Most of my recent research focuses on the model of lattice self-avoiding trails, analogous to self-avoiding walks but featuring bond-avoidance rather than site-avoidance. This research has produced a number of unexpected results. I studied the collapse of trails in three dimension, showing that it exhibits non-standard scaling where there is currently no theoretical prediction available. I also explored how the phase diagram changes when we introduce a second parameter such as stiffness or nearest-neighbour interactions.
Lattice trails are not only an important problem in the study of polymeric systems. The geometry of intersecting paths lies at the base of our theoretical understanding of many completely different phenomena, for example it is connected to the role of disorder in the Anderson metal-insulator transitions in two dimensions. The problem of characterising the collapse transition of trails is considered an open problem in the field, with contrasting predictions, both theoretical and numerical.
I developed an algorithm to compute the Tutte polynomial of a graph, an object of central importance in both mathematics and physics. My algorithm extends the traditional approach based on the transfer matrix with the computer-science concept of tree-decomposition. The result is two-fold: a software package to compute the Tutte polynomial of a given graph faster than any other, and a very general framework that can be easily adapted to compute many other kinds of graph polynomials.
This work is of lasting importance because it allows the many researchers working on the chromatic polynomial to study its properties, to state conjectures and to test hypotheses in a fraction of the time required before.
Using my algorithm I investigated the distribution of the roots of the chromatic polynomial, over the ensemble of planar graphs. This is an area of very active research given the numerous conjectures on the possible loci of such roots that get stated and often disproven. The results of this study have been published in Journal of Physics A (DOI 10.1088/1751-8113/43/38/385001).
My PhD research was devoted to the study of combinatorial problems through the use of a formalism based on the Grassmann calculus. This is called a “fermionic” formalism in reference to the Pauli principle, which states that two half-spin particles (fermions) of the same kind cannot occupy the same physical state. This principle is reflected in the nilpotency of algebraic variables. Using this formalism I obtained formulas for enumerating the forest configurations (subgraphs without cycles) one can draw on a graph. My method is able to deal with forests on a more general structure, a hypergraph, extending the results previously only available for ordinary graphs. I proved that the asymptotic behaviour of the number of forests on a hypergraph depends on the ratio between the number of connected components in the forest (trees, in this case) and the number of vertices in the hypergraph. This fact gives rise to a phase transition in the canonical ensemble, where a fugacity parameter controls the number of trees. Furthermore I proved that this phase transition is consequence of the spontaneous breaking of a hidden supersymmetry of the model and is associated with the appearance of a “giant component”.