Fractional Gaussian Fields (FGF’s) are a family of stochastic processes that can be defined on any compact manifold M and model the behavior of random oscillations on M. FGF’s can be used to model for the behavior of bosons in particle physics, but they also find use in exploiting the connection between random matrix theory and number theory to prove the Riemann Hypothesis. For these reasons, studying the extreme values of these fields is of great interest to a variety of subjects, and yet computing the distributions of the maxima and minima of the FGF are challenging, open problems. We define discrete fractional gaussian fields on manifolds M that converge to the continuous fractional field, and by numerically simulating the discrete fractional gaussian fields, we generate conjectures for the behavior of the extreme values of the continuous field.

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In this poster, we begin by defining the Fractional Gaussian Field (FGF) X_s on S1 with parameter s. For s>1/4, we index our Gaussian random variables by points on the circle S1, and for s\leq 1/4 we index these random variables by smooth, real-valued functions on S1. Using methods from Fourier analysis and probability, we prove that these definitions make sense (i.e. that the series defining the FGF pointwise converges almost surely). We then prove that the sample paths of X_s are almost surely Lipschitz when s>3/4, Holder with exponent 1/k for s>1/2k, and k times continuously differentiable for s>(k+1)/2. We then prove the analogous results on the d dimensional torus Td.

The theory of FGF’s on manifolds is currently a topic of interest to cosmologists studying cosmic microwave background radiation and particle physicists studying Liouville quantum gravity. We outline a procedure for defining the FGF on an arbitrary compact Riemannian manifold (M,g), using tools from functional analysis and probability. Our current interest is in regularity conditions for paths of FGF on the sphere S2, which are more difficult to study due to the sphere’s positive Ricci curvature.

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Petersen and Tenner’s depth measures the distance of a permutation from its symmetric group’s identity and is known to be bound below by the average of two other measures of distance, length and reflection length. Permutations where this bound is an equality (i.e., permutations that are shallow) are known to not be characterizable by the containment and avoidance of patterns. However, using Hadjicostas and Monico’s recursive definition for shallowness, we show that the subset of shallow permutations which avoid 4231 can be characterized in this way. Such permutations are the ones which avoid 4231, 3412, 34521, 54123, 365214, 541632, 7652143, and 5476321.

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