Guus Regts

Zero-free regions of partition functions with applications to algorithms and graph limits

Based on a technique of Barvinok [4,5,6] and Barvinok and Soberón [8,9] we identify a class of edge-coloring models whose partition functions do not evaluate to zero on bounded degree graphs. Subsequently we give a quasi-polynomial time approximation scheme for computing these partition functions. As another application we show that the normalised partition functions of these models are continuous with respect to the Benjamini-Schramm topology on bounded degree graphs. We moreover give quasi-polynomial time approxi-mation schemes for evaluating a large class of graph polynomials, including the Tutte polynomial, on bounded degree graphs.

Compact orbit spaces in Hilbert spaces and limits of edge-colouring models

Let G be a group of orthogonal transformations of a real Hilbert space H. Let R and W be bounded G-stable subsets of H. Let .R be the seminorm on H defined by xR:=suprR|r,x| for xH. We show that if W is weakly compact and the orbit space Rk/G is compact for each kN, then the orbit space W/G is compact when W is equipped with the norm topology induced by .R.As a consequence we derive the existence of limits of edge-colouring models which answers a question posed by Lovsz. It forms the edge-colouring counterpart of the graph limits of Lovsz and Szegedy, which can be seen as limits of vertex-colouring models.In the terminology of de la Harpe and Jones, vertex- and edge-colouring models are called spin models and vertex models respectively.

A Precise Threshold For Quasi-Ramsey Numbers

We consider the variation of Ramsey numbers introduced by Erdos and Pach [J. Graph Theory, 7 (1983), pp. 137--147], where instead of seeking complete or independent sets we only seek a

Computing the Number of Induced Copies of a Fixed Graph in a Bounded Degree Graph

t

A Tutte polynomial for non-orientable maps

-homogeneous set, a vertex subset that induces a subgraph of minimum degree at least

On a Ramsey-type problem of Erdős and Pach

t

The rank of edge connection matrices and the dimension of algebras of invariant tensors

or the complement of such a graph. For any

Partition functions from orthogonal and symplectic group invariants

\nu > 0

On the existence of real R-matrices for virtual link invariants

and positive integer

Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials

k