Expansion of the effective action around non-Gaussian theories

This paper derives the Feynman rules for the diagrammatic perturbation expansion of the effective action around an arbitrary solvable problem. The perturbation expansion around a Gaussian theory is well known and composed of one-line irreducible diagrams only. For the expansions around an arbitrary, non-Gaussian problem, we show that a more general class of irreducible diagrams remains in addition to a second set of diagrams that has no analogue in the Gaussian case. The effective action is central to field theory, in particular to the study of phase transitions, symmetry breaking, effective equations of motion, and renormalization. We exemplify the method on the Ising model, where the effective action amounts to the Gibbs free energy, recovering the Thouless-Anderson-Palmer mean-field theory in a fully diagrammatic derivation. Higher order corrections follow with only minimal effort compared to existing techniques. Our results show further that the Plefka expansion and the high-temperature expansion are special cases of the general formalism presented here.Comment: 37 pages, published versio

List homomorphism problems for signed graphs

We consider homomorphisms of signed graphs from a computational perspective. In particular, we study the list homomorphism problem seeking a homomorphism of an input signed graph $(G,\sigma)$, equipped with lists $L(v) \subseteq V(H), v \in V(G)$, of allowed images, to a fixed target signed graph $(H,\pi)$. The complexity of the similar homomorphism problem without lists (corresponding to all lists being $L(v)=V(H)$) has been previously classified by Brewster and Siggers, but the list version remains open and appears difficult. We illustrate this difficulty by classifying the complexity of the problem when $H$ is a tree (with possible loops). The tools we develop will be useful for classifications of other classes of signed graphs, and we illustrate this by classifying the complexity of irreflexive signed graphs in which the unicoloured edges form some simple structures, namely paths or cycles. The structure of the signed graphs in the polynomial cases is interesting, suggesting they may constitute a nice class of signed graphs analogous to the so-called bi-arc graphs (which characterize the polynomial cases of list homomorphisms to unsigned graphs).Comment: various changes + rewritten section on path- and cycle-separable graphs based on a new conference submission (split possible in future

On the automorphism group of generalized Baumslag-Solitar groups

A generalized Baumslag-Solitar group (GBS group) is a finitely generated group $G$ which acts on a tree with all edge and vertex stabilizers infinite cyclic. We show that Out(G) either contains non-abelian free groups or is virtually nilpotent of class at most 2. It has torsion only at finitely many primes. One may decide algorithmically whether Out(G) is virtually nilpotent or not. If it is, one may decide whether it is virtually abelian, or finitely generated. The isomorphism problem is solvable among GBS groups with Out(G) virtually nilpotent. If $G$ is unimodular (virtually $F_n \times Z$), then Out(G) is commensurable with a semi-direct product $Z^k \rtimes Out(H)$ with $H$ virtually free

Quotients and subgroups of Baumslag-Solitar groups

We determine all generalized Baumslag-Solitar groups (finitely generated groups acting on a tree with all stabilizers infinite cyclic) which are quotients of a given Baumslag-Solitar group BS(m,n), and (when BS(m,n) is not Hopfian) which of them also admit BS(m,n) as a quotient. We determine for which values of r,s one may embed BS(r,s) into a given BS(m,n), and we characterize finitely generated groups which embed into some BS(n,n).Comment: Final version, to appear in Journal of Group Theor

Multi-Colour Braid-Monoid Algebras

We define multi-colour generalizations of braid-monoid algebras and present explicit matrix representations which are related to two-dimensional exactly solvable lattice models of statistical mechanics. In particular, we show that the two-colour braid-monoid algebra describes the Yang-Baxter algebra of the critical dilute A-D-E models which were recently introduced by Warnaar, Nienhuis, and Seaton as well as by Roche. These and other solvable models related to dense and dilute loop models are discussed in detail and it is shown that the solvability is a direct consequence of the algebraic structure. It is conjectured that the Yang-Baxterization of general multi-colour braid-monoid algebras will lead to the construction of further solvable lattice models.Comment: 32 page