Titlebook: Elliptic Extensions in Statistical and Stochastic Systems; Makoto Katori Book 2023 The Author(s), under exclusive license to Springer Natu

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978-981-19-9526-2The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023

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https://doi.org/10.1007/978-981-19-9527-9q-extensions and elliptic extensions; Probability theory and stochastic processes; Statistical physics

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Brownian Motion and Theta Functions,ngle delta function. Then we consider the Brownian motion on a unit circle, which is regarded as a one-dimensional torus and is denoted by .. Two different formulas of the transition probability are given, both of which are expressed using the theta function with different nomes. The equivalence of

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Biorthogonal Systems of Theta Functions and Macdonald Denominators,? Rosengren and Schlosser gave seven kinds of answers to this fundamental question by introducing seven infinite series of spaces of theta functions associated with the irreducible reduced affine root systems, ., ., ., ., ., ., ., and ., .. Here . indicates the degree of the elliptic analogues of po

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Doubly Periodic Determinantal Point Processes,ument . of these functions to complex variable . and define seven types of orthonormal . theta functions . in the fundamental domain . in ., which is given by a . rectangular domain. Then seven types of DPPs are introduced so that the correlation functions are expressed by the orthonormal functions.