77 research outputs found
Scaling limits for the critical Fortuin-Kastelyn model on a random planar map II: local estimates and empty reduced word exponent
We continue our study of the inventory accumulation introduced by Sheffield
(2011), which encodes a random planar map decorated by a collection of loops
sampled from the critical Fortuin-Kasteleyn (FK) model. We prove various
\emph{local estimates} for the inventory accumulation model, i.e., estimates
for the precise number of symbols of a given type in a reduced word sampled
from the model. Using our estimates, we obtain the scaling limit of the
associated two-dimensional random walk conditioned on the event that it stays
in the first quadrant for one unit of time and ends up at a particular position
in the interior of the first quadrant. We also obtain the exponent for the
probability that a word of length sampled from the inventory accumulation
model corresponds to an empty reduced word, which is equivalent to an
asymptotic formula for the partition function of the critical FK planar map
model. The estimates of this paper will be used in a subsequent paper to obtain
the scaling limit of the lattice walk associated with a finite-volume FK planar
map.Comment: 49 pages, 2 figures; final version published in EJP. Changes include
significantly approved exposition and relation to partition functio
Random surfaces and Liouville quantum gravity
Liouville quantum gravity (LQG) surfaces are a family of random fractal surfaces which can be thought of as the canonical models of random two-dimensional Riemannian manifolds, in the same sense that Brownian motion is the canonical model of a random path. LQG surfaces are the continuum limits of discrete random surfaces called random planar maps.
In this expository article, we discuss the definition of random planar maps and LQG, the sense in which random planar maps converge to LQG, and the motivations for studying these objects. We also mention several open problems.
We do not assume any background knowledge beyond that of a second-year mathematics graduate student
The Poisson Integral Formula and Representations of SU(1,1)
We present a new proof of the Poisson integral formula for harmonic functions using the methods of representation theory. In doing so, we exhibit the irreducible subspaces and unitary structure of a representation of the group SU(1,1) of 2 x 2 complex generalized special unitary matrices. Our arguments illustrate a technique that can be used to prove similar reproducing formulas in higher dimensions and for other classes of functions. Our paper should be accessible to readers with minimal knowledge of complex analysis
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