×

zbMATH — the first resource for mathematics

Quantum ergodicity on graphs: from spectral to spatial delocalization. (English) Zbl 1423.58021
Let \(G=(V,E)\) be a finite graph. Fix an interval \(I\subset\mathbb{R}\). The authors examine the extent to which the eigenfunctions of the adjacency matrix are localized or delocalized. One says that one has spectral delocalication if there is purely absolutely continuous spectrum in \(I\). The authors consider a sequence of finite graphs which have discrete Schrödinger operators which have a local weak limit model which is a random rooted tree which has a random discrete Schrödinger model and study when quantum ergodicity pertains; this is a form of spatial delocalization of the eigenvalues of the finite graphs which aproximate the weak model. Roughly speaking, this implies that the eigenfunctions become equi-distributed in phase space. The results of the paper apply to graphs which converge to the Anderson model of a regular tree.
The main result of the paper can be roughly stated as follows: “If a large finite system is close in the Benjamini-Schramm topology to an infinite system having a purely absolutely continuous spectrum in an interval I, then the eigenfunctions with eigenvalues lying in I of the finite system satisfy quantum ergodicity.”
Section 1 is an introduction to the subject. Section 2 provides some basic identities. Section 3 discusses the non-backtracking quantum variance. Section 4 provides a bound on the non-backtracking quantum variance. Section 5 presents an invariance property of the quantum variance. Section 6 involves a stationary Markov chain. Section 7 deals with the spectral gap and mixing. Section 8 treats transition matrices with phases. Section 9 uses these results to establish one of the results of the paper. In Section 10, one returns to the original eigenfunctions by showing it suffices to consider the non-backtracking quantum variance to prove quantum ergodicity. In Appendix A, basic facts concerning the Benjamini-Schramm topology are given.

MSC:
58J51 Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] author homepage, Benjamini-Schramm convergence and pointwise convergence of the spectral measure
[2] De Luca, A.; Altshuler, B. L.; Kravtsov, V. E.; Scardicchio, A., Anderson localization on the Bethe lattice: Nonergodicity of extended states, Phys. Rev. Lett., 113, 046806 pp., (2014)
[3] De Luca, A.; Altshuler, B. L.; Kravtsov, V. E.; Scardicchio, A., Support set of random wave-functions on the Bethe lattice, (2013)
[4] Aizenman, Michael; Warzel, Simone, Absolutely continuous spectrum implies ballistic transport for quantum particles in a random potential on tree graphs, J. Math. Phys.. Journal of Mathematical Physics, 53, 095205-15, (2012) · Zbl 1278.81090
[5] Aizenman, Michael; Shamis, Mira; Warzel, Simone, Resonances and partial delocalization on the complete graph, Ann. Henri Poincar\'e. Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics, 16, 1969-2003, (2015) · Zbl 1329.81192
[6] Aldous, David; Lyons, Russell, Processes on unimodular random networks, Electron. J. Probab.. Electronic Journal of Probability, 12, 54-1454, (2007) · Zbl 1131.60003
[7] Anantharaman, Nalini, Quantum ergodicity on regular graphs, Comm. Math. Phys.. Communications in Mathematical Physics, 353, 633-690, (2017) · Zbl 1368.58015
[8] Anantharaman, Nalini, Some relations between the spectra of simple and non-backtracking random walks, (2017) · Zbl 1386.60134
