21 research outputs found
Moderate deviations for the chemical distance in Bernoulli percolation
In this paper, we establish moderate deviations for the chemical distance in
Bernoulli percolation. The chemical distance between two points is the length
of the shortest open path between these two points. Thus, we study the size of
random fluctuations around the mean value, and also the asymptotic behavior of
this mean value. The estimates we obtain improve our knowledge of the
convergence to the asymptotic shape. Our proofs rely on concentration
inequalities proved by Boucheron, Lugosi and Massart, and also on the
approximation theory of subadditive functions initiated by Alexander.Comment: 19 pages, in english. A french version, entitled "D\'eviations
mod\'er\'ees de la distance chimique" is also availabl
Duality between Spin networks and the 2D Ising model
The goal of this paper is to exhibit a deep relation between the partition
function of the Ising model on a planar trivalent graph and the generating
series of the spin network evaluations on the same graph. We provide
respectively a fermionic and a bosonic Gaussian integral formulation for each
of these functions and we show that they are the inverse of each other (up to
some explicit constants) by exhibiting a supersymmetry relating the two
formulations. We investigate three aspects and applications of this duality.
First, we propose higher order supersymmetric theories which couple the
geometry of the spin networks to the Ising model and for which supersymmetric
localization still holds. Secondly, after interpreting the generating function
of spin network evaluations as the projection of a coherent state of loop
quantum gravity onto the flat connection state, we find the probability
distribution induced by that coherent state on the edge spins and study its
stationary phase approximation. It is found that the stationary points
correspond to the critical values of the couplings of the 2D Ising model, at
least for isoradial graphs. Third, we analyze the mapping of the correlations
of the Ising model to spin network observables, and describe the phase
transition on those observables on the hexagonal lattice. This opens the door
to many new possibilities, especially for the study of the coarse-graining and
continuum limit of spin networks in the context of quantum gravity.Comment: 35 page