Week 22.09.2025 – 28.09.2025

The talk will begin with a brief review of Quantum Chromodynamics (QCD) and the Confinement problem. Lattice Gauge Theory (LGT) provides a non-perturbative formulation of QCD, which has led to good numerical results for the low-lying hadron spectra. Yet, an analytical understanding of QCD is not available. I will then discuss several gauge theories which have some of the key features of QCD. One of them is based on the gauge/gravity duality and is described by the warped deformed conifold background of type IIB string theory. This theory exhibits confinement, and the quark-antiquark potential is similar to that found in LGT.

The 1+1 dimensional gauge theories have also served as useful models of quark confinement. I will revisit the classic Schwinger model and its lattice Hamiltonian formulation. A mass shift between the lattice and continuum definitions of mass, which is motivated by chiral symmetry, is shown to lead to improved results. I will also present the zero-temperature phase diagram of the two-flavor Schwinger model at theta=pi, which exhibits dimensional transmutation and spontaneous breaking of charge conjugation. Finally, I will discuss the 2D SU(N) gauge theory coupled to an adjoint multiplet of Majorana fermions. This model has a rich topological structure. I will introduce a Hamiltonian lattice approach to this gauge theory, in which one can compute the spectrum, the string tension, and other observables. The talk will end with some surprising exact results for this model.

The random connection model (RCM) built on a Poisson process of intensity $\lambda$ can be constructed by site percolation with parameter $1/N$ on an RCM of intensity $N\lambda$. We will see that as $N \to \infty$, the RCM constructed in this way behaves more and more like a site percolation cluster on a vertex transitive graph, in a way we will make precise.
We call this method "asymptotic transitivity" and we will apply it to extend the recent exploration methods of Vanneuville to the RCM. In particular, for the subcritical RCM with any connection function, we will see that the probability the cluster of the origin has size at least $n$ decays exponentially as $n$ increases. To our knowledge this is the first proof of the exponential decay of the volume for any long-range percolation model.