Plenary session 5

Friday 7/18, 9:00-12:00, Congress Centre

Exchangeability in Continuum Random Trees

Minmin Wang (University of Sussex)

De Fenetti’s Theorem states that all \(\mathbb N\)-indexed exchangeable sequences of real-valued random variables are mixings of i.i.d sequences. For real-valued random processes with exchangeable increments on \([0, 1]\), Kallenberg’s result \([1]\) provides a complete characterisation of these processes via another mixing relationship.

Continuum random trees are random tree-like metric spaces that arise naturally as scaling limits of various models of discrete random trees. In this talk, we will focus in particular on two subclasses of continuum random trees: the so-called stable trees and inhomogeneous continuum random trees. An analogue of Kallenberg’s Theorem for continuum random trees first appeared as a claim in a paper \([2]\) by Aldous, Miermont and Pitman. They suggested that, in much the same way that a stable bridge process on \([0, 1]\) is a mixing of certain extremal exchangeable processes, stable trees are mixings of inhomogeneous continuum random trees.

We present an outline of a rigorous argument supporting this claim, based on a novel construction that applies to both classes of trees. We will also briefly discuss some implications of this result on critical random graphs.

The talk is based on the paper \([3]\).

Bibliography

\([1]\) O. Kallenberg. “Canonical representations and convergence criteria for processes with interchangeable increments.” Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, vol. 27, 1973, pp. 23–36.

\([2]\) D. Aldous, G. Miermont and J. Pitman. “The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin’s local time identity." Probab. Theory Related Fields, vol. 129, no. 2, 2004, pp. 182–218.

\([3]\) M. Wang. “Stables trees as mixings of inhomogeneous continuum random trees.” Stochastic Process. Appl., vol. 175, 2024.

Boundary traces of diffusions

Mateusz Kwaśnicki (Wrocław University of Science and Technology)

In 1958 Spitzer \([7]\) showed that the 1-D Cauchy process (i.e., the symmetric 1-stable Lévy process) can be obtained from the paths of the reflected 2-D Brownian motion in the half-plane \(\mathbb H = \mathbb R \times [0, \infty)\) by retaining only points on the boundary \(\mathbb R \times \{0\}\), and rescaling time. A decade later Molchanov and Ostrovski \([6]\) proved that all symmetric stable Lévy processes arise similarly as boundary traces of suitable reflected diffusions.

In PDEs, it is a classical result that the generator of the Cauchy process, \(-(-\Delta)^{1/2}\), is the Dirichlet-to-Neumann map for the Laplacian in \(\mathbb H\). In 2008, Caffarelli and Silvestre \([1]\) generalised this by identifying the generator of the symmetric stable Lévy process with the Dirichlet-to-Neumann map corresponding to the second-order elliptic operator which is the generator of the diffusion in \([6]\).

These results illustrate a general principle: the generator \(K\) of the boundary trace of a reflected diffusion with generator \(L\) often coincides with the Dirichlet-to-Neumann map for \(L\). This idea has been explored further by Assing, Benjamini, Chen, Fukushima, Herman, Hsu, Kim, Kolsrud, Molchanov, Motoo, Nagasawa, Rohde, Sato, Song, Ueno, Vondraček, Ying, and others.

Caffarelli and Silvestre posed a natural question: which translation-invariant operators arise as Dirichlet-to-Neumann maps for elliptic operators in \(\mathbb H\)? Probabilistically: which Lévy processes can be realised as boundary traces of reflected diffusions? An answer to the PDE version was given in my joint paper with Mucha \([5]\) for symmetric operators (in arbitrary dimension), and extended in \([2]\) to non-symmetric operators. However, the corresponding probabilistic result does not follow directly, known results on Dirichlet-to-Neumann maps and boundary traces are too restrictive. A full characterisation of Lévy processes that are traces of reflected diffusions in \(\mathbb H\) was given in \([3]\), using a variety of probabilistic tools. The corresponding result for isotropic Lévy processes in higher dimensions appeared in \([4]\).

In my talk I will state the problem more rigorously, survey the literature, and outline the main results of \([3]\) and \([4]\).

