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Daniel Munoz George

Publications and source records attributed to Daniel Munoz George.

8 recordsLinked to original sources
arXiv2026

Asymptotic infinitesimal freeness of covariance matrices

We consider $n\times n$ covariance matrices $M=\frac{1}{n}XX^*$ where $X=(x_{i,j})$ is a matrix whose entries are independent complex random variables with $\mathbb{E}(x_{i,j})=0$ and $\mathbb{E}(|x_{i,j}|^2)=1$. We derive a $\frac{1}{n}$ expansion of the mixed moments, $\frac{1}{n}\mathbb{E}(\Tr(M^{(r_1)}\cdots M^{(r_q)}))$, of the form $a_0+a_1\frac{1}{n}+O(\frac{1}{n^2})$. This permits us to find explicit formulas for the moments and infinitesimal moments of several covariance matrices where we allow repetition. As an application of our formulas, we derive asymptotic freeness and infinitesimal freeness of independent covariance matrices under a fourth-moment condition. This generalizes previous results for the Wishart ensemble in which $x_{i,j}$ is complex Gaussian.

math.PR↗
arXiv2026

Probabilities of random monomial ideals associated to large graphs

Inspired by the Erdős Rényi model, we propose a new model for freesquare random monomial ideals generated by edges and covers of a graph. This permit us to investigate the conditions of normality for which we obtain asymptotic results. We also elaborate on asymptotic results for other invariants such as the Krull dimension (for which we obtain threshold function), the regularity and the $v$-number.

math.AC↗
arXiv2025

Ultra high order cumulants and quantitative CLT for polynomials in Random Matrices

From the study of the high order freeness of random matrices, it is known that the order $r$ cumulant of the trace of a polynomial of $N$-dimensional GUE/GOE is of order $N^{2-r}$ if $r$ is fixed. In this work, we extend the study along three directions. First, we also consider generally distributed Wigner matrices with subexponential entries. Second, we include the deterministic matrices into discussion and consider arbitrary polynomials in random matrices and deterministic matrices. Third, more importantly, we consider the ultra high order cumulants in the sense that $r$ is arbitrary, i.e., could be $N$ dependent. Our main results are the upper bounds of the ultra high order cumulants, for which not only the $N$-dependence but also the $r$-dependence become significant. These results are then used to derive three types of quantitative CLT for the trace of any given self-adjoint polynomial in these random matrix variables: a CLT with a Cramér type correction, a Berry-Esseen bound, and a concentration inequality which captures both the Gaussian tail in the small deviation regime and $M$-dependent tail in the large deviation regime, where $M$ is the degree of the polynomial. In contrast to the second order freeness which implies the CLT for linear eigenvalue statistics of polynomials in random matrices, our study on the ultra high order cumulants leads to the quantitative versions of the CLT.

math.PR↗
arXiv2025

Second order free cumulants: product, commutator, and anti-commutator

Given two second order free random variables $a$ and $b$, we study the second order free cumulants of their product $ab$, their commutator $ab-ba$, and their anti-commutator $ab+ba$. Let $(κ_n^a)_{n\geq 1}$ and $(κ_{n,m}^a)_{n,m\geq 1}$ denote the sequence of free cumulants of first and second order, respectively, of a random variable $a$ in a second order non-commutative probability space $(\mathcal{A},φ,φ^2)$. Given $a$ and $b$ two second order freely independent random variables, we provide formulas to compute each of the cumulants $(κ_{n,m}^{ab})_{n,m\geq 1}$, $(κ_{n,m}^{ab-ba})_{n,m\geq 1}$, and $(κ_{n,m}^{ab+ba})_{n,m\geq 1}$ in terms of the individual cumulants $(κ_{n}^{a})_{n\geq 1}$, $(κ_{n,m}^{a})_{n,m\geq 1}$, $(κ_{n}^{b})_{n\geq 1}$, and $(κ_{n,m}^{b})_{n,m\geq 1}$. For $n=m=1$ our formulas read: \begin{align*} κ_{1,1}^{ab} &= κ_{2}^{a}κ_{2}^{b} +κ_{1,1}^{a}(κ_{1}^{b})^2+κ_{1,1}^{b}(κ_{1}^{a})^2,\\ κ_{1,1}^{ab-ba} &= 2κ_{2}^{a}κ_{2}^{b},\\ κ_{1,1}^{ab+ba} &= 2κ_{2}^{a}κ_{2}^{b} +4κ_{1,1}^{a}(κ_{1}^{b})^2+4κ_{1,1}^{b}(κ_{1}^{a})^2. \end{align*} In general, our formulas express the cumulants $κ_{n,m}^{ab}$, $κ_{n,m}^{ab-ba}$, and $κ_{n,m}^{ab+ba}$ as sums indexed by special subsets of non-crossing partitioned permutations. The formulas for the commutator and anti-commutator where not studied before, while the formula for the product was only known in the case the where the individual second order free cumulants vanish. As an application, we compute explicitly the cumulants of the anti-commutator and product of two second order free semicircular variables.

math.OA↗
arXiv2025

Third Order Cumulants of products

We provide a formula for the third order free cumulants of products as entries. We apply this formula to find the third order free cumulants of various Random Matrix Ensambles including product of Ginibre Matrices and Wishart matrices, both in the Gaussian case.

math.PR↗
arXiv2025

Asymptotic limit of cumulants and higher order free cumulants of complex Wigner matrices

We compute the fluctuation moments $α_{m_1,\dots,m_r}$ 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 $α_{m_1,\dots,m_r}$ in terms of simpler quantities $κ_{m_1,\dots,m_r}$ 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.

math.PR↗
arXiv2023

Third order moments of complex Wigner matrices

We compute the third order moments of a complex Wigner matrix. We provide a formula for the third order moments $α_{m_1,m_2,m_3}$ in terms of quotient graphs $T_{m_1,m_2,m_3}^π$ where $π$ 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.

math.PR↗