Hamiltonian paths are self-avoiding random walks that visit all sites of a given lattice. We consider various configuration exponents of Hamiltonian walks drawn on random planar maps. Estimates from exact enumerations are compared with predictions based on the Knizhnik-Polyakov-Zamolodchikov (KPZ) relations, as applied to exponents on the regular hexagonal lattice. Astonishingly, when the maps...
We construct the continuum analogue of the chemical distance metric in
lattice models such as percolation. The chemical distance metric is the
graph distance induced by the percolation clusters. It is known that for
critical percolation, the lengths have non-trivial scaling behaviour,
however it is very difficult to find the exact scaling exponent. (This
is one of the questions from...
In this talk, I will consider the interface separating +1 and -1 spins in the critical planar Ising model with Dobrushin boundary conditions perturbed by an external magnetic field. I will prove that this interface has a scaling limit. This result holds when the Ising model is defined on a bounded and simply connected subgraph of $\delta\mathbb{Z}^2$, with $\delta > 0$. I will show that if the...
