The paper concerns lattice triangulations, that is, triangulations of the integer points in a polygon in @math whose vertices are also integer points. Lattice triangulations have been studied extensively both as geometric objects in their own right and by virtue of applications in algebraic geometry. Our focus is on random triangulations in which a triangulation @math has weight @math , where @math is a positive real parameter, and @math is the total length of the edges in @math . Empirically, this model exhibits a "phase transition" at @math (corresponding to the uniform distribution): for @math very large regions of aligned edges appear. We substantiate this picture as follows. For @math we show that the mixing time is exponential. These are apparently the first rigorous quantitative results on the structure and dynamics of random lattice triangulations.
We mention finally that our mixing time result for the special case of 1-dimensional regions with @math is related to work of Greenberg, Pascoe and Randall @cite_21 on lattice paths. Those authors use a similar path coupling argument, but for a different probability distribution on lattice paths: in their model paths are biased according to the area under the path, while in ours the bias depends on the excursions of the path from a fixed line.
@cite_21
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Oliveira conjectured that the order of the mixing time of the exclusion process with @math -particles on an arbitrary @math -vertex graph is at most that of the mixing-time of @math independent particles. We verify this up to a constant factor for @math -regular graphs when each edge rings at rate @math in various cases: (1) when @math or (2) when @math the spectral-gap of a single walk is @math and @math or (3) when @math for some constant @math . In these cases our analysis yields a probabilistic proof of a weaker version of Aldous' famous spectral-gap conjecture (resolved by ). We also prove a general bound which is at worst @math , which is within a @math factor from Oliveira's conjecture when @math . As applications we get new mixing bounds: (a) @math for expanders, (b) order @math for the hypercube @math and (c) order @math for vertex-transitive graphs of moderate growth and for supercritical percolation on a fixed dimensional torus. We analyze the mixing behavior of the biased exclusion process on a path of length n as the bias ( _n ) tends to 0 as (n ). We show that the sequence of chains has a pre-cutoff, and interpolates between the unbiased exclusion and the process with constant bias. As the bias increases, the mixing time undergoes two phase transitions: one when ( _n ) is of order 1 n, and the other when ( _n ) is order ( n n ). Given a finite graph @math , a vertex of the lamplighter graph @math consists of a zero-one labeling of the vertices of @math , and a marked vertex of @math . For transitive @math we show that, up to constants, the relaxation time for simple random walk in @math is the maximal hitting time for simple random walk in @math , while the mixing time in total variation on @math is the expected cover time on @math . The mixing time in the uniform metric on @math admits a sharp threshold, and equals @math multiplied by the relaxation time on @math , up to a factor of @math . For @math , the lamplighter group over the discrete two dimensional torus, the relaxation time is of order @math , the total variation mixing time is of order @math , and the uniform mixing time is of order @math . For @math when @math , the relaxation time is of order @math , the total variation mixing time is of order @math , and the uniform mixing time is of order @math . In particular, these three quantities are of different orders of magnitude. Monotonic surfaces spanning finite regions of Zd arise in many contexts, including DNA-based self-assembly, card-shuffling and lozenge tilings. We explore how we can sample these surfaces when the distribution is biased to favor higher surfaces. We show that a natural local chain is rapidly mixing with any bias for regions in Z2, and for bias λ > d2 in Zd, when d > 2. Moreover, our bounds on the mixing time are optimal on d-dimensional hyper-cubic regions. The proof uses a geometric distance function and introduces a variant of path coupling in order to handle distances that are exponentially large.