arXiv subjects

Patrick Oliveira Santos

Publications and source records attributed to Patrick Oliveira Santos.

10 recordsLinked to original sources

arXiv2026

A Centroid Framework for Operator-Valued Haagerup Inequalities

We introduce a centroid-based block decomposition of the left-regular representation and use it to prove operator-valued Haagerup inequalities for finitely generated groups with centroid maps. The decomposition separates the three centroid growth conditions into distinct operator-norm contributions and, under a factorization hypothesis for the associated Schur multipliers, yields complementary lower bounds. For median groups of rank $ν$, the centroid blocks are related to spherical operators, giving two-sided estimates with the explicit factor $\binom{ν+r}{r}$; this applies in particular to right-angled Artin groups and to groups acting on finite-rank $\operatorname{CAT}(0)$ cube complexes. For finite-rank coarse median groups, we construct a coarse iterate median with error independent of the cardinality of the input set and obtain spherical-block estimates with factor $o(r^{1+ν/2+\varepsilon})$ for every $\varepsilon>0$.

math.OA↗

arXiv2026

Log-Sobolev under random monotone censoring

We show that the logarithmic Sobolev inequality of the Boolean cube is stable under random monotone censoring. More precisely, if $A_n\subseteq \{0,1\}^n$ is chosen uniformly among all monotone subsets, then the logarithmic Sobolev constant of the censored walk on $A_n$ is of order $n$ with high probability. As a consequence, several analytic and probabilistic properties of the Boolean cube persist for a typical monotone subset: the censored semigroup is hypercontractive, the uniform measure on $A_n$ satisfies Gaussian concentration for Lipschitz observables, and the associated walk mixes in time $O(n\log n)$. The latter proves a conjectured mixing bound of Ding and Mossel for almost all monotone sets. The result is genuinely typical rather than universal. We construct monotone sets of density bounded away from zero whose logarithmic Sobolev constant is of order $n^2$. To prove the result, we establish a sharp logarithmic Sobolev inequality for Hamming caps and combine it with a harmonic extension argument transferring this inequality to monotone sets lying between nearby caps, together with a structural theorem of Korshunov on random monotone sets.

math.PR↗

arXiv2025

Khintchine inequalities, trace monoids and Turán-type problems

We prove scalar and operator-valued Khintchine inequalities for mixtures of free and tensor-independent semicircle variables, interpolating between classical and free Khintchine-type inequalities. Specifically, we characterize the norm of sums of $G$-independent semicircle variables in terms of the spectral radius of the Cayley graph associated with the trace monoid determined by the graph $G$. Our approach relies on a precise correspondence between closed paths in trace monoids and the norms of such operator sums. This correspondence uncovers connections between non-commutative probability, combinatorial group theory, and extremal graph theory. In particular, we formulate Turán-type extremal problems that govern maximal norm growth under classical commutation constraints, and identify the extremal configurations. We hope that the methods and connections developed here will be useful in the study of non-commutative structures constrained by combinatorial symmetries.

math.OA↗

arXiv2025

Central limit theorem for $ε$-independent products and higher-order tensors

We establish a central limit theorem (CLT) for families of products of $ε$-independent random variables. We utilize graphon limits to encode the evolution of independence and characterize the limiting distribution. Our framework subsumes a wide class of dependency structures and includes, as a special case, a CLT for higher-order tensor products of free random variables. Our results extend earlier findings and recover as a special case a recent tensor-free CLT, which was obtained through the development of a tensor analogue of free probability. In contrast, our approach is more direct and provides a unified and concise derivation of a more general CLT via graphon convergence.

math.PR↗

arXiv2024

Graphon-Theoretic Approach to Central Limit Theorems for $ε$-Independence

We establish a central limit theorem for the sum of $ε$-independent random variables, extending both the classical and free probability setting. Central to our approach is the use of graphon limits to characterize the limiting distribution, which depends on the asymptotic structure of the underlying graphs governing $ε$-independence. This framework yields a range of exotic limit laws, interpolating between free and classical cases and allowing for mixtures such as free and classical convolutions of the semi-circle and Gaussian distributions. We provide a complete characterization of the limiting law, captured as the distribution of an operator on the full Fock space. We extend our main result to the multivariate setting as well as the non-identically distributed case. The proof provides insights into the combinatorial structure of $ε$-independence, shedding light on a natural connection between the graph of crossings of the partitions involved and the graphon limit of the graphs of $ε$-independence.

