MARK KAC SEMINAR

Conservation laws play a very fundamental role in physics. Thermodynamics and hydrodynamics are founded on conservation principles. For many years I have worked on integrable or solvable models. Thus I often heard the statement that solvable models have extensively many conserved quantities. But I never got to know them for any model. This made me curious to see them explicitly.

Onno Huygens worked with me as a student on some version of the XXZ chain. This is a solvable generalization of the one-dimensional Heisenberg model for ferromagnetism, in which the interaction is anisotropic in spin space. Out of curiosity I asked Onno to calculate the first 6 or so of its list of conserved quantities. He succeeded, to get 8 of them. But as to be expected, they get more complicated as you go down the list. So much so that it is hard to learn anything from it.

Therefore I asked Onno to find common properties. For a few months he found every week a few new properties all his 8 conserved quantities had in common. Until at some day he walked into my office saying "I got them". He meant that the properties he had found, were enough to define the series, albeit as an algorithm, not as a closed form expression. Of course we had no proof that the subsequent terms in the list still had the propertied Onno had found. With massive computer power he was able to find number 9 and 10, and they were correctly predicted by his algorithm.

Later, when Onno had moved on to other subjects, I discovered a possiblity to write the list as a fairly simple closed form expression. This helped to find a method to prove that they do indeed represent conserved quantities. Still hard labor, but it worked.

In this talk I will of course introduce the XXZ chain. Then I will illustrate the above history written by showing examples of the ingredients of each of the steps in the progress.

One-dimensional long-range models have captured considerable attention within the Statistical Mechanics community, especially since F. Dyson demonstrated the presence of the phase transitions for long-range Ising models in the low-temperature regime. In 2017, A. Johansson, A. Öberg, and M. Pollicott studied the Dyson model on the half-line $\mathbb{Z}$ and established that it also exhibits a phase transition, with a phase diagram similar to that of Dyson's classical model on $\mathbb{Z}$.

In this talk, I discuss the relationship between half-line and whole-line (classical) Gibbs states for one-dimensional systems in a general setup. Notably, the findings discussed apply to both ferromagnetic and antiferromagnetic Dyson models. Additionally, the talk addresses the problem of the existence and regularity of the principal eigenfunction of the Perron-Frobenius transfer operator for potentials that fall outside the studied classes in Thermodynamic Formalism.

The run-and-tumble particle process is a toy model for active particles, which are particles that use internal energy to move in a preferred direction. We model this as a multi-layer process, where each layer represents an internal state, to ensure that we are working with a Markov process. In this talk, we will investigate the scaling limits of the empirical measure of this model, with special focus on the large deviations. A main tool here is to introduce a weakly asymmetric version of the model, which produces the correct deviating paths for the large deviation principle. This talk is based on a joint work with Frank Redig and Elena Pulvirenti.