Friday, May 29, 2026
10:00 – 10:45 AM Ofer Zeitouni, Weizmann Institute of Science and Courant Institute of Mathematical Sciences, NYU
The Liouville Model in the L^1 Phase: Coupling and Extremes Values
We establish a strong coupling between the Liouville model and the Gaussian free field on the two dimensional torus in the L^1 regime, such that the difference of the two fields is a Hölder continuous function. The coupling originates from a Polchinski renormalisation group approach, which was previously used to prove analogous results for other Euclidean field theories in dimension two. Our main observations for the Liouville model are that the Polchinski flow has a definite sign and can be controlled well thanks to an FKG argument. The coupling allows to relate extreme values of the Liouville model and the Gaussian free field, and as an application we show that the global maximum of the Liouville field converges in distribution to a randomly shifted Gumbel distribution. Joint work with Michael Hofstetter.
10:45 – 11:15 AM Morning Coffee Break
11:15 AM – 12:00 PM Gordon Slade, University of British Columbia
A Random Walk Approach to High-dimensional Critical Phenomena
The study of critical phenomena in lattice statistical mechanical models such as percolation and spin systems has a long history in both physics and mathematics. A central problem is to prove existence and calculate the values of the critical exponents that govern the universal behaviour of the model at and near its critical point. Detailed proofs are typically available only for models in dimension two, or above an upper critical dimension where mean-field behaviour is observed. We present a new, unified, generic, probabilistic, and relatively very simple proof of mean-field critical behaviour for high-dimensional models containing a small parameter. The results apply to spin systems and self-avoiding walk in dimensions above 4, percolation in dimensions above 6, and lattice trees in dimensions above 8. In particular, we obtain a proof of the triangle condition (introduced by Aizenman and Newman in 1984) for spread-out percolation in dimensions above 6, without using the lace expansion. Minimal model-specific data is required for the applications. This is joint work with Hugo Duminil-Copin, Aman Markar and Romain Panis.
12:00 – 2:00 PM Lunch Break
2:00 – 2:45 PM Takashi Hara, Kyushu University
Timescale for Macroscopic Equilibration in Free Fermion Systems
Nearly a century ago, von Neumann [1] presented an idea concerning the foundations of quantum statistical mechanics: Any initial state of an isolated quantum system evolves into an equilibrium, so that quantum dynamics alone, without any ad hoc assumptions, can explain thermalization. His fascinating idea had somewhat been forgotten for a long time, but researches in this direction are now flourishing. In particular, we can safely claim (although not completely mathematically rigorously) that a pure initial state will evolve into an equilibrium state under quantum dynamics, in the long run.
However, the issue of the TIMESCALE required for macroscopic equilibration has not been well addressed. For some cases, it is possible to show that the system equilibrates if we wait for as long as as the dimension of the Hilbert space of the system (= order of the exponential of the Avogadro number; well much much longer than the age of our universe!). There are other cases where we have some results, but realistic timescales has not been proven (for macroscopic equilibration).
We have recently succeeded [2] in establishing the optimal (and physically satisfactory) equilibration timescale, which is the order of the system’s linear size, for FREE Fermion systems. Our result, currently restricted to free systems, is not spectacular; but we hope this will shed some light on deeper understanding of time scales in more nontrivial systems.
(My talk will be divided into two parts. In the first part, I explain the problem and give a concise --- and necessarily incomplete --- summary of what has been accomplished by many people. In the second part, I present our recent result mentioned above. This is based on joint work with Tatsuhiko Koike.)
[1] J. von Neumann, Beweis des Ergodensatzes und des H-Theorems in der neuen Mechanik, Zeitschrift fuer Physik 57, 30 (1929)
[2] T. Hara and T. Koike, Timescale for macroscopic equilibration in isolated quantum systems: a rigorous derivation for free fermions, arXiv:2602.14096
2:45 – 3:15 PM Afternoon Coffee Break
3:15 – 4:00 PM Daniel Stein, Courant Institute of Mathematical Sciences, NYU
Ground State Stability, Excitations and Multiplicity in the Edwards-Anderson Model
We investigate the stability of ground states in the classical Edwards-Anderson Ising spin glass in dimensions two and higher against perturbations of a single coupling. We will discuss how ground state stability is related to fluctuations in the energy difference, restricted to finite volumes, between two infinite-volume spin glass ground states. We find that if incongruent ground states exist in any dimension, these fluctuations grow with the volume. These results are used to prove that in the appropriate setting the two-dimensional Edwards-Anderson Ising spin glass has just a single pair of globally spin-reversed ground states. We further show that a type of excitation above an infinite-volume ground state, whose interface with the ground state is space-filling and whose energy remains O（1）independently of the restricted volume, cannot exist in any dimension.