IS28: Random matrices and combinatorial structures

date: 7/15/2025, time: 14:00-15:30, room: IM 605

Organizer: Elia Bisi (University of Florence)

Chair: Elia Bisi (University of Florence)

Operator norm for random matrices with general variance profile

Dimitris Cheliotis (National and Kapodistrian University of Athens)

We discuss the convergence of the operator norm of variance profile random matrices to the largest element of the support of the limiting empirical spectral distribution under very general assumptions for the variance profile of the matrices. We prove that it is sufficient for the entries of the matrix to have finite only the 4-th moment in order for the convergence to hold in probability. Our approach covers the cases of random symmetric or non-symmetric matrices whose variance profile is given by a step or a continuous function, random band matrices whose bandwidth is proportional to their dimension, random Gram triangular matrices and more. Based on joint work with Michail Louvaris.

Can One Hear the Shape of a Random Matrix?

Fabio Deelan Cunden (Università degli Studi di Bari & INFN Bari)

In this talk, I will discuss a class of ‘\(\lambda\)-shaped random matrices’ – random matrices whose entires are independent and identically distributed within the boxes of a fixed Young diagram \(\lambda\), and zero elsewhere. I will present results describing the limiting spectral distribution of these matrices, as the diagram is scaled by a growing factor \(N\). The moments of the limiting law are a generalisation of the Catalan numbers, and enumerate a new class of combinatorial structures called \(\lambda\)-plane trees: plane trees with vertex labels ‘compatible’ with \(\lambda\). I will conclude with a natural and intriguing question: can one hear the shape of a random matrix? Based on joint works with Elia Bisi, Ivailo Hartarsky, Marilena Ligabò, and Stephan Wagner.

The spectrum of dense kernel-based random graphs

Michele Salvi (University of Rome, Tor Vergata)

We study a broad class of inhomogeneous spatial random graphs, including long-range and scale-free percolation and preferential attachment-like models (see session IS07 for more details!). Vertices are placed on the discrete d-dimensional torus and are equipped with heavy tailed random weights. The probability of linking any pair of vertices decays in their distance but increases as a function of the weights. We focus on the adjacency matrix of such graphs in the dense regime and prove that, as the size of the torus goes to infinity, the empirical spectral distribution converges. The corresponding limiting measure is given by an operator-valued semicircle law that we show to be absolutely continuous and to have finite second moment, even when the weights have infinite variance. We characterize its Stieltjes transform by a fixed point equation in an appropriate Banach space.

Based on a joint work with A. Cipriani, R. S. Hazra and N. Malhotra.