[9] Anantharaman, Nalini; Le Masson, Etienne, Quantum ergodicity on large regular graphs, Duke Math. J.. Duke Mathematical Journal, 164, 723-765, (2015) · Zbl 1386.58015
[10] Anantharaman, Nalini; Sabri, Mostafa, Poisson kernel expansions for Schr\"odinger operators on trees, J. Spectr. Theory. Journal of Spectral Theory, 9, 243-268, (2019) · Zbl 1410.81022
[11] Anantharaman, Nalini; Sabri, Mostafa, Quantum ergodicity for the Anderson model on regular graphs, J. Math. Phys.. Journal of Mathematical Physics, 58, 091901-10, (2017) · Zbl 1376.82091
[12] Backhausz, A.; Szegedy, B., On the almost eigenvectors of random regular graphs, (2016)
[13] Bauerschmidt, Roland; Knowles, Antti; Yau, Horng-Tzer, Local semicircle law for random regular graphs, Comm. Pure Appl. Math.. Communications on Pure and Applied Mathematics, 70, 1898-1960, (2017) · Zbl 1372.05194
[14] Bauerschmidt, Roland; Huang, Jiaoyang; Knowles, Antti; Yau, Horng-Tzer, Bulk eigenvalue statistics for random regular graphs, Ann. Probab.. The Annals of Probability, 45, 3626-3663, (2017) · Zbl 1379.05098
[15] Bauerschmidt, Roland; Huang, Jiaoyang; Yau, H. T., Local Kesten–McKay law for random regular graphs, (2019) · Zbl 1414.05260
[16] Benjamini, Itai; Schramm, Oded, Recurrence of distributional limits of finite planar graphs, Electron. J. Probab.. Electronic Journal of Probability, 6, 23-13, (2001) · Zbl 1010.82021
[17] Benjamini, Itai, Coarse Geometry and Randomness, Lecture Notes in Math., 2100, viii+129 pp., (2013) · Zbl 1282.05001
[18] Berry, M.; Tabor, M., Level clustering in the regular spectrum, Proc. Royal Soc. A, 356, 375-394, (1977) · Zbl 1119.81395
[19] Bohigas, O.; Giannoni, M.-J.; Schmit, C., Characterization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett.. Physical Review Letters, 52, 1-4, (1984) · Zbl 1119.81326
[20] Bohigas, Oriol; Giannoni, Marie-Joya; Schmit, Charles, Spectral fluctuations, random matrix theories and chaotic motion. Stochastic Processes in Classical and Quantum Systems, Lecture Notes in Phys., 262, 118-138, (1986)
[21] Bordenave, Charles, On quantum percolation in finite regular graphs, Ann. Henri Poincar\'e. Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics, 16, 2465-2497, (2015) · Zbl 1332.82054
[22] Bourgade, P.; Yau, H.-T., The eigenvector moment flow and local quantum unique ergodicity, Comm. Math. Phys.. Communications in Mathematical Physics, 350, 231-278, (2017) · Zbl 1379.58014
[23] Brooks, Shimon; Lindenstrauss, Elon, Non-localization of eigenfunctions on large regular graphs, Israel J. Math.. Israel Journal of Mathematics, 193, 1-14, (2013) · Zbl 1317.05110
[24] Brooks, Shimon; Le Masson, Etienne; Lindenstrauss, Elon, Quantum ergodicity and averaging operators on the sphere, Int. Math. Res. Not. IMRN. International Mathematics Research Notices. IMRN, 6034-6064, (2016) · Zbl 1404.35307
[25] Colin de Verdi\`“ere, Y., Ergodicit\'”e et fonctions propres du laplacien, Comm. Math. Phys.. Communications in Mathematical Physics, 102, 497-502, (1985) · Zbl 0592.58050
[26] Diaconis, Persi; Stroock, Daniel, Geometric bounds for eigenvalues of Markov chains, Ann. Appl. Probab.. The Annals of Applied Probability, 1, 36-61, (1991) · Zbl 0731.60061
[27] Dumitriu, Ioana; Pal, Soumik, Sparse regular random graphs: spectral density and eigenvectors, Ann. Probab.. The Annals of Probability, 40, 2197-2235, (2012) · Zbl 1255.05173
