A Unitary Extension of Virtual Permutations

Abstract

Analogously to the space of virtual permutations [5], we define projective limits of isometries: these sequences of unitary operators are natural in the sense that they minimize the rank norm between successive matrices of increasing sizes. The space of virtual isometries we construct this way may be viewed as a natural extension of the space of virtual permutations of Kerov, Olshanski and Vershik ([5]) as well as the space of virtual isometries of Neretin ([9]). We then derive with purely probabilistic methods an almost sure convergence for these random matrices under the Haar measure: for a coherent Haar measure on virtual isometries, the smallest normalized eigenangles converge a.s. to a point process whose correlation function is given by the sine kernel. This almost sure convergence actually holds for a larger class of measures as is proved by Borodin and Olshanski ([1]). We give a different proof, probabilist in the sense that it makes use of martingale arguments and shows how the eigenangles interlace when going from dimension n to n+ 1. Our method also proves that for some universal constant ε > 0, the rate of convergence is almost surely dominated by n −ε when the dimension n goes to infinity.

Sum rules and large deviations for spectral measures on the unit circle

“…, e n ), then R 1 maps e onto Ue and is a reflection since the rank of R 1 − I N is the same as the rank of U − (1 ⊕ Ξ p N ) which is at most p (see Prop. 2.5 in [9]). More generally, for k ≥ 2, R k is a reflection mapping e k onto R † k−1 R † k−2 .…”

Hua–Pickrell diffusions and Feller processes on the boundary of the graph of spectra

“…If ξ (n) N → ∞, then f ξ (n) → 0 and moreover since we have uniformly in n and m the bound from the constraints (6):…”

The Limiting Characteristic Polynomial of Classical Random Matrix Ensembles

“…We call it the limiting characteristic polynomial for the following reason: if U(n) is the group of n × n unitary matrices, endowed with Haar measure, and g is a random element of U(N), then for ξ n (s) := det(e i2πs/n − g) det(1 − g) , it was shown in [CNN17] that the random analytic function ξ n tends in distribution to ξ ∞ in the topology of uniform convergence on compact sets. The proof in that paper proceeds from the machinery of virtual isometries, special to the unitary group developed in [BNN12]. Such a result is closely related to, though does not follow only from, the fact that the rescaled eigenangles of a random unitary matrix tend locally to a Sine process.…”