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Applied Math Seminar Shape Fluctuations for Faceted Crystals |
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As a common feature crystals at low temperatures have plane facets joined to rounded pieces. Within a specific statistical mechanics model we discuss the fluctuations relative to the macroscopic shape. Of particular interest are the fluctuations of the ledge bordering the facet. We prove that, on the appropriate scale it is non-Gaussian and governed by the Airy process. The single point distribution is the Tracy-Widom distribution as known from the largest eigenvalue of GUE random matrices. Our results rely on determinantal processes, a connection which for the model under study was pointed out by Okounkov and Reshetikhin. |