[28] Erd\Hos, L\'aszl\'o; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun, Spectral statistics of Erd\Hos-R\'enyi graphs I: Local semicircle law, Ann. Probab.. The Annals of Probability, 41, 2279-2375, (2013) · Zbl 1272.05111
[29] Erd\Hos, L\'aszl\'o; Schlein, Benjamin; Yau, Horng-Tzer, Local semicircle law and complete delocalization for Wigner random matrices, Comm. Math. Phys.. Communications in Mathematical Physics, 287, 641-655, (2009) · Zbl 1186.60005
[30] Erd\Hos, L\'aszl\'o; Schlein, Benjamin; Yau, Horng-Tzer, Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices, Ann. Probab.. The Annals of Probability, 37, 815-852, (2009) · Zbl 1175.15028
[31] Geisinger, Leander, Convergence of the density of states and delocalization of eigenvectors on random regular graphs, J. Spectr. Theory. Journal of Spectral Theory, 5, 783-827, (2015) · Zbl 1384.60024
[32] Keating, J. P., Quantum graphs and quantum chaos. Analysis on Graphs and its Applications, Proc. Sympos. Pure Math., 77, 279-290, (2008) · Zbl 1153.81501
[33] Klein, Abel, Extended states in the Anderson model on the Bethe lattice, Adv. Math.. Advances in Mathematics, 133, 163-184, (1998) · Zbl 0899.60088
[34] Klenke, Achim, Probability Theory. A Comprehensive Course, Universitext, xii+638 pp., (2014) · Zbl 1295.60001
[35] Marklof, Jens, Pair correlation densities of inhomogeneous quadratic forms, Ann. of Math. (2). Annals of Mathematics. Second Series, 158, 419-471, (2003) · Zbl 1106.11018
[36] Meyer, Carl D., Matrix Analysis and Applied Linear Algebra, xii+718 pp., (2000) · Zbl 0962.15001
[37] Parry, William; Pollicott, Mark, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Ast\'erisque, 187-188, 268 pp., (1990) · Zbl 0726.58003
[38] Salez, J., Some implications of local weak convergence for sparse random graphs
[39] Schumacher, Christoph; Schwarzenberger, Fabian, Approximation of the integrated density of states on sofic groups, Ann. Henri Poincar\'e. Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics, 16, 1067-1101, (2015) · Zbl 1316.82015
[40] Sarnak, Peter, Arithmetic quantum chaos. The Schur Lectures (1992), Israel Math. Conf. Proc., 8, 183-236, (1995) · Zbl 0831.58045
[41] Sarnak, Peter, Values at integers of binary quadratic forms. Harmonic Analysis and Number Theory, CMS Conf. Proc., 21, 181-203, (1997) · Zbl 0911.11032
[42] Smilansky, Uzy, Quantum chaos on discrete graphs, J. Phys. A. Journal of Physics. A. Mathematical and Theoretical, 40, F621-F630, (2007) · Zbl 1124.81024
[43] Smilansky, Uzy, Discrete graphs—a paradigm model for quantum chaos. Chaos, Prog. Math. Phys., 66, 97-124, (2013) · Zbl 1319.81047
[44] \vSnirel\cprimeman, A. I., Ergodic properties of eigenfunctions, Uspehi Mat. Nauk. Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk, 29, 181-182, (1974) · Zbl 0324.58020
[45] Tikhonov, K. S.; Mirlin, A. D.; Skvortsov, M. A., Anderson localization and ergodicity on random regular graphs, Phys. Rev. B, 94, 220203 pp., (2016)
[46] Tran, Linh V.; Vu, Van H.; Wang, Ke, Sparse random graphs: eigenvalues and eigenvectors, Random Structures Algorithms. Random Structures & Algorithms, 42, 110-134, (2013) · Zbl 1257.05089
[47] Zelditch, Steven, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J.. Duke Mathematical Journal, 55, 919-941, (1987) · Zbl 0643.58029
[48] Zelditch, Steven, Quantum ergodicity of \(C^*\) dynamical systems, Comm. Math. Phys.. Communications in Mathematical Physics, 177, 507-528, (1996) · Zbl 0856.58019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.