A Pfaffian version of the Borodin-Okounkov formula.

par Pierre Lazag (Louvain)

mardi 17 févr. 2026, 09:50 → 10:50 Europe/Paris

Amphi Schwartz

The Borodin-Okounkov formula is an equality between a Toeplitz determinant of size NxN and a Fredholm determinant of an operator acting on l^2(N,N+1,...), which in particular implies the strong Szegö theorem, giving a precise asymptotic of a Toeplitz determinant. This identity provides an equality between the multiplicative statistics for the unitary group on the one hand, and the distribution of the first row of a random Young diagram on the other, a corollary being a central limit theorem for the unitary group. The original proof by Borodin and Okounkov relies on Gessel's formula, expressing the distribution of the first row of a Young diagram (distributed according to a measure called a Schur measure) as a Toeplitz determinant, as well as on a Theorem by Okounkov establishing that Schur measures are determinantal point processes. In this talk, I will review the proof by Borodin and Okounkov, and then show how it adapts to the Pfaffian case by considering Pfaffian measures on Young diagrams introduced by Borodin and Rains. Time permitting, I will discuss multiplicative statistics on the orthogonal group. This is work in progress in collaboration with Alexander Bufetov.