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About Pure and Applied Mathematics

Foundations of Computational Mathematics, Santander 2005 (London Mathematical Society Lecture Note Series)

We study the (two-parameter) Segal--Bargmann transform $\mx{B}_{s,t}^N$ 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 $\mx{B}_{s,t}$ as $N\to\infty$, which can be identified as an operator on the space of complex Laurent polynomials. We introduce the space of {\em trace Laurent 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 $\mx{B}_{s,t}^N$ to its limit $\mx{B}_{s,t}$. We characterize the operator $\mx{B}_{s,t}$ through its inverse action on the standard polynomial basis. Finally, we show that, in the case $s=t$, the limit transform $\mx{B}_{t,t}$ is the free unitary Segal--Bargmann transform $\mathscr{G}^t$ introduced by Biane.

The Large-N Limit of the Segal--Bargmann Transform on U(N)

The Peterson-Thom conjecture asserts that any diffuse, amenable subalgebra of a free group factor is contained in a unique maximal amenable subalgebra. This conjecture is motivated by related results in Popa's deformation/rigidity theory and Peterson-Thom's results on L^{2}-Betti numbers. We present an approach to this conjecture in terms of so-called strong convergence of random matrices by formulating a conjecture which is a natural generalization of the Haagerup-Thorbjornsen theorem whose validity would imply the Peterson-Thom conjecture. This random matrix conjecture is related to recent work of Collins-Guionnet-Parraud.

A random matrix approach to the Peterson-Thom conjecture

We study $N$-ary non-commutative notions of independence, which are given by trees and which generalize free, Boolean, and monotone independence. For every rooted subtree $\mathcal{T}$ of the $N$-regular tree, we define the $\mathcal{T}$-free product of $N$ non-commutative probability spaces and we define the $\mathcal{T}$-free additive convolution of $N$ non-commutative laws. These $N$-ary convolution operations form a topological symmetric operad which includes the free, Boolean, monotone, and anti-monotone convolutions, as well as the orthogonal and subordination convolutions. Using the operadic framework, the proof of convolution identities (such as the relation between free, monotone, and subordination convolutions studied by Lenczewski) can be reduced to combinatorial manipulations of trees. We also develop a theory of $\mathcal{T}$-free independence that closely parallels the free, Boolean, and monotone cases, provided that the root vertex has more than one neighbor. In particular, we study the case where the root vertex of $\mathcal{T}$ has $n$ children and each other vertex has $d$ children, and we relate the $\mathcal{T}$-free convolution powers to free and Boolean convolution powers and the Belinschi-Nica semigroup.

An Operad of Non-commutative Independences Defined by Trees

Let $(X_1,\dots,X_m)$ be self-adjoint non-commutative random variables distributed according to the free Gibbs law given by a sufficiently regular convex and semi-concave potential $V$, and let $(S_1,\dots,S_m)$ be a free semicircular family. We show that conditional expectations and conditional non-microstates free entropy given $X_1$, \dots, $X_k$ arise as the large $N$ limit of the corresponding conditional expectations and entropy for the random matrix models associated to $V$. Then by studying conditional transport of measure for the matrix models, we construct an isomorphism $\mathrm{W}^*(X_1,\dots,X_m) \to \mathrm{W}^*(S_1,\dots,S_m)$ which maps $\mathrm{W}^*(X_1,\dots,X_k)$ to $\mathrm{W}^*(S_1,\dots,S_k)$ for each $k = 1, \dots, m$, and which also witnesses the Talagrand inequality for the law of $(X_1,\dots,X_m)$ relative to the law of $(S_1,\dots,S_m)$.

