Cutting up graphs revisited - a short proof of Stallings' structure theorem

This is a new and short proof of the main theorem of classical structure tree theory. Namely, we show the existence of certain automorphism-invariant tree-decompositions of graphs based on the principle of removing finitely many edges. This was first done in "Cutting up graphs" by M.J. Dunwoody. The main ideas are based on the paper "Vertex cuts" by M.J. Dunwoody and the author. We extend the theorem to a detailed combinatorial proof of J.R. Stallings' theorem on the structure of finitely generated groups with more than one end.Comment: 12 page

Tree indiscernibilities, revisited

We give definitions that distinguish between two notions of indiscernibility for a set \{a_\eta \mid \eta \in \W\} that saw original use in \cite{sh90}, which we name \textit{\s-} and \textit{\n-indiscernibility}. Using these definitions and detailed proofs, we prove \s- and \n-modeling theorems and give applications of these theorems. In particular, we verify a step in the argument that TP is equivalent to TP$_1$ or TP$_2$ that has not seen explication in the literature. In the Appendix, we exposit the proofs of \citep[{App. 2.6, 2.7}]{sh90}, expanding on the details.Comment: submitte

The marginally stable Bethe lattice spin glass revisited

Bethe lattice spins glasses are supposed to be marginally stable, i.e. their equilibrium probability distribution changes discontinuously when we add an external perturbation. So far the problem of a spin glass on a Bethe lattice has been studied only using an approximation where marginally stability is not present, which is wrong in the spin glass phase. Because of some technical difficulties, attempts at deriving a marginally stable solution have been confined to some perturbative regimes, high connectivity lattices or temperature close to the critical temperature. Using the cavity method, we propose a general non-perturbative approach to the Bethe lattice spin glass problem using approximations that should be hopeful consistent with marginal stability.Comment: 23 pages Revised version, hopefully clearer that the first one: six pages longe

Harmonic analysis of finite lamplighter random walks

Recently, several papers have been devoted to the analysis of lamplighter random walks, in particular when the underlying graph is the infinite path $\mathbb{Z}$. In the present paper, we develop a spectral analysis for lamplighter random walks on finite graphs. In the general case, we use the $C_2$-symmetry to reduce the spectral computations to a series of eigenvalue problems on the underlying graph. In the case the graph has a transitive isometry group $G$, we also describe the spectral analysis in terms of the representation theory of the wreath product $C_2\wr G$. We apply our theory to the lamplighter random walks on the complete graph and on the discrete circle. These examples were already studied by Haggstrom and Jonasson by probabilistic methods.Comment: 29 page