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Description
Fluctuations of observables provide unique insights into the nature of classical or quantum systems, and their study stands as a cornerstone of both theoretical and experimental science. Generalized fluctuations, or cumulants, provide information beyond the mean and variance of an observable. In this talk I will introduce a systematic method to determine the behavior of cumulants of local observables as the studied spatial region becomes large. The analysis reveals that the expansion is closely tied to the geometric characteristics of the region and its boundary, with coefficients given by convex moments of the connected correlation function: the latter is integrated against intrinsic volumes of convex polytopes built from the coordinates, which can be interpreted as average shadows. I will discuss in depth several possible applications of these results, including to the quantum Hall effect. Reference: 2408.08364.
