Statistical Physics seminars

Large recurrent networks are important models in several fields, including neuroscience, machine learning, physics, and applied mathematics. Yet their dynamics are difficult to study directly, because high-dimensional nonlinear systems can exhibit rich behavior that is hard to summarize in terms of individual trajectories. In this talk, I will discuss an approach that seeks to understand such dynamics through the structure of the network’s equilibria. I will focus on a random balanced network of threshold-linear units that undergoes a transition from a single stable equilibrium to extensive chaos as the disorder strength crosses a critical value. Using a combination of Kac–Rice theory, replica calculations, numerical root-finding, and dynamical mean-field theory, we show that the chaotic regime contains an exponentially large number of equilibria. These equilibria are all saddles, but with only a fractionally small number of unstable directions. Surprisingly, despite the completely random connectivity, the equilibria are not scattered randomly through phase space. Instead, they are strongly correlated and confined to a comparatively small region. The chaotic attractor lies within this same region, suggesting a direct geometric link between the organization of unstable equilibria and the collective structure of the dynamics. This picture helps explain why networks with extensive chaos can nevertheless display dynamics dominated by a relatively small number of collective modes. More broadly, the results suggest that the geometry of equilibria provides a useful complementary perspective to dynamical mean-field theory for understanding high-dimensional neural dynamics. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2026-05-27. Recording duration: 00:46:40.

Jessica Howard presents optimal transport as a common language for neural-network ensembles, statistical field theories and exact renormalisation-group flows. Viewing a field theory as a probability distribution over functions makes transport methods useful for numerical nonperturbative calculations and analytical insights. The talk connects these ideas to particle-physics data analysis and to feature learning and training dynamics in asymptotically wide networks.

Optimal transport is a mathematical method to define a distance between probability distributions. This is particularly useful in various domains, including physics, biology, machine learning, and economics, among others. After introducing the Optimal Transport (OT) problem at finite temperature, we show how it can be formulated as a statistical physics problem. This approach allows us to derive very efficient algorithms to effectively compute the distance between two probability distributions. The a priori unrelated Schrödinger bridge (SB) problem is presented, and it is shown to be a dynamical version of the optimal transport problem. Indeed, the Schrodinger bridge looks for the most probable path in probability distribution space, which connects two given probabilities. The Schrodinger bridge problem, originally devised for freely diffusing particles, can be generalized to the case of interacting particles. It can be formulated in terms of functional integrals over bosonic fields, which allows us to derive partial differential equations that characterize the most probable paths in probability space. CARL VAN VREESWIJK MEMORIAL LECTURE 2026. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2026-04-15. Recording duration: 00:55:05.

Vijay Balasubramanian connects quantum chaos with the complexity of time evolution. One approach bounds the smallest circuit needed to simulate physical dynamics; another measures how quickly and ergodically a state explores its accessible Hilbert space. Examples from spin chains, billiards, many-body systems and random matrices distinguish chaotic and integrable behaviour. Applications address geometric conjectures for complexity in quantum gravity and possible failure of effective field theory near spacelike singularities.

Thomas Schuster uses quantum-information dynamics to study complexity in increasingly coherent experimental systems. A classical polynomial-time algorithm for circuits with uncorrected noise establishes limits on their possible quantum advantage. A second result shows that universal random-matrix behaviour can emerge exponentially faster than previously expected in local circuits and time evolution, with implications for benchmarking, cryptographic protocols and fundamental physics.

Chi-Fang (Anthony) Chen presents a general framework for preparing quantum thermal states. It combines algorithms and analysis into a quantum counterpart of Markov chain Monte Carlo and models physical system–bath interactions. The work addresses the unresolved end-to-end preparation problem that limits calculations of thermal equilibrium properties and practical many-body quantum simulation.