Bibliography

\([1]\) L. Caffarelli, L. Silvestre. “An extension problem related to the fractional Laplacian.” Comm. Partial Differ. Equ., vol. 32, no. 7, 2007, pp. 1245–1260.

\([2]\) M. Kwaśnicki. “Harmonic extension technique for non-symmetric operators with completely monotone kernels.” Calc. Var. Partial Differ. Equ., vol. 61, no. 202, 2022, pp. 1–40.

\([3]\) M. Kwaśnicki. “Boundary traces of shift-invariant diffusions in half-plane.” Ann. Inst. Henri Poincaré Probab. Statist., vol. 59, no. 1, 2023, pp. 411–436.

\([4]\) M. Kwaśnicki. “Harmonic extension technique: probabilistic and analytic perspectives.” Unpublished lecture notes, arXiv:2409.19118.

\([5]\) M. Kwaśnicki, J. Mucha. “Extension technique for complete Bernstein functions of the Laplace operator.” J. Evol. Equ., vol. 18, no. 3, 2018, pp. 1341–1379.

\([6]\) S.A. Molchanov, E. Ostrovskii. “Symmetric stable processes as traces of degenerate diffusion processes.” Theor. Prob. Appl., vol. 14, no. 1, 1969, pp. 128–131.

\([7]\) F. Spitzer. “Some theorems concerning 2-dimensional Brownian motion.” Trans. Amer. Math. Soc., vol. 87, 1958, pp. 187–197.

On the norm of random matrices with a tensor structure

Benoit Collins (Kyoto University)

The notion of strong convergence in random matrix theory refers to when the operator norm of a model of random matrices converges to the operator norm of the limit object. In general, one takes a sequence of (non-commuting) random matrices and tries to check for any non-commuting polynomial in these matrices, if its operator norm converges towards the operator norm of the limit object. Important milestones in this theory include the seminal work of Haagerup-Thorbjørnsen \([1]\), the work of Collins-Male \([2]\), and joint works with Bordenave \([3,4,5]\).

In particular, the early strong convergence results did not allow us to understand models with tensor structure, and considerable progress has been made in \([4,5]\). We will describe in particular the following main result of \([5]\): Consider operators of type \[\text{Re} (a_0\otimes 1_N+ \sum_{i=1}^d a_i\otimes U_i),\] where \(a_i\) are \(n\times n\) matrices and \(U_i\) are \(N\times N\) iid Haar unitaries. Then, with high probability, the operator norm of this random matrix is close to the norm of the same operator when \(U_i\) are replaced by free Haar unitaries in the free group factor. In addition, there exists \(a>0\) such that the above estimate is uniform over all possible choices of \(a_i\) as long as \(n\le \exp (N^a)\).

We will present an outline of the operator algebraic tools needed to prove this result, and time allowing, we will discuss applications, in particular to a proof of the Peterson-Thom conjecture, as well as more recent developments \([6]\).

Bibliography

\([1]\) U. Haagerup and S. Thorbjornsen. A new application of random matrices: \(\text{Ext}(C_\text{red}^*(F_2))\) is not a group. Ann. of Math. (2) 162 (2005), no. 2, 711–775.

\([2]\) B. Collins and C. Male. The strong asymptotic freeness of Haar and deterministic matrices. Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 1, 147–163.

\([3]\) C. Bordenave and B. Collins. Eigenvalues of random lifts and polynomials of random permutation matrices. Ann. of Math. (2) 190 (2019), no. 3, 811–875.

\([4]\) C. Bordenave and B. Collins. Strong asymptotic freeness for independent uniform variables on compact groups associated to non-trivial representations. Invent. Math. 237 (2024), no. 1, 221—273

\([5]\) C. Bordenave and B. Collins. Norm of matrix-valued polynomials in random unitaries and permutations. arXiv:2304.05714

\([6]\) Z. Chen, A. Garza-Vargas, and R. van Handel. Random matrices with independent entries: beyond non-crossing partitions. Ann. of Math. (2) 200 (2024), no. 1, 1–92.