Rare-event statistics and modular invariance

S.K. Nechaev^a, b, K. Polovnikov^c, d
^aLandau Institute for Theoretical Physics, Russian Academy of Sciences, ul. Kosygina 2, Moscow, 119334, Russian Federation
^bLaboratoire J.-V. Poncelet, Bolshoi Vlasevskii per. 11, Moscow, 119002, Russian Federation
^cLomonosov Moscow State University, Faculty of Physics, Leninskie Gory 1 build. 2, Moscow, 119991, Russian Federation
^dSkolkovo Institute of Science and Technology, Bolshoy Boulevard 30, bld. 1, Moscow, 121205, Russian Federation

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/
Web of Science

000429883000006
Scopus

2-s2.0-85045749757
ADS NASA

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

References (22) ↓ Cited by (8) Similar articles (10)

Beanland K, Roberts J W, Stevenson C Am. Math. Monthly 116 531 (2009)
Trifonov V et al. Sci. Rep. 1 191 (2011)
Planat M, Eckert C IEEE Trans. Ultrasonics Ferroelectrics Frequency Control 47 1173 (2000)
Lundholm D Phys. Rev. A 96 012116 (2017); Lundholm D arXiv:1608.05067
Middendorf M, Ziv E, Wiggins C H Proc. Natl. Acad. Sci. USA 102 3192 (2005)
Avetisov V, Krapivsky P L, Nechaev S J. Phys. A 49 035101 (2016)
Hardy G H, Wright E M An Introduction To The Theory Of Numbers (New York: Oxford Univ. Press, 1979)
Ford L R Am. Math. Monthly 45 586 (1938)
Coxeter H S M Am. Math. Monthly 75 5 (1968)
Chandrasekharan K Elliptic Functions (Berlin: Springer-Verlag, 1985)
Epstein P Math. Ann. 56 615 (1903)
Siegel C L Lectures On Advanced Analytic Number Theory (Mumbai: Tata Institute of Fundamental Research, 1961)
Motohashi Y Proc. Jpn. Acad. 44 614 (1968)
Dyson F J Phys. Rev. 92 1331 (1953)
Domb C et al. Phys. Rev. 115 18 (1959)
Nieuwenhuizen Th M, Luck J M J. Stat. Phys. 41 745 (1985)
Borwein J M, Borwein P B J. Number Theory 43 293 (1993)
Comtet A, Nechaev S J. Phys. A 31 5609 (1998)
Mezard M, Parisi G, Virasoro M A Spin Glass Theory And Beyond (World Scientific Lecture Notes in Physics) Vol. 9 (Singapore: World Scientific, 1987)
Frauenfelder H Protein Structure: Molecular and Electronic Reactivity. Proc. of a Conf., April 10 - 13, 1985, Philadelphia, Pa., USA (Eds R Austin et al.) (New York: Springer-Verlag, 1987) p. 245
Nechaev S, Polovnikov K Soft Matter 13 1420 (2017)
Levitov L S Europhys. Lett. 14 533 (1991)