Abstract
We give three different proofs of the main result of Anantharaman and Le Masson (Duke Math J 164(4):723–765, 2015), establishing quantum ergodicity—a form of delocalization—for eigenfunctions of the laplacian on large regular graphs of fixed degree. These three proofs are much shorter than the original one, quite different from one another, and we feel that each of the four proofs sheds a different light on the problem. The goal of this exploration is to find a proof that could be adapted for other models of interest in mathematical physics, such as the Anderson model on large regular graphs, regular graphs with weighted edges, or possibly certain models of non-regular graphs. A source of optimism in this direction is that we are able to extend the last proof to the case of anisotropic random walks on large regular graphs.
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References
Anantharaman N., Le Masson E.: Quantum ergodicity on large regular graphs. Duke Math. J. 164(4), 723–765 (2015)
Aomoto K.: Spectral theory on a free group and algebraic curves. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31(2), 297–318 (1984)
Bass H.: The Ihara–Selberg zeta function of a tree lattice. Int. J. Math. 3(6), 717–797 (1992)
Benjamini I., Schramm O.: Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6(23), 13 (2001)
Berkolaiko G., Keating J.P., Smilansky U.: Quantum ergodicity for graphs related to interval maps. Commun. Math. Phys. 273(1), 137–159 (2007)
Berkolaiko G., Keating J.P., Winn B.: No quantum ergodicity for star graphs. Commun. Math. Phys. 250(2), 259–285 (2004)
Brammall M., Winn B.: Quantum ergodicity for quantum graphs without back-scattering. Ann. Henri. Poincaré. 17(6), 1353–1382 (2016)
Brooks, S., Le Masson, É., Lindenstrauss, E.: Quantum ergodicity and averaging operators on the sphere. Int. Math. Res. Not. 2016(19), 6034–6064 (2016)
Cowling M., Setti A.G.: The range of the Helgason–Fourier transformation on homogeneous trees. Bull. Aust. Math. Soc. 59(2), 237–246 (1999)
Colin de Verdière, Y.: Ergodicité et fonctions propres du laplacien. Commun. Math. Phys. 102(3), 497–502 (1985)
Colin de Verdière, Y.: Semi-classical measures on quantum graphs and the gauss map of the determinant manifold. Ann. H. Poincaré, to appear (2013)
Elon, Y.: Eigenvectors of the discrete Laplacian on regular graphs—a statistical approach. J. Phys. A 41(43):435203, 17 (2008)
Figà-Talamanca A., Steger T.: Harmonic analysis for anisotropic random walks on homogeneous trees. Mem. Am. Math. Soc. 110(531), xii+68 (1994)
Froese R., Hasler D., Spitzer W.: Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schrödinger operators on graphs. J. Funct. Anal. 230(1), 184–221 (2006)
Gnutzmann S., Keating J.P., Piotet F.: Eigenfunction statistics on quantum graphs. Ann. Phys. 325(12), 2595–2640 (2010)
Gross J.L.: Every connected regular graph of even degree is a Schreier coset graph. J. Combin. Theory Ser. B 22(3), 227–232 (1977)
Keating J.P., Marklof J., Winn B.: Value distribution of the eigenfunctions and spectral determinants of quantum star graphs. Commun. Math. Phys. 241(2-3), 421–452 (2003)
Kesten H.: Symmetric random walks on groups. Trans. Am. Math. Soc. 92, 336–354 (1959)
Lubetzky E., Peres Y.: Cutoff on all Ramanujan graphs. Geom. Funct. Anal. 26, 1190 (2016)
Lubotzky A., Phillips R., Sarnak P.: Ramanujan graphs. Combinatorica 8(3), 261–277 (1988)
McKay B.D.: The expected eigenvalue distribution of a large regular graph. Linear Algebra Appl. 40, 203–216 (1981)
Šnirel’man, A.I.: Ergodic properties of eigenfunctions. Uspehi Mat. Nauk 29(6(180), 181–182 (1974)
Zelditch S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55(4), 919–941 (1987)
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Communicated by H.-T. Yau
