# Seminars & Events for Mathematical Physics Seminar

##### The Slow Bond Conjecture

The slow bond model is a variant of the 1-dimensional Totally Asymmetric Exclusion Process, where the bond at the origin rings at a slower rate r<1. Janowsky and Lebowitz conjectured that for all values of r<1, the single slow bond produces a macroscopic change in the current of the system. Equivalently, this can be viewed as a pinning problem in last passage

##### Growth rates of unbounded orbits in non-periodic twist maps and a theorem by Neishtadt

We consider twist maps on the plane (like the ping-pong map) with non-periodic angles, where typically bounded and unbounded motions co-exist. For the latter case we prove a theorem which shows that in the analytic setting the growth rate is at most logarithmic, and furthermore an example of a system is given where all orbits grow at this rate. Moreover, we determine the optimal growth rate for a ping-pong with finite regularity. (This is joint work with R. Ortega.)

##### Gravitational allocation to uniform points on the sphere

Given n uniform points on the surface of a two-dimensional sphere, how can we partition the sphere fairly among them ? "Fairly" means that each region has the same area. It turns out that if the given points apply a two-dimensional gravity force to the rest of the sphere, then the basins of attraction for the resulting gradient flow yield such a partition-with exactly equal areas, no matter how the points are distributed. (See the cover of the AMS Notices at http://www.ams.org/publications/journals/notices/201705/rnoti-cvr1.pdf and Dacorogna-Moser (1990) .) We show that this partition minimizes, up to a bounded factor, the average distance between points in the same cell.

##### Macroscopic loops in the loop O(n) model

A loop configuration on the hexagonal (honeycomb) lattice is a finite subgraph of the lattice in which every vertex has degree 0 or 2, so that every connected component is isomorphic to a cycle. The loop O(n) model on the hexagonal lattice is a random loop configuration, with the energy of of a loop configuration taken to be linear in the number of edges and the number of loops. I will discuss the resulting phase structure of the loop O(n) model, focusing on recent results about the non-existence of macroscopic loops for large n, and about the existence of macroscopic loops on a critical line when n is between 1 and 2.

Talk based on joint works with Hugo Duminil-Copin, Alexander Glazman, Ron Peled and Wojciech Samotij.

##### Rare Region Effects and Many-Body Localization

Certain strongly disordered many-body quantum systems are incapable of reaching thermal equilibrium. The nature of this so-called many-body localized (MBL) phase has recently been an active area of research. The phenomenon can be understood through perturbative approximations, but rare regions with weak disorder (Griffiths regions) have the potential to bypass barriers to thermalization. I show that these effects do not destroy the MBL phase in one dimension, under a natural assumption on eigenvalue statistics. In higher dimensions, Griffiths effects may indeed restore thermalization on a very long time scale.

##### Quantum Markov Semigroups with detailed balance as gradient flow for relative entropy and entropy production inequalities

Semigroups of completely positive trace preserving maps satisfying a certain detailed balance condition are gradient flow driven by dissipation of the quantum relative entropy with respect to a non-commutative analog of the 2-Wasserstein metric on the space of probability densities on Euclidean space. As in the classical case, this way of viewing the evolution equations solved by these semigroups leads to sharp entropy production inequalities. This perspective has resolved some recent conjectures in quantum information theory. This is joint work with Jan Maas.

##### Hydrodynamics of integrable classical and quantum systems

Discussed is the Euler-type hydrodynamics for one-dimensional integrable quantum systems, as the Lieb-Liniger delta Bose gas and the XXZ chain. Of particular interest are domain wall initial states. We will use classical hard rods as an illustration of the underlying structure.