The large- N limit of the Segal–Bargmann transform onUN

Abstract

We study the (two-parameter) Segal-Bargmann transform B N s,t on the unitary group U N , for large N . Acting on matrix valued functions that are equivariant under the adjoint action of the group, the transform has a meaningful limit G s,t as N → ∞, which can be identified as an operator on the space of complex Laurent polynomials. We introduce the space of trace polynomials, and use it to give effective computational methods to determine the action of the heat operator, and thus the Segal-Bargmann transform. We prove several concentration of measure and limit theorems, giving a direct connection from the finite-dimensional transform B N s,t to its limit G s,t . We characterize the operator G s,t through its inverse action on the standard polynomial basis. Finally, we show that, in the case s = t, the limit transform G t,t is the "free Hall transform" G t introduced by Biane.This highlights the fact that the Segal-Bargmann transform does not preserve the space of polynomial functions of a U N -variable; in general, it maps such functions to trace polynomials. Definition 1.7. Let C[u, u −1 ] denote the algebra of Laurent polynomials in a single variable u:C[u, u −1 ] = k∈Z a k u k : a k ∈ C, a k = 0 for all but finitely-many k ,(1.11)

Heat Kernel Empirical Laws on $${\mathbb {U}}_N$$ U N and $${\mathbb {GL}}_N$$ GL N

“…If we restrict this map to HP, cancellations like this do not occur; nevertheless, the map is still not one-to-one, due to the Cayley-Hamilton theorem, as explained in [17,Section 2.4]. Nevertheless, restricted to HP n for some n ∈ N, the map is one-to-one for all sufficiently large N (depending on n).…”

“…We adhere to the notation we used in [17]; in [10], f N was denoted θ N f . Let ρ N be a probability measure supported in M nor N .…”

“…In this section, we summarize the relevant results from the author's recent joint paper [17], in addition to giving some estimates of the involved constants.…”

“…We now introduce two families of polynomials {Q ± ε : ε ∈ E } and {R ± ε,δ : ε, δ ∈ E } in P that were defined in [17,Theorem 3.25]. They will be important in the proof of Proposition 3.19.…”

“…In [17,Theorem 3.25] where these polynomials are introduced, these features are also left vague: It is an existence proof. It is, in fact, possible to track through and give precise (but lengthy and tedious) definitions of all the involved constants and substrings, but this would not serve a useful purpose.…”