Trace formulae and spectral statistics for discrete Laplacians on regular graphs (I)

Journal of Physics A: Mathematical and Theoretical

Volumn 42, Issue 41, 2009, Pages

  Oren, Idan ^a   Godel, Amit ^a   Smilansky, Uzy ^a,b  

^a WEIZMANN INSTITUTE OF SCIENCE   (Israel)

^b CARDIFF UNIVERSITY   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 70449565304     PISSN: 17518113     EISSN: 17518121     Source Type: Journal    
DOI: 10.1088/1751-8113/42/41/415101     Document Type: Article

References (44)

1. Imry Y 1997 Introduction to Mesoscopic Physics (Oxford: Oxford University Press)
2. Grimmett G 1999 Grundlehren der mathematischen Wissenschaften Percolation vol 321 2nd edn (Berlin: Springer)
3. Grimmett G and Stirzaker D 2001 Probability and Random Processes (Oxford: Oxford University Press)
4. Jakobson D, Miller S, Rivin I and Rudnick Z 1999 Level spacings for regular graphs IMA Vol. Math. Appl. 109 317-29
5. Bohigas O, Giannoni M-J and Schmit C 1984 Characterization of chaotic quantum spectra and universality of level fluctuation laws Phys. Rev. Lett. 52 1-4
6. Gutzwiller M C 1971 J. Math. Phys. 12 343
7. Berry M V 1985 Semiclassical theory of spectral rigidity Proc. R. Soc. A 400 229-51
8. Sieber M and Richter K 2001 Correlations between periodic orbits and their Rle in spectral statistics Phys. Scr. T90 128-33
9. Heusler S, Muller S, Braun P and Haake F 2004 Universal spectral form factor for chaotic dynamics J. Phys. A: Math. Gen. 37 L31
10. Hashimoto K 1989 Zeta functions of finite graphs and representations of p-adic groups automorphic forms and geometry of arithmetic varieties (Adv. Stud. Pure Math. vol 15) (Boston, MA: Academic) pp 211-80
11. Avron J E, Raveh A and Zur B 1988 Adiabtaic quantum transport in multiply connected systems Rev. Mod. Phys. 60 873-915
12. Kesten H 1959 Symmetric random walks on groups Trans. Am. Math. Soc. 92 336-54
13. McKay B D 1981 The expected eigenvalue distribution of a random labelled regular graph Linear Algebr. Appl. 40 203-16
14. Brooks R 1991 The spectral geometry of k-regular graphs J. Anal. 57 120-51
15. Alon N, Benjamini I, Lubetzky E and Sodin S 2006 Non-backtracking random walks mix faster arXiv:math/0610550v1
16. Harrison J M, Smilansky U and Winn B 2007 Quantum graphs where back-scattering is prohibited J. Phys. A: Math. Theor. 40 14181-93
17. Smilansky U 2007 Quantum chaos on discrete graphs J. Phys. A: Math. Theor. 40 F621-30
18. Kottos T and Smilansky U 1997 Quantum chaos on graphs Phys. Rev. Lett. 79 4794-7
19. Kottos T and Smilansky U 1999 Periodic orbit theory and spectral statistics for quantum graphs Ann. Phys. 274 76-124
20. Roth J P 1985 Lecture Notes in Mathematics: Theorie de Potential ed A Dold and B Eckmann (Berlin: Springer) p 521
21. Bartholdi L 1999 Counting paths in graphs Enseign. Math. 45 83-131
22. Bass H 1992 The Ihara-Selberg zeta function of a tree lattice Int. J. Math. 3 717-97
23. Mizuno H and Sato I 2005 A new proof of Bartholdi's theorem J. Algebr. Comb. 22 259-71
24. Murty M Ram 2003 Ramanjuan graphs J. Ramanujan Math. Soc. 18 1-20
25. Mizuno H and Sato I 2008 The scattering matrix of a graph Electron. J. Comb. 15 R96
26. Shnirelman A I 1974 Ergodic properties of eigenfunctions Uspehi Mat. Nauk 29 181-2
27. Blümel R and Smilansky U 1990 Random matrix description of chaotic scattering: semi classical approach. Phys. Rev. Lett. 64 241-4
28. Gutzwiller M 1991 Chaos in Classical and Quantum Mechanics (New York: Springer)
29. Haake F 2001 Quantum Signatures Of Chaos (Berlin: Springer)
30. Stoeckmann H J 1990 Quantum Chaos-An Introduction (Cambridge: Cambridge University press)
31. Gnutzmann S and Smilansky U 2006 Quantum graphs: applications to quantum chaos and universal spectral statistics Adv. Phys. 55 527-625
32. Bollobas B 1985 Random Graphs (London: Academic)
33. Chung F R K 1997 Spectral graph theory Regional Conference Series in Mathematics vol 92 (Providence, RI: American Mathematical Society)
34. Terras A A 1999 Fourier analysis on finite groups and applications London Mathematical Society Student Texts vol 43 (Cambridge: Cambridge University Press)
35. Terras A A 2002 Arithmetic quantum chaos IAS/Park City Math. Ser. 12 333-75
36. Hurt N E 2001 The prime geodesic theorem and quantum mechanics on finite volume graphs: a review Rev. Math. Phys. 13 1459-503
37. Ihara Y 1966 On discrete subgroups of the two by two projective linear group over a p-adic field J. Mat. Soc. Japan 18 219-35
38. Stark H M and Terras A A 1996 Zeta functions of finite graphs and coverings Adv. Math. 121 124-65
39. Kotani M and Sunada T 2000 Zeta functions on finite graphs J. Math. Sci. Univ. Tokyo 7 7-25
40. Stark H M 1999 Multipath zeta functions of graphs Emerging Applications of Number Theory (Minneapolis, MN, 1996) (IMA Vol. Math. Appl. vol 109) (New York: Springer) pp 601-15
41. Janson S, Łuczak T and Rucińki A 2000 Random Graphs (New York: Wiley)
42. Mnëv P 2007 Discrete path integral approach to the Selberg trace formula for regular graphs Commun. Math. Phys. 274 233-41
43. Hoory S, Linial N and Wigderson A 2006 Expander graphs and their applications Bull. Am. Math. Soc. 43 439-561
44. Sodin S 2009 The Tracy-Widom law for some sparse random matrices arXiv:0903.4295v2 [math-ph]