Skip to main content

Mixing in High-Dimensional Expanders


We establish a generalization of the Expander Mixing Lemma for arbitrary (finite) simplicial complexes. The original lemma states that concentration of the Laplace spectrum of a graph implies combinatorial expansion (which is also referred to as mixing, or pseudo-randomness). Recently, an analogue of this lemma was proved for simplicial complexes of arbitrary dimension, provided that the skeleton of the complex is complete. More precisely, it was shown that a concentrated spectrum of the simplicial Hodge Laplacian implies a similar type of pseudo-randomness as in graphs. In this paper we remove the assumption of a complete skeleton, showing that simultaneous concentration of the Laplace spectra in all dimensions implies pseudo-randomness in any complex. We discuss various applications and present some open questions.

Hide All
[1] Aharoni R., Berger E. and Meshulam R. (2005) Eigenvalues and homology of flag complexes and vector representations of graphs. Geom. Funct. Anal. 15 555566.
[2] Alon N. (1986) Eigenvalues and expanders. Combinatorica 6 8396.
[3] Alon N. and Chung F. R. K. (1988) Explicit construction of linear sized tolerant networks. Discrete Math. 72 1519.
[4] Alon N. and Milman V. D. (1985) λ1, isoperimetric inequalities for graphs, and superconcentrators. J. Combin. Theory Ser. B 38 7388.
[5] Beigel R., Margulis G. and Spielman D. A. (1993) Fault diagnosis in a small constant number of parallel testing rounds. In Proc. Fifth Annual ACM Symposium on Parallel Algorithms and Architectures, ACM, pp. 21–29.
[6] Bilu Y. and Linial N. (2006) Lifts, discrepancy and nearly optimal spectral gap. Combinatorica 26 495519.
[7] Cartwright D. I., Solé P. and Żuk A. (2003) Ramanujan geometries of type Ãn . Discrete Math. 269 3543.
[8] Cohen E., Mubayi D., Ralli P. and Tetali P. (2016) Inverse expander mixing for hypergraphs. Electron. J. Combin. 23 P2.20.
[9] Dodziuk J. (1976) Finite-difference approach to the Hodge theory of harmonic forms. Amer. J. Math. 98 79104.
[10] Dodziuk J. (1984) Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Amer. Math. Soc. 284 787794.
[11] Duval A., Klivans C. and Martin J. (2009) Simplicial matrix-tree theorems. Trans. Amer. Math. Soc. 361 60736114.
[12] Eckmann B. (1944) Harmonische Funktionen und Randwertaufgaben in einem Komplex. Commentarii Mathematici Helvetici 17 240255.
[13] Fox J., Gromov M., Lafforgue V., Naor A. and Pach J. (2012) Overlap properties of geometric expanders. J. Reine Angew. Math. 671 4983.
[14] Friedman J. (1998) Computing Betti numbers via combinatorial Laplacians. Algorithmica 21 331346.
[15] Friedman J. (2008) A Proof of Alon's Second Eigenvalue Conjecture and Related Problems , Vol. 908 of Memoirs of the American Mathematical Society, AMS.
[16] Friedman J. and Pippenger N. (1987) Expanding graphs contain all small trees. Combinatorica 7 7176.
[17] Garland H. (1973) p-adic curvature and the cohomology of discrete subgroups of p-adic groups. Ann. of Math. 97 375423.
[18] Golubev K. (2016) On the chromatic number of a simplicial complex. Combinatorica.
[19] Golubev K. and Parzanchevski O. (2014) Spectrum and combinatorics of Ramanujan triangle complexes. arXiv:1406.6666
[20] Gromov M. (2010) Singularities, expanders and topology of maps, part 2: From combinatorics to topology via algebraic isoperimetry. Geom. Funct. Anal. 20 416526.
[21] Gundert A. and Szedlák M. (2014) Higher dimensional Cheeger inequalities. In SOCG'14: Symposium on Computational Geometry, ACM, pp. 181188.
[22] Gundert A. and Wagner U. (2012) On Laplacians of random complexes. In SOCG'12: Symposium on Computational Geometry, ACM, pp. 151160.
[23] Gundert A. and Wagner U. (2013) On expansion and spectral properties of simplicial complexes. PhD thesis, ETH Zürich, Switzerland. Dissertation ETH no. 21600 of Anna Gundert.
[24] Hoffman A. J. (1970) On eigenvalues and colorings of graphs. Graph Theory and its Appl. (Proc. Advanced Sem., Math. Research Center, Univ. of Wisconsin, Madison, Wis., 1969), Academic Press, New York, pp. 7991.
[25] Horak D. and Jost J. (2013) Spectra of combinatorial Laplace operators on simplicial complexes. Adv. Math. 244 303336.
[26] Kook W., Reiner V. and Stanton D. (2000) Combinatorial Laplacians of matroid complexes. J. Amer. Math. Soc. 13 129148.
[27] Li W. C. W. (2004) Ramanujan hypergraphs. Geom. Funct. Anal. 14 380399.
[28] Linial N. and Meshulam R. (2006) Homological connectivity of random 2-complexes. Combinatorica 26 475487.
[29] Lubotzky A., Phillips R. and Sarnak P. (1988) Ramanujan graphs. Combinatorica 8 261277.
[30] Lubotzky A., Samuels B. and Vishne U. (2005) Ramanujan complexes of type Ãd . Israel J. Math. 149 267299.
[31] Marcus A., Spielman D. A. and Srivastava N. (2015) Interlacing families I: Bipartite Ramanujan graphs of all degrees. Ann. of Math. 182 307325.
[32] Margulis G. A. (1988) Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators. Problemy Peredachi Informatsii 24 5160.
[33] Matouşek J. and Wagner U. (2014) On Gromov's method of selecting heavily covered points. Discrete Comput. Geom. 52 133.
[34] Mukherjee S. and Steenbergen J. (2016) Random walks on simplicial complexes and harmonics. Random Struct. Alg. 49 379405.
[35] Pach J. (1998) A Tverberg-type result on multicolored simplices. Comput. Geom. 10 7176.
[36] Parzanchevski O. and Rosenthal R. (2017) Simplicial complexes: Spectrum, homology and random walks. Random Struct. Alg. 50 225261.
[37] Parzanchevski O., Rosenthal R. and Tessler R. J. (2016) Isoperimetric inequalities in simplicial complexes. Combinatorica 36 195227.
[38] Puder D. (2015) Expansion of random graphs: New proofs, new results. Inventio. Math. 201 845908.
[39] Sarveniazi A. (2007) Explicit construction of a Ramanujan (n 1, n 2,. . ., n d-1) -regular hypergraph. Duke Math. J. 139 141171.
[40] Spielman D. A. and Teng S. H. (2011) Spectral sparsification of graphs. SIAM J. Comput. 40 9811025.
[41] Tanner R. M. (1984) Explicit concentrators from generalized n-gons. SIAM J. Algebraic Discrete Methods 5 287.
[43] Żuk A. (1996) La propriété (T) de Kazhdan pour les groupes agissant sur les polyèdres. CR Acad. Sci. Sér. 1 Math. 323 453458.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 39 *
Loading metrics...

Abstract views

Total abstract views: 193 *
Loading metrics...

* Views captured on Cambridge Core between 17th May 2017 - 18th January 2018. This data will be updated every 24 hours.