-1}$, $\sigma_n=s_1(n)\circ s_2(n)\circ s_3(n)\circ s_2(n)^{-1}$. Moreover, we restrict the set of possible lengths of the cycles of the $s_i(n)$'s: we fix sets $A_1,..., A_k$ of positive integers and suppose that for all $n$, for all $i$, $s_i(n)$ is uniformly distributed on the set of permutations of $\{1,..., n\}$ which have all their cycle lengths in $A_i$. For all positive integer $l$, we are going to give asymptotics, as $n$ goes to infinity, on the number $N_l(\sigma_n)$ of cycles of length $l$ of $\sigma_n$. We shall also consider the joint distribution of the random vectors $(N_1(\sigma_n),..., N_l(\sigma_n))$. We first prove that the order of $w$ in a certain quotient of the free group with generators $g_1,..., g_k$ determines the rate of growth of the random variables $N_l(\sigma_n)$ as $n$ goes to infinity. We also prove that in many cases, the distribution of $N_l(\sigma_n)$ converges to a Poisson law with parameter $1/l$ and that the random variables $N_1(\sigma_n),N_2(\sigma_n), ...$ are asymptotically independent. We notice the surprising fact that from this point of view, many things happen as if $\sigma_n$ were uniformly distributed on the $n$-th symmetric group.Comment: 28 page

Rectangular random matrices. Related convolution

We characterize asymptotic collective behaviour of rectangular random matrices, the sizes of which tend to infinity at different rates: when embedded in a space of larger square matrices, independent rectangular random matrices are asymtotically free with amalgamation over a subalgebra. Therefore we can define a "rectangular free convolution", linearized by cumulants and by an analytic integral transform, called the "rectangular R-transform".Comment: 36 pages, to appear in PTR