18 research outputs found

    Dynamics of Lattice Triangulations on Thin Rectangles

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    We consider random lattice triangulations of n×kn\times k rectangular regions with weight Î»âˆŁÏƒâˆŁ\lambda^{|\sigma|} where λ>0\lambda>0 is a parameter and âˆŁÏƒâˆŁ|\sigma| denotes the total edge length of the triangulation. When λ∈(0,1)\lambda\in(0,1) and kk is fixed, we prove a tight upper bound of order n2n^2 for the mixing time of the edge-flip Glauber dynamics. Combined with the previously known lower bound of order exp⁥(Ω(n2))\exp(\Omega(n^2)) for λ>1\lambda>1 [3], this establishes the existence of a dynamical phase transition for thin rectangles with critical point at λ=1\lambda=1

    Improved Mixing for the Convex Polygon Triangulation Flip Walk

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    Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings

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    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

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    A common feature shared by many natural objects arising in probability theory is that they tend to be very “rough”, as opposed to the “smooth” 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 “random geometries” has seen spectacular advances on a number of fronts

    A polynomial upper bound for the mixing time of edge rotations on planar maps

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    We consider a natural local dynamic on the set of all rooted planar maps with nn 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 nn-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 n−11/2n^{-11/2}. 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

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    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

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    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
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