Todd Kemp

Wigner chaos and the fourth moment

We prove that a normalized sequence of multiple Wigner integrals (in a fixed order of free Wigner chaos) converges in law to the standard semicircular distribution if and only if the corresponding sequence of fourth moments converges to 2, the fourth moment of the semicircular law. This extends to the free probabilistic, setting some recent results by Nualart and Peccati on characterizations of central limit theorems in a fixed order of Gaussian Wiener chaos. Our proof is combinatorial, analyzing the relevant noncrossing partitions that control the moments of the integrals. We can also use these techniques to distinguish the first order of chaos from all others in terms of distributions; we then use tools from the free Malliavin calculus to give quantitative bounds on a distance between different orders of chaos. When applied to highly symmetric kernels, our results yield a new transfer principle, connecting central limit theorems in free Wigner chaos to those in Gaussian Wiener chaos. We use this to prove a new free version of an important classical theorem, the Breuer–Major theorem.

Hypercontractivity in Non-Commutative Holomorphic Spaces

We prove an analog of Janson’s strong hypercontractivity inequality in a class of non-commutative “holomorphic” algebras. Our setting is the q-Gaussian algebras Γq associated to the q-Fock spaces of Bozejko, Kümmerer and Speicher, for q ∈ [−1,1]. We construct subalgebras , a q-Segal-Bargmann transform, and prove Janson’s strong hypercontractivity for r an even integer.

Liberation of projections

Abstract We study the liberation process for projections: ( p , q ) ↦ ( p t , q ) = ( u t p u t ⁎ , q ) where u t is a free unitary Brownian motion freely independent from { p , q } . Its action on the operator-valued angle q p t q between the projections induces a flow on the corresponding spectral measures μ t ; we prove that the Cauchy transform of the measure satisfies a holomorphic PDE. We develop a theory of subordination for the boundary values of this PDE, and use it to show that the spectral measure μ t possesses a piecewise analytic density for any t > 0 and any initial projections of trace 1 2 . We us this to prove the Unification Conjecture for free entropy and information in this trace 1 2 setting.

Three Proofs of the Makeenko–Migdal Equation for Yang–Mills Theory on the Plane

We give three short proofs of the Makeenko–Migdal equation for the Yang–Mills measure on the plane, two using the edge variables and one using the loop or lasso variables. Our proofs are significantly simpler than the earlier pioneering rigorous proofs given by Lévy and by Dahlqvist. In particular, our proofs are “local” in nature, in that they involve only derivatives with respect to variables adjacent to the crossing in question. In an accompanying paper with Gabriel, we show that two of our proofs can be adapted to the case of Yang–Mills theory on any compact surface.

The Makeenko–Migdal Equation for Yang–Mills Theory on Compact Surfaces

We prove the Makeenko–Migdal equation for two-dimensional Euclidean Yang–Mills theory on an arbitrary compact surface, possibly with boundary. In particular, we show that two of the proofs given by the first, third, and fourth authors for the plane case extend essentially without change to compact surfaces.

Duality in Segal–Bargmann spaces

Abstract For α > 0 , the Bargmann projection P α is the orthogonal projection from L 2 ( γ α ) onto the holomorphic subspace L hol 2 ( γ α ) , where γ α is the standard Gaussian probability measure on C n with variance ( 2 α ) − n . The space L hol 2 ( γ α ) is classically known as the Segal–Bargmann space. We show that P α extends to a bounded operator on L p ( γ α p / 2 ) , and calculate the exact norm of this scaled L p Bargmann projection. We use this to show that the dual space of the L p -Segal–Bargmann space L hol p ( γ α p / 2 ) is an L p ′ Segal–Bargmann space, but with the Gaussian measure scaled differently: ( L hol p ( γ α p / 2 ) ) ⁎ ≅ L hol p ′ ( γ α p ′ / 2 ) (this was shown originally by Janson, Peetre, and Rochberg). We show that the Bargmann projection controls this dual isomorphism, and gives a dimension-independent estimate on one of the two constants of equivalence of the norms.

Resolvents of ℛ-diagonal operators

We consider the resolvent (λ-a) -1 of any ℛ-diagonal operator a in a II 1 -factor. Our main theorem (Theorem 1.1) gives a universal asymptotic formula for the norm of such a resolvent. En route to its proof, we calculate the ℛ-transform of the operator |λ - c| 2 where c is Voiculescus circular operator, and we give an asymptotic formula for the negative moments of |ℛ - a| 2 for any ℛ-diagonal a. We use a mixture of complex analytic and combinatorial techniques, each giving finer information where the other can give only coarse detail. In particular, we introduce partition structure diagrams in Section 4, a new combinatorial structure arising in free probability.

The spectral edge of unitary Brownian motion

The Brownian motion

The minimum Rényi entropy output of a quantum channel is locally additive

(U^N_t)_{t\ge 0}