20 research outputs found
How quickly can we sample a uniform domino tiling of the 2L x 2L square via Glauber dynamics?
TThe prototypical problem we study here is the following. Given a square, there are approximately ways to tile it with
dominos, i.e. with horizontal or vertical rectangles, where
is Catalan's constant [Kasteleyn '61, Temperley-Fisher '61]. A
conceptually simple (even if computationally not the most efficient) way of
sampling uniformly one among so many tilings is to introduce a Markov Chain
algorithm (Glauber dynamics) where, with rate , two adjacent horizontal
dominos are flipped to vertical dominos, or vice-versa. The unique invariant
measure is the uniform one and a classical question [Wilson
2004,Luby-Randall-Sinclair 2001] is to estimate the time it takes to
approach equilibrium (i.e. the running time of the algorithm). In
[Luby-Randall-Sinclair 2001, Randall-Tetali 2000], fast mixin was proven:
for some finite . Here, we go much beyond and show that . Our result applies to rather general domain
shapes (not just the square), provided that the typical height
function associated to the tiling is macroscopically planar in the large
limit, under the uniform measure (this is the case for instance for the
Temperley-type boundary conditions considered in [Kenyon 2000]). Also, our
method extends to some other types of tilings of the plane, for instance the
tilings associated to dimer coverings of the hexagon or square-hexagon
lattices.Comment: to appear on PTRF; 42 pages, 9 figures; v2: typos corrected,
references adde
The mixing time of the lozenge tiling Glauber dynamics
The broad motivation of this work is a rigorous understanding of reversible,
local Markov dynamics of interfaces, and in particular their speed of
convergence to equilibrium, measured via the mixing time . In the
-dimensional setting, , this is to a large extent mathematically
unexplored territory, especially for discrete interfaces. On the other hand, on
the basis of a mean-curvature motion heuristics and simulations, one expects
convergence to equilibrium to occur on time-scales of order in any dimension, with the lattice mesh.
We study the single-flip Glauber dynamics for lozenge tilings of a finite
domain of the plane, viewed as -dimensional surfaces. The stationary
measure is the uniform measure on admissible tilings. At equilibrium, by the
limit shape theorem, the height function concentrates as around a
deterministic profile , the unique minimizer of a surface tension
functional. Despite some partial mathematical results, the conjecture
has been proven, so far, only in the situation where
is an affine function. In this work, we prove the conjecture under the
sole assumption that the limit shape contains no frozen regions
(facets).Comment: 31 pages, 2 figure
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DIMERS AND IMAGINARY GEOMETRY
We present a general result which shows that the winding of the branches in a
uniform spanning tree on a planar graph converge in the limit of fine mesh size
to a Gaussian free field. The result holds true assuming only convergence of
simple random walk to Brownian motion and a Russo-Seymour-Welsh type crossing
estimate. As an application, we prove universality of the fluctuations of the
height function associated to the dimer model, in several situations. This
includes the case of lozenge tilings with boundary conditions lying in a plane,
and Temperleyan domains in isoradial graphs (recovering a recent result of Li).
The robustness of our approach, which is a key novelty of this paper, comes
from the fact that the exactly solvable nature of the model plays only a minor
role in the analysis. Instead, we rely on a connection to imaginary geometry,
where the limit of a uniform spanning tree is viewed as a set of flow lines
associated to a Gaussian free field
DIMERS AND IMAGINARY GEOMETRY
We present a general result which shows that the winding of the branches in a
uniform spanning tree on a planar graph converge in the limit of fine mesh size
to a Gaussian free field. The result holds true assuming only convergence of
simple random walk to Brownian motion and a Russo-Seymour-Welsh type crossing
estimate. As an application, we prove universality of the fluctuations of the
height function associated to the dimer model, in several situations. This
includes the case of lozenge tilings with boundary conditions lying in a plane,
and Temperleyan domains in isoradial graphs (recovering a recent result of Li).
The robustness of our approach, which is a key novelty of this paper, comes
from the fact that the exactly solvable nature of the model plays only a minor
role in the analysis. Instead, we rely on a connection to imaginary geometry,
where the limit of a uniform spanning tree is viewed as a set of flow lines
associated to a Gaussian free field
The dimer model on Riemann surfaces, I
We develop a framework to study the dimer model on Temperleyan graphs
embedded on a Riemann surface with finitely many holes and handles. We extend
Temperley's bijection to this setting and show that the dimer model can be
understood in terms of an object which we call Temperleyan forests. Extending
our earlier work to the setup of Riemann surfaces, we show that if the
Temperleyan forest has a scaling limit then the fluctuations of the height
one-form of the dimer model also converge.
Furthermore, if the Riemann surface is either a torus or an annulus, we show
that Temperleyan forests reduce to cycle-rooted spanning forests and show
convergence of the latter to a conformally invariant, universal scaling limit.
This generalises a result of Kassel--Kenyon. As a consequence, the dimer height
one-form fluctuations also converge on these surfaces, and the limit is
conformally invariant. Combining our results with those of Dub\'edat, this
implies that the height one-form on the torus converges to the compactified
Gaussian free field, thereby settling a question in \cite{DubedatGheissari}.
This is the first part in a series of works on the scaling limit of the dimer
model on general Riemann surfaces.Comment: 75 page
Critical Exponents on Fortuin-Kasteleyn Weighted Planar Maps
In this paper we consider random planar maps weighted by the self-dual Fortuin--Kasteleyn model with parameter . Using a bijection due to Sheffield and a connection to planar Brownian motion in a cone we obtain rigorously the value of the critical exponent associated with the length of cluster interfaces, which is shown to be where is the SLE parameter associated with this model. We also derive the exponent corresponding to the area enclosed by a loop which is shown to be 1 for all values of . Applying the KPZ formula we find that this value is consistent with the dimension of SLE curves and SLE duality.
Communicated by H.-T. YauNathanaël Berestycki: Supported in part by EPSRC grants EP/L018896/1 and EP/I03372X/1.
Benoît Laslier: Supported in part by EPSRC grant EP/I03372X/1.
Gourab Ray: Supported in part by EPSRC grant EP/I03372X/1