11,259 research outputs found
Liouville Brownian motion
We construct a stochastic process, called the Liouville Brownian motion,
which is the Brownian motion associated to the metric ,
and is a Gaussian Free Field. Such a process is
conjectured to be related to the scaling limit of random walks on large planar
maps eventually weighted by a model of statistical physics which are embedded
in the Euclidean plane or in the sphere in a conformal manner. The construction
amounts to changing the speed of a standard two-dimensional Brownian motion
depending on the local behavior of the Liouville measure
"". We prove that the associated Markov
process is a Feller diffusion for all and that for all
, the Liouville measure is invariant under
. This Liouville Brownian motion enables us to introduce a
whole set of tools of stochastic analysis in Liouville quantum gravity, which
will be hopefully useful in analyzing the geometry of Liouville quantum
gravity.Comment: Published at http://dx.doi.org/10.1214/15-AOP1042 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Polytopal realizations of finite type -vector fans
This paper shows the polytopality of any finite type -vector fan,
acyclic or not. In fact, for any finite Dynkin type , we construct a
universal associahedron with the property
that any -vector fan of type is the normal fan of a
suitable projection of .Comment: 27 pages, 9 figures; Version 2: Minor changes in the introductio
The facial weak order and its lattice quotients
We investigate a poset structure that extends the weak order on a finite
Coxeter group to the set of all faces of the permutahedron of . We call
this order the facial weak order. We first provide two alternative
characterizations of this poset: a first one, geometric, that generalizes the
notion of inversion sets of roots, and a second one, combinatorial, that uses
comparisons of the minimal and maximal length representatives of the cosets.
These characterizations are then used to show that the facial weak order is in
fact a lattice, generalizing a well-known result of A. Bj\"orner for the
classical weak order. Finally, we show that any lattice congruence of the
classical weak order induces a lattice congruence of the facial weak order, and
we give a geometric interpretation of their classes. As application, we
describe the facial boolean lattice on the faces of the cube and the facial
Cambrian lattice on the faces of the corresponding generalized associahedron.Comment: 40 pages, 13 figure
On the heat kernel and the Dirichlet form of Liouville Brownian Motion
In \cite{GRV}, a Feller process called Liouville Brownian motion on
has been introduced. It can be seen as a Brownian motion evolving in a random
geometry given formally by the exponential of a (massive) Gaussian Free Field
and is the right diffusion process to consider regarding
2d-Liouville quantum gravity. In this note, we discuss the construction of the
associated Dirichlet form, following essentially \cite{fuku} and the techniques
introduced in \cite{GRV}. Then we carry out the analysis of the Liouville
resolvent. In particular, we prove that it is strong Feller, thus obtaining the
existence of the Liouville heat kernel via a non-trivial theorem of Fukushima
and al.
One of the motivations which led to introduce the Liouville Brownian motion
in \cite{GRV} was to investigate the puzzling Liouville metric through the eyes
of this new stochastic process. One possible approach was to use the theory
developed for example in \cite{stollmann,sturm1,sturm2}, whose aim is to
capture the "geometry" of the underlying space out of the Dirichlet form of a
process living on that space. More precisely, under some mild hypothesis on the
regularity of the Dirichlet form, they provide an intrinsic metric which is
interpreted as an extension of Riemannian geometry applicable to non
differential structures. We prove that the needed mild hypotheses are satisfied
but that the associated intrinsic metric unfortunately vanishes, thus showing
that renormalization theory remains out of reach of the metric aspect of
Dirichlet forms.Comment: 31 page
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