Thomas Krajewski

Recipe theorem for the Tutte polynomial for matroids, renormalization group-like approach

Using a quantum field theory renormalization group-like differential equation, we give a new proof of the recipe theorem for the Tutte polynomial for matroids. The solution of such an equation is in fact given by some appropriate characters of the Hopf algebra of isomorphic classes of matroids, characters which are then related to the Tutte polynomial for matroids. This Hopf algebraic approach also allows to prove, in a new way, a matroid Tutte polynomial convolution formula appearing in [W. Kook, V. Reiner, D. Stanton, A convolution formula for the Tutte polynomial, J. Combin. Theory Ser. B 76 (1999) 297-300] and [G. Etienne, M. Las Vergnas, External and internal elements of a matroid basis, Discrete Math. 179 (1998) 111-119].

Polchinski’s exact renormalisation group for tensorial theories: Gaußian universality and power counting

In this paper, we use the exact renormalisation in the context of tensor models and group field theories. As a byproduct, we rederive Gausian universality for random tensors and provide a general power counting for Abelian tensorial field theories with a closure constraint, leading us to a only five renormalisable theories.

Exact Renormalisation Group Equations and Loop Equations for Tensor Models

In this paper, we review some general formulations of exact renormalisation group equations and loop equations for tensor models and tensorial group field theories. We illustrate the use of these equations in the derivation of the leading order expectation values of observables in tensor models. Furthermore, we use the exact renormalisation group equations to establish a suitable scaling dimension for interactions in Abelian tensorial group field theories with a closure constraint. We also present analogues of the loop equations for tensor models.

Wigner law for matrices with dependent entries—a perturbative approach

We show that Wigner semi-circle law holds for Hermitian matrices with dependent entries, provided the deviation of the cumulants from the normalised Gaussian case obeys a simple power law bound in the size of the matrix. To establish this result, we use replicas interpreted as a zero-dimensional quantum field theoretical model whose effective potential obey a renormalisation group equation.

Power counting and scaling for tensor models

Random tensors are natural generalisations of matrix models related to random geometries of dimension D. Here, we revisit the large N limit of tensor models and the power counting of tensorial group field theories using a renormalisation group equation.

An extension of the Bollobás-Riordan polynomial for vertex partitioned ribbon graphs: definition and universality

Abstract In this paper we are interested in vertex partitioned ribbon graphs, which are a generalization of ribbon graphs that are studied in some theoretical physics models. We define a Hopf algebra of vertex partitioned ribbon graphs, then go on to describe how a natural generalization of the Bollobas-Riordan polynomial arises from this Hopf algebra. Using some appropriate Hopf algebraic characters we also prove the universality of our polynomial