11,658 research outputs found
The -invariant massive Laplacian on isoradial graphs
We introduce a one-parameter family of massive Laplacian operators
defined on isoradial graphs, involving elliptic
functions. We prove an explicit formula for the inverse of , the
massive Green function, which has the remarkable property of only depending on
the local geometry of the graph, and compute its asymptotics. We study the
corresponding statistical mechanics model of random rooted spanning forests. We
prove an explicit local formula for an infinite volume Boltzmann measure, and
for the free energy of the model. We show that the model undergoes a second
order phase transition at , thus proving that spanning trees corresponding
to the Laplacian introduced by Kenyon are critical. We prove that the massive
Laplacian operators provide a one-parameter
family of -invariant rooted spanning forest models. When the isoradial graph
is moreover -periodic, we consider the spectral curve of the
characteristic polynomial of the massive Laplacian. We provide an explicit
parametrization of the curve and prove that it is Harnack and has genus . We
further show that every Harnack curve of genus with
symmetry arises from such a massive
Laplacian.Comment: 71 pages, 13 figures, to appear in Inventiones mathematica
Processes on Unimodular Random Networks
We investigate unimodular random networks. Our motivations include their
characterization via reversibility of an associated random walk and their
similarities to unimodular quasi-transitive graphs. We extend various theorems
concerning random walks, percolation, spanning forests, and amenability from
the known context of unimodular quasi-transitive graphs to the more general
context of unimodular random networks. We give properties of a trace associated
to unimodular random networks with applications to stochastic comparison of
continuous-time random walk.Comment: 66 pages; 3rd version corrects formula (4.4) -- the published version
is incorrect --, as well as a minor error in the proof of Proposition 4.10;
4th version corrects proof of Proposition 7.1; 5th version corrects proof of
Theorem 5.1; 6th version makes a few more minor correction
Estimating the inverse trace using random forests on graphs
Some data analysis problems require the computation of (regularised) inverse
traces, i.e. quantities of the form \Tr (q \bI + \bL)^{-1}. For large
matrices, direct methods are unfeasible and one must resort to approximations,
for example using a conjugate gradient solver combined with Girard's trace
estimator (also known as Hutchinson's trace estimator). Here we describe an
unbiased estimator of the regularized inverse trace, based on Wilson's
algorithm, an algorithm that was initially designed to draw uniform spanning
trees in graphs. Our method is fast, easy to implement, and scales to very
large matrices. Its main drawback is that it is limited to diagonally dominant
matrices \bL.Comment: Submitted to GRETSI conferenc
- …