18 research outputs found
Dynamics of Lattice Triangulations on Thin Rectangles
We consider random lattice triangulations of rectangular regions
with weight where is a parameter and
denotes the total edge length of the triangulation. When
and is fixed, we prove a tight upper bound of order
for the mixing time of the edge-flip Glauber dynamics. Combined with the
previously known lower bound of order for [3],
this establishes the existence of a dynamical phase transition for thin
rectangles with critical point at
Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings
We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall, and Spencer in 2002. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2^{-s}, (a+1)2^{-s}] x [b2^{-t}, (b+1)2^{-t}] for a,b,s,t nonnegative integers. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n^{4.09}), which implies that the mixing time is at most O(n^{5.09}). We complement this by showing that the relaxation time is at least Omega(n^{1.38}), improving upon the previously best lower bound of Omega(n*log n) coming from the diameter of the chain
Stochastic Analysis: Geometry of Random Processes
A common feature shared by many natural objects arising in probability theory is that they tend to be very âroughâ, as opposed to the âsmoothâ objects usually studied in other branches of mathematics. It is however still desirable to understand their geometric properties, be it from a metric, a topological, or a measure-theoretic perspective. In recent years, our understanding of such ârandom geometriesâ has seen spectacular advances on a number of fronts
A polynomial upper bound for the mixing time of edge rotations on planar maps
We consider a natural local dynamic on the set of all rooted planar maps with
edges that is in some sense analogous to "edge flip" Markov chains, which
have been considered before on a variety of combinatorial structures
(triangulations of the -gon and quadrangulations of the sphere, among
others). We provide the first polynomial upper bound for the mixing time of
this "edge rotation" chain on planar maps: we show that the spectral gap of the
edge rotation chain is bounded below by an appropriate constant times
. In doing so, we provide a partially new proof of the fact that the
same bound applies to the spectral gap of edge flips on quadrangulations, which
makes it possible to generalise a recent result of the author and Stauffer to a
chain that relates to edge rotations via Tutte's bijection
New physics in and with gravity
At the center of black holes, the theory of general relativity breaks down. The resolution of such singularities could require a theory of quantum gravity which describes the fundamental nature of space-time at shortest distances. In this thesis, we explore the tensor model approach to quantum gravity and inspect its relation to other theories of quantum gravity, such as, e.g., asymptotic safety,
through a universal continuum limit. Even though at microscopic distances, general relativity breaks down, at large distances this theory is highly successful. We
will inspect how one of the predictions of general relativity, gravitational waves, can help us to learn more about new physics beyond the Standard Model
Phase Transitions
As in the past, the workshop brought together researchers with a background in physics, partial differential equations and continuum mechanics and statistical mechanics. Equilibrium and dynamic phase transitions were discussed. A wide range of systems from solid-solid phase transitions to the quantum Curie Weiss model were considered