Geodesic
Hecke graphs, Ramanujan graphs and generalized duality transformations for lattice spin systems
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Classical and modern mathematics in the wake of Boris Nikolaevich Delone, Tome 275 (2011), pp. 295-300 
M. I. Monastyrsky. Hecke graphs, Ramanujan graphs and generalized duality transformations for lattice spin systems. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Classical and modern mathematics in the wake of Boris Nikolaevich Delone, Tome 275 (2011), pp. 295-300. http://geodesic.mathdoc.fr/item/TM_2011_275_a18/
@article{TM_2011_275_a18,
     author = {M. I. Monastyrsky},
     title = {Hecke graphs, {Ramanujan} graphs and generalized duality transformations for lattice spin systems},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {295--300},
     year = {2011},
     volume = {275},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2011_275_a18/}
}
TY  - JOUR
AU  - M. I. Monastyrsky
TI  - Hecke graphs, Ramanujan graphs and generalized duality transformations for lattice spin systems
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2011
SP  - 295
EP  - 300
VL  - 275
UR  - http://geodesic.mathdoc.fr/item/TM_2011_275_a18/
LA  - en
ID  - TM_2011_275_a18
ER  - 
%0 Journal Article
%A M. I. Monastyrsky
%T Hecke graphs, Ramanujan graphs and generalized duality transformations for lattice spin systems
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2011
%P 295-300
%V 275
%U http://geodesic.mathdoc.fr/item/TM_2011_275_a18/
%G en
%F TM_2011_275_a18

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

I discuss two related subjects: (1) Hecke surfaces and $k$-regular graphs, (2) duality transformations for lattice spin models. Each of them is related to deep mathematical and physical theories, and at first glance, they have nothing in common. However, it became evident in recent years that there exist deep internal relations between these two problems. Especially interesting (and mysterious) is the role of Hecke groups in this context. I consider the following relevant example: Hecke graphs and Ramanujan graphs.

[1] Huxley M.N., “Exceptional eigenvalues and congruence subgroups”, The Selberg trace formula and related topics, Contemp. Math., 53, Amer. Math. Soc., Providence, RI, 1986, 341–349 | DOI | MR

[2] Lubotzky A., Phillips R., Sarnak P., “Ramanujan graphs”, Combinatorica., 8:3 (1988), 261–277 | DOI | MR | Zbl

[3] Babai L., “Spectra of Cayley graphs”, J. Comb. Theory B, 27 (1979), 180–189 | DOI | MR | Zbl

[4] Diaconis P., Shahshahani M., “Generating a random permutation with random transpositions”, Ztschr. Wahrscheinlichkeitstheor. Verw. Geb., 57 (1981), 159–179 | DOI | MR | Zbl

[5] Brooks R., Makover E., “Random construction of riemann surfaces”, J. Diff. Geom., 68:1 (2004), 121–157 | MR | Zbl

[6] Belyi G.V., “O rasshireniyakh Galua maksimalnogo krugovogo polya”, Izv. AN SSSR. Ser. mat., 43:2 (1979), 267–276 ; Belyi G.V., “On Galois extensions of a maximal cyclotomic field”, Math. USSR. Izv., 14:2 (1980), 247–256 | MR | Zbl | DOI | MR | Zbl

[7] Brooks R., Makover E., “Belyi surfaces”, Entire functions in modern analysis, Israel Math. Conf. Proc., 15, Bar-Ilan Univ., Ramat-Gan, 2001, 37–46 | MR | Zbl

[8] Buchstaber V.M., Monastyrsky M.I., “Generalized Kramers–Wannier duality for spin systems with non-commutative symmetry”, J. Phys. A: Math. and Gen., 36:28 (2003), 7679–7692 | DOI | MR | Zbl

[9] Murty M.Ram., “Ramanujan graphs”, J. Ramanujan Math. Soc., 18:1 (2003), 33–52 | MR | Zbl

[10] Monastyrsky M.I., “Generalized Kramers–Wannier duality for spin systems with non-commutative symmetry”, Proc. 4th Summer School in Modern Mathematical Physics, Belgrade (Serbia), Sept. 3–14, 2006, Inst. Phys., Belgrade, 2007, 283–296

[11] Sarnak P., Xue X., “Bounds for multiplicities of automorphic representations”, Duke Math. J., 64:1 (1991), 207–227 | DOI | MR | Zbl

[12] Brooks R., Monastyrsky M., “$k$-Regular graphs and Hecke surfaces”, Geometry, spectral theory, groups, and dynamics, Contemp. Math., 387, Amer. Math. Soc., Providence, RI, 2005, 65–74 | DOI | MR | Zbl

[13] Monastyrskii M.I., “Poverkhnosti Gekke i preobrazovaniya dualnosti v reshetochnykh spinovykh sistemakh”, TMF, 163:3 (2010), 505–512 | DOI | MR