Week 08.09.2025 – 14.09.2025

We introduce classes of restricted walks, surfaces and their generalisations. For example, self-osculating walks (SOWs) are supersets of self-avoiding walks (SAWs) where edges are still not allowed cross but may 'kiss' at a vertex. They are analogous to osculating polygons introduced in (Jensen and Guttmann, 1998) except that they are not required to be closed. By adapting the 'automata' method of (Pönitz and Tittmann, 2000), we find upper bounds for the connective constant for SOWs on the square and triangular lattices to be ≤ 2.73911 and ≤ 4.44931, respectively.

In analogy, we also introduce self-osculating surfaces (SOSs), a superset of self-avoiding surfaces (SASs) and can be generated from fixed polyominoids (XDs). We further generalise and define self-avoiding k-manifolds (SAMs) and its supersets self-osculating k-manifolds (SOMs) in the d-dim hypercubic lattice and (d, k)-XDs. By adapting the concatenation procedure (van Rensburg and Whittington, 1989), we prove that their growth constants exist, and an explicit form for their upper and lower bounds.

The upper bounds can be improved by adapting the 'twig' method, originally developed for polyominoes (Eden, 1961, Klarner and Rivest, 1973). For the cubic lattice, we find improved upper bounds for the growth constant of SASs as ≤ 17.11728.

In this talk I will discuss the projected ensemble, the collection of local post-measurement wavefunctions of a quantum many-body state. I will focus on states arising from evolution under generic isolated quantum dynamics, and describe the universal limiting distributions the projected ensemble attains, which depend on the interplay of conserved quantities and information obtained from the measurements.This amounts to a more refined version of quantum equilibration beyond standard thermalization of local observables, which has been dubbed “deep thermalization”. I will further explain how these limiting distributions are predicted by generalized maximum entropy principles rooted in quantum information theory. At the same time, I will also present a class of quantum dynamics where depending on the choice of measurements, the projected ensemble undergoes a phase transition from a maximally entropic one to a minimally entropic one, signatures of which are not detectable at the level of the density matrix. This constitutes a novel form of ergodicity-breaking, characterized not by the failure of the system to regularly thermalize, but rather by its failure to deep thermalize.