Strong convergence: a short survey

2025/10/14 by van Handel, Ramon · 3 citations
#Combinatorics (math.CO) #Differential Geometry (math.DG) #FOS: Mathematics #Operator Algebras (math.OA) #Probability (math.PR) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2510.12520

Abstract

A family of random matrices is said to converge strongly to a limiting family of operators if the operator norm of every noncommutative polynomial of the matrices converges to that of the limiting operators. Recent developments surrounding the strong convergence phenomenon have led to new progress on important problems in random graphs, geometry, operator algebras, and applied mathematics. We review classical and recent results in this area, and their applications to various areas of mathematics.

Citations

• Modern aspects of Markov chains: entropy, curvature and the cutoff phenomenon
• Spectral gap with polynomial rate for Weil-Petersson random surfaces
• Pauli Measurements Are Near-Optimal for Single-Qubit Tomography
• On the spectral stability of finite coverings
• Strong Convergence of Multiplicative Brownian Motions on the General Linear Group
• The strong convergence phenomenon
• Strong convergence to operator-valued semicirculars
• Spectral gap with polynomial rate for random covering surfaces
• Spectral gap of random covers of negatively curved noncompact surfaces
• Strong convergence of uniformly random permutation representations of surface groups
• Strong convergence of unitary and permutation representations of discrete groups
• Pauli measurements are not optimal for single-copy tomography
• On the spectral gap of negatively curved surface covers
• Matrix Chaos Inequalities and Chaos of Combinatorial Type
• A new approach to strong convergence II. The classical ensembles
• The spectrum of a tensor of random and deterministic matrices
• Optimal quantum (tensor product) expanders from unitary designs
• Strong asymptotic freeness of Haar unitaries in quasi-exponential dimensional representations
• Strong convergence for tensor GUE random matrices
• Matrix Concentration Inequalities and Free Probability II. Two-sided Bounds and Applications
• A new approach to strong convergence
• Consequences of the random matrix solution to the Peterson-Thom conjecture
• Strongly convergent unitary representations of right-angled Artin groups
• Amenability for unitary groups of simple monotracial C*-algebras
• Norm of matrix-valued polynomials in random unitaries and permutations
• Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps I
• A note on quantum expanders
• Asymptotic expansion of smooth functions in deterministic and iid Haar unitary matrices, and application to tensor products of matrices
• Strongly convergent unitary representations of limit groups
• Strong convergence of tensor products of independent G.U.E. matrices
• Universality and sharp matrix concentration inequalities
• Near optimal spectral gaps for hyperbolic surfaces
• Additivity violation of quantum channels via strong convergence to semi-circular and circular elements
• Asymptotic expansion of smooth functions in polynomials in deterministic matrices and iid GUE matrices
• A random matrix approach to the Peterson-Thom conjecture
• On the operator norm of non-commutative polynomials in deterministic matrices and iid Haar unitary matrices
• The Kikuchi Hierarchy and Tensor PCA
• Quantum Chaos on random Cayley graphs of \rm SL₂[ℤ/pℤ]
• Spectral properties of polynomials in independent Wigner and\n deterministic matrices
• Cutoff on all Ramanujan graphs
• Second-Order Matrix Concentration Inequalities
• A statistical model for tensor PCA
• Quasitraces on exact C*-algebras are traces
• Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus
• The norm of polynomials in large random and deterministic matrices
• Asymptotic expansions for the Gaussian Unitary Ensemble
• Group cocycles and the ring of affiliated operators
• Free diffusions and Matrix models with strictly convex interaction
• A Random Matrix Approach to the Lack of Projections in C*_red(F₂)
• A proof of Alon's second eigenvalue conjecture and related problems
• Non-commutative Polynomials of Independent Gaussian Random Matrices. The Real and Symplectic Cases
• The topological version of free entropy

Cited by

• Uniform amenability at infinity
• Nearly optimal spectral gaps for random Belyi surfaces
• An RDT based confirmation of Lehner's formula for Kronecker-Gaussian matrices

Related

• The strong convergence phenomenon
• A new approach to strong convergence
• A new approach to strong convergence II. The classical ensembles
• Strong convergence to operator-valued semicirculars
• Strong convergence of tensor products of independent G.U.E. matrices