Alex Roberts | Searcharxiv

arXiv2026

Existence of Decreasing Nambu Solutions to the Rainbow Ladder Gap Equation of QCD by Cone Compression

Studying Nambu solutions of the rainbow-ladder gap equation in QCD at zero temperature and chemical potential, we prove that the mass function emerges continuously from zero as the interaction strength is increased past the critical point for all positive, asymptotically perturbative kernels almost everywhere continuous in $L^1$ using the Krasnosel'skii-Guo Cone Compression Theorem. We prove that the coupled system of equations must have a positive, continuous Nambu solution with decreasing mass function for all current quark masses for a class of models which includes the physical point of a popular model of QCD by using a hybrid Krasnosel'skii-Schauder Fixed Point Theorem.

math-ph↗

arXiv2018

Exceptional graphs for the random walk

If $\mathcal{W}$ is the simple random walk on the square lattice $\mathbb{Z}^2$, then $\mathcal{W}$ induces a random walk $\mathcal{W}_G$ on any spanning subgraph $G\subset \mathbb{Z}^2$ of the lattice as follows: viewing $\mathcal{W}$ as a uniformly random infinite word on the alphabet $\{\mathbf{x}, -\mathbf{x}, \mathbf{y}, -\mathbf{y} \}$, the walk $\mathcal{W}_G$ starts at the origin and follows the directions specified by $\mathcal{W}$, only accepting steps of $\mathcal{W}$ along which the walk $\mathcal{W}_G$ does not exit $G$. For any fixed subgraph $G \subset \mathbb{Z}^2$, the walk $\mathcal{W}_G$ is distributed as the simple random walk on $G$, and hence $\mathcal{W}_G$ is almost surely recurrent in the sense that $\mathcal{W}_G$ visits every site reachable from the origin in $G$ infinitely often. This fact naturally leads us to ask the following: does $\mathcal{W}$ almost surely have the property that $\mathcal{W}_G$ is recurrent for \emph{every} subgraph $G \subset \mathbb{Z}^2$? We answer this question negatively, demonstrating that exceptional subgraphs exist almost surely. In fact, we show more to be true: exceptional subgraphs continue to exist almost surely for a countable collection of independent simple random walks, but on the other hand, there are almost surely no exceptional subgraphs for a branching random walk.

math.PR↗