math.PR↗

arXiv2024

Central Limit Theorem for tensor products of free variables

We establish a central limit theorem for tensor product random variables $c_k:=a_k \otimes a_k$, where $(a_k)_{k \in \mathbb{N}}$ is a free family of variables. We show that if the variables $a_k$ are centered, the limiting law is the semi-circle. Otherwise, the limiting law depends on the mean and variance of the variables $a_k$ and corresponds to a free interpolation between the semi-circle law and the classical convolution of two semi-circle laws.

math.PR↗

arXiv2024

On spectral outliers of inhomogeneous symmetric random matrices

Sharp conditions for the presence of spectral outliers are well understood for Wigner random matrices with iid entries. In the setting of inhomogeneous symmetric random matrices (i.e., matrices with a non-trivial variance profile), the corresponding problem has been considered only recently. Of special interest is the setting of sparse inhomogeneous matrices since sparsity is both a key feature and a technical obstacle in various aspects of random matrix theory. For such matrices, the largest of the variances of the entries has been used in the literature as a natural proxy for sparsity. We contribute sharp conditions in terms of this parameter for an inhomogeneous symmetric matrix with sub-Gaussian entries to have outliers. Our result implies a ``structural'' universality principle: the presence of outliers is only determined by the level of sparsity, rather than the detailed structure of the variance profile.

math.PR↗

arXiv2023

Limiting spectral distribution of random self-adjoint quantum channels

We study the limiting spectral distribution of quantum channels whose Kraus operators are sampled as $n\times n$ random Hermitian matrices satisfying certain assumptions. We show that when the Kraus rank goes to infinity with n, the limiting spectral distribution (suitably rescaled) of the corresponding quantum channel coincides with the semi-circle distribution. When the Kraus rank is fixed, the limiting spectral distribution is no longer the semi-circle distribution. It corresponds to an explicit law, which can also be described using tools from free probability.

quant-ph↗

arXiv2023

A combinatorial view on star moments of regular directed graphs and trees

We investigate the method of moments for $d$-regular digraphs and the limiting $d$-regular directed tree $T_d$ as the number of vertices tends to infinity, in the same spirit as McKay (Linear Algebra Appl., 1981) for the undirected setting. In particular, we provide a combinatorial derivation of the formula for the star moments (from a root vertex $o\in T_d$) $$M_d(w)\qquad:=\sum_{\substack{v_0,v_1\ldots,v_{k-1},v_k\in T_d\\v_0=v_k=o}} A^{w_1}(v_0,v_1)A^{w_2}(v_1,v_2) \cdots A^{w_k}(v_{k-1},v_k)$$ with $A$ the adjacency matrix of $T_d$, where $w:=w_1\cdots w_k$ is any word on the alphabet $\{1,{*}\}$ and $A^*$ is the adjoint matrix of $A$. Our analysis highlights a connection between the non-zero summands of $M_d(w)$ and the non-crossing partitions of $\{1,\ldots,k\}$ which are in some sense compatible with $w$.

math.CO↗

arXiv2023

Almost sharp covariance and Wishart-type matrix estimation

Let $X_1,..., X_n \in \mathbb{R}^d$ be independent Gaussian random vectors with independent entries and variance profile $(b_{ij})_{i \in [d],j \in [n]}$. A major question in the study of covariance estimation is to give precise control on the deviation of $\sum_{j \in [n]}X_jX_j^T-\mathbb{E} X_jX_j^T$. We show that under mild conditions, we have \begin{align*} \mathbb{E} \left\|\sum_{j \in [n]}X_jX_j^T-\mathbb{E} X_jX_j^T\right\| \lesssim \max_{i \in [d]}\left(\sum_{j \in [n]}\sum_{l \in [d]}b_{ij}^2b_{lj}^2\right)^{1/2}+\max_{j \in [n]}\sum_{i \in [d]}b_{ij}^2+\text{error}. \end{align*} The error is quantifiable, and we often capture the $4$th-moment dependency already presented in the literature for some examples. The proofs are based on the moment method and a careful analysis of the structure of the shapes that matter. We also provide examples showing improvement over the past works and matching lower bounds.

math.ST↗