Simple geometric arguments based on constructing the Euclid orchard are presented that explain the equivalence of various types of distributions that result from rare-event statistics. In particular, the spectral density of the exponentially weighted ensemble of linear polymer chains is examined for its number-theoretic properties. It can be shown that the eigenvalue statistics of the corresponding adjacency matrices in the sparse regime show a peculiar hierarchical structure and are described by the popcorn (Thomae) function discontinuous in the dense set of rational numbers. Moreover, the spectral edge density distribution exhibits Lifshitz tails, reminiscent of 1D Anderson localization. Finally, a continuous approximation for the popcorn function is suggested based on the Dedekind-function, and the hierarchical ultrametric structure of the popcorn-like distributions is demonstrated to be related to hidden SL(2, Z) modular symmetry.

Keywords: modular form, popcorn function, Dedekind function, spectrum of sparse matrix, Euclid's orchard, SL(2, Z) modular group, Lifshitz tails, Anderson localization PACS: 02.30.−f, 02.50.−r, 05.40.−a (all) DOI: 10.3367/UFNe.2017.01.038106 URL: https://ufn.ru/en/articles/2018/1/h/ 000429883000006 2-s2.0-85045749757 2018PhyU...61...99N Citation: Nechaev S K, Polovnikov K "Rare-event statistics and modular invariance" Phys. Usp. 61 99–104 (2018) BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks Received: 24th, February 2017, accepted: 18th, January 2017 Оригинал: Нечаев С К, Половников К «Статистика редких событий и модулярная инвариантность» УФН 188 106–112 (2018); DOI: 10.3367/UFNr.2017.01.038106 I.M. Lifshits is author of Physics-Uspekhi