Conditional Expectation, Entropy, and Transport for Convex Gibbs Laws in Free Probability

We present an alternative approach to the theory of free Gibbs states with convex potentials. Instead of solving SDE's, we combine PDE techniques with a notion of asymptotic approximability by trace polynomials for a sequence of functions on $M_N(\mathbb{C})_{sa}^m$ to prove the following. Suppose $\mu_N$ is a probability measure on on $M_N(\mathbb{C})_{sa}^m$ given by uniformly convex and semi-concave potentials $V_N$, and suppose that the sequence $DV_N$ is asymptotically approximable by trace polynomials. Then the moments of $\mu_N$ converge to a non-commutative law $\lambda$. Moreover, the free entropies $\chi(\lambda)$, $\underline{\chi}(\lambda)$, and $\chi^*(\lambda)$ agree and equal the limit of the normalized classical entropies of $\mu_N$.

https://msp.org/apde/2020/13-8/apde-v13-n8-p02-s.pdf

An elementary approach to free entropy theory for convex potentials

We formulate a free probabilistic analog of the Wasserstein manifold on $\mathbb{R}^d$ (the formal Riemannian manifold of smooth probability densities on $\mathbb{R}^d$), and we use it to study smooth non-commutative transport of measure. The points of the free Wasserstein manifold $\mathscr{W}(\mathbb{R}^{*d})$ are smooth tracial non-commutative functions $V$ with quadratic growth at $\infty$, which correspond to minus the log-density in the classical setting. The space of smooth tracial non-commutative functions used here is a new one whose definition and basic properties we develop in the paper; they are scalar-valued functions of self-adjoint $d$-tuples from arbitrary tracial von Neumann algebras that can be approximated by trace polynomials. The space of non-commutative diffeomorphisms $\mathscr{D}(\mathbb{R}^{*d})$ acts on $\mathscr{W}(\mathbb{R}^{*d})$ by transport, and the basic relationship between tangent vectors for $\mathscr{D}(\mathbb{R}^{*d})$ and tangent vectors for $\mathscr{W}(\mathbb{R}^{*d})$ is described using the Laplacian $L_V$ associated to $V$ and its pseudo-inverse $\Psi_V$ (when defined). Following similar arguments to arXiv:1204.2182, arXiv:1701.00132, and arXiv:1906.10051 in the new setting, we give a rigorous proof for the existence of smooth transport along any path $t \mapsto V_t$ when $V$ is sufficiently close $(1/2) \sum_j \operatorname{tr}(x_j^2)$, as well as smooth triangular transport.

Tracial smooth functions of non-commuting variables and the free Wasserstein manifold.

We adapt the theory of chordal Loewner chains to the operator-valued matricial upper-half plane over a $C^*$-algebra $\mathcal{A}$. We define an $\mathcal{A}$-valued chordal Loewner chain as a subordination chain of analytic self-maps of the $\mathcal{A}$-valued upper half-plane, such that each $F_t$ is the reciprocal Cauchy transform of an $\mathcal{A}$-valued law $\mu_t$, such that the mean and variance of $\mu_t$ are continuous functions of $t$. 
We relate $\mathcal{A}$-valued Loewner chains to processes with $\mathcal{A}$-valued free or monotone independent independent increments just as was done in the scalar case by Bauer ("Lowner's equation from a non-commutative probability perspective", J. Theoretical Prob., 2004) and Scheisinger ("The Chordal Loewner Equation and Monotone Probability Theory", Inf. Dim. Anal., Quantum Probability, and Related Topics, 2017). 
We show that the Loewner equation $\partial_t F_t(z) = DF_t(z)[V_t(z)]$, when interpreted in a certain distributional sense, defines a bijection between Lipschitz mean-zero Loewner chains $F_t$ and vector fields $V_t(z)$ of the form $V_t(z) = -G_{\nu_t}(z)$ where $\nu_t$ is a generalized $\mathcal{A}$-valued law. 
Based on the Loewner equation, we derive a combinatorial expression for the moments of $\mu_t$ in terms of $\nu_t$. We also construct non-commutative random variables on an operator-valued monotone Fock space which realize the laws $\mu_t$. Finally, we prove a version of the monotone central limit theorem which describes the behavior of $F_t$ as $t \to +\infty$ when $\nu_t$ has uniformly bounded support.

Operator-Valued Chordal Loewner Chains and Non-Commutative Probability

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