L3

The dynamical Phi^4_3 model is a stochastic partial differential equation that arises in quantum field theory and statistical physics. Owing to the singular nature of the driving noise and the presence of a nonlinear term, the equation is inherently ill-posed. Nevertheless, it can be given a rigorous meaning, for example, through the framework of regularity structures. On compact domains, standard arguments show that any solution converges to the equilibrium state described by the unique invariant measure. Extending this result to infinite volume is highly nontrivial: even for the lattice version of the model, uniqueness holds only in the high-temperature regime, whereas at low temperatures multiple phases coexist.

We prove that, when the mass is sufficiently large or the coupling constant sufficiently small (that is, in the high-temperature regime), all solutions of the dynamical Phi^4_3 model in infinite volume converge exponentially fast to the unique stationary solution, uniformly over all initial conditions. In particular, this result implies that the invariant measure of the dynamics is unique, exhibits exponential decay of correlations, and is invariant under translations, rotations, and reflections.

Joint work with Martin Hairer, Jaeyun Yi, and Wenhao Zhao.

TBA

Fracture is usually treated as an outcome to be avoided; here we see it as something we may write into a lattice's microstructure. Maxwell lattices sit at the edge of mechanical stability, where robust topological properties provide a way on how stress localises and delocalises across the structure with directional preference. Building on this, we propose a direct relationship between lattice topology and damage propagation. We identify a set of topology- and geometry-dependent parameters that gives a simple, predictive framework for nonideal Maxwell lattices and their damage processes. We will discuss how topological polarisation and domain walls steer and arrest damage in a repeatable way. Experiments confirm the theoretical predicted localisation and the resulting tuneable progression of damage and show how this control mechanism can be used to enhance dissipation and raise the apparent fracture energy.