Week 29.09.2025 – 05.10.2025

A $q$-state spin system is defined by an underlying $n$-vertex graph~$G$ together with one or more parameters. Configurations of a spin system are assignments $\sigma:V(G)\to[q]$ of spins to the vertices (or sometimes edges) of~$G$. Each configuration has a defined weight, which, when normalised by the partition function of the system, specifies the Gibbs distribution on configurations. Generally, as some parameter (sometimes called `temperature') is varied, the system undergoes a phase transition. On one side, the (complex) zeros of the partition function avoid the real axis and Glauber (single site) dynamics mixes rapidly; on the other, zeros approach the real axis and the mixing time of Glauber dynamics is exponential in the size of the underlying graph.

A rare exception to this picture is the monomer-dimer model, which does not exhibit a phase transition at any non-zero temperature (Heilmann and Lieb). However, it is possible to obtain interesting examples from other models --- antiferromagnetic Ising model, hard-core model --- by suitably restricting the underlying graph~$G$. I'll explore this phenomenon mainly in the context of the hard-core model, as it has the most intricate and fascinating behaviour.

We study the stability of competitive Lotka–Volterra systems on large networks. Extending May’s classical result, we show that instability arises from highly connected outlier nodes rather than simply from system size, and we derive degree-based thresholds that guarantee coexistence. To quantify robustness, we introduce the notion of critical coupling—the maximum interaction strength a system can withstand before losing stability—and demonstrate that network clustering enhances stability, suggesting potential adaptive benefits of such structures. Finally, we show that these dynamics can efficiently approximate solutions to hard graph problems such as the maximum independent set, drawing a direct connection between ecological stability and the classical Motzkin–Straus theorem.

Recent developments have revealed that black holes near extremality exhibit large quantum fluctuations in their geometry, marking a controllable breakdown of semiclassical quantum field theory in curved spacetime. In this talk, I will discuss how these fluctuations can be revealed through scattering waves off the black hole. In particular, we find that extremely cold black holes become transparent to low-frequency light or gravitational radiation. These effects provide concrete signatures of quantum gravity at play in near-extremal regimes.

Suppose we wish to estimate a finite-dimensional parameter but we don't want to restrict ourselves to a finite-dimensional model. This is called semiparametric inference. An exciting aspect of this paradigm is that we might be able to leverage state-of-the-art machine learning algorithms to estimate our high-dimensional nuisance parameters and still obtain statistical guarantees (e.g. a 95% confidence interval). This approach has been especially popular in the field of causal inference in recent years. To attain these nice inferential properties, however, we will generally need to carefully tailor our inference to the target estimand. This can be problematic for nonparametric Bayesian inference, which focuses on good performance for the whole data-generating distribution, possibly at the expense of low-dimensional parameters of interest. To remedy this, we introduce a simple, computationally efficient procedure that corrects the marginal posterior of our target estimand, yielding a debiased and calibrated one-step posterior.