Random simplicial complexes SAMSI Combinatorial Probability Virtual Opening Workshop January 19, 2021 Handbook of Discrete & Computational Geometry Book chapter survey on “Random simplicial complexes” arXiv:1607.07069 “I predict a new subject of statistical topology. Rather than count the number of holes, Betti numbers, etc., one will be more interested in the distribution of such objects on noncompact manifolds as one goes out to infinity.” –Isadore Singer, 2004 Why stochastic topology? Randomness models nature • physics: black holes, quantum gravity, etc. Randomness models nature • statistics: topological data analysis Randomness models nature • mathematics: models for “typical” objects “Many simplicial complexes that arise in combinatorics are homotopy equivalent to a wedge of spheres. I have often wondered if perhaps there is some deeper explanation for this.” ––Robin Forman The bouquet-of-spheres conjecture proposes a measure-theoretic explanation for this phenomenon. Conjecture. If X(n, p) is the clique complex of a random graph , and if n−1/k ≪ p ≪ n−1/(k+1), then w.h.p. X is homotopy equivalent to a wedge sum of k-dimensional spheres. Theorem. If X(n, p) is the clique complex of a random graph , and if n−1/k ≪ p ≪ n−1/(k+1), then w.h.p. X is rationally homotopy equivalent to a wedge sum of k-dimensional spheres. Kahle, Matthew. Sharp vanishing thresholds for cohomology of random flag complexes. Ann. of Math. (2) 179 (2014), no. 3, 1085–1107. Another point of view: the probabilistic method gives existence proofs, often when we are short of constructions. The probabilistic method has now been applied in many area of mathematics. • Extremal graph theory • Ramsey theory • Expander graphs • Linear algebra • Geometric group theory Random graphs Define G(n, p) to be a random graph on vertex set [n]= 1, 2,...,n ,where each edge has probability{ p,independently.} Usually p = p(n) and n . !1 We say that event happens with high probability (w.h.p.) if the probability 1. ! In random graph theory, one is often interested in thresholds for various graph properties. Theorem. (Erdős–Rényi) If (1 + ✏) log n p ≥ n then w.h.p. G(n, p) is connected. If (1 ✏) log n p − n then w.h.p. G(n, p) is disconnected. Erdős, P.; Rényi, A. On random graphs. I. Publ. Math. Debrecen 6 (1959), 290–297. Theorem. (Erdős–Rényi) If log n + c p = n then c e− [G(n, p) is connected] e− . P ! At p =1/n, • giant component appears, • cycles appear, At p =1/n, • arbitrary topological minors exist, • G(n,p) is not planar. At p = log n/n, • G(n,p) has no isolated vertices, • G(n,p) is connected, • G(n,p) is an expander. Random 2-complexes β1 = 160, β2 = 0 The random 2-complex Y Y (n, p) is a random ⇠ n simplicial complex with n vertices, 2 edges, and each 2-dimensional face appears with probability p, independently.<latexit sha1_base64="2EMnCxWdE49oSAjwn+B86LndJRg=">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</latexit> p = c/n 2.754 ... c 0 0.5 2.433 ... p = c/n Cohen, D.; Costa, A.; Farber, M.; Kappeler, T. Topology of random 2- complexes. Discrete Comput. Geom. 47 (2012), no. 1, 117–149. 1-collapsible 2.754 ... c 0 0.5 2.433 ... p = c/n Cooley, Oliver; Kang, Mihyun; Person, Yury Largest components in random hypergraphs. Combin. Probab. Comput. 27 (2018), no. 5, 741–762. giant subcomplex 2.754 ... c 0 0.5 2.433 ... p = c/n Aronshtam, Lior; Linial, Nathan; Łuczak, Tomasz; Meshulam, Roy Collapsibility and vanishing of top homology in random simplicial complexes. Discrete Comput. Geom. 49 (2013), no. 2, 317–334. not 1-collapsible 2.754 ... c 0 0.5 2.433 ... p = c/n Aronshtam, L.; Linial, N. The threshold for d-collapsibility in random complexes. Random Structures Algorithms 48 (2016), no. 2, 260–269. not 1-collapsible 2.754 ... c 0 0.5 2.433 ... p = c/n Aronshtam, Lior; Linial, Nathan When does the top homology of a random simplicial complex vanish? Random Structures Algorithms 46 (2015), no. 1, 26–35. H2(Y,R) =0 6 2.754 ... c 0 0.5 2.433 ... p = c/n Linial, Nathan; Peled, Yuval On the phase transition in random simplicial complexes. Ann. of Math. (2) 184 (2016), no. 3, 745–773. H2(Y,R) =0 6 2.754 ... c 0 0.5 2.433 ... Torsion experiments n=75 Torsion experiments n=16, d=5 Torsion experiments Torsion experiments Conjecture: at p = 2.7538 / n , • there is a torsion burst in H 1 ( Y ( n, p )) , and at its peak it is Cohen–Lenstra distributed, • Y(n,p) is not embeddable in R 4 , and • ⇡1(Y (n, p)) goes from free to not free. p = c/n 2.754 ... c 0 0.5 2.433 ... 2logn At p = 2 log n/n, • no isolated edges • H 1(Y,Z/`)=0 (1) Linial, Nathan; Meshulam, Roy Homological connectivity of random 2-complexes. Combinatorica 26 (2006), no. 4, 475–487.1 (2) Meshulam, R.; Wallach, N. Homological connectivity of random k-dimensional complexes. Random Structures Algorithms 34 (2009), no. 3, 408–417. • H 1(Y,R)=0 Hoffman, Christopher. Kahle, Matthew, Paquette, Elliot Spectral graps of random graphs and applications. IMRN. (2019), rnz077.] • H1(Y,Z)=0 (1) Hoffman, Christopher; Kahle, Matthew; Paquette, Elliot The threshold for integer homology in random d-complexes. Discrete Comput. Geom. 57 (2017), no. 4, 810–823. (2) Łuczak, Tomasz; Peled, Yuval Integral homology of random simplicial complexes. Discrete Comput. Geom. 59 (2018), no. 1, 131–142. The fundamental group of Y(n,p) p = c/n 2.754 ... c 0 2.433 ... 2logn pn p = c/n [Newman, Andrew Freeness of the random fundamental group. J. Topol. Anal. 12 (2020), no. 1, 29–35.] ⇡1(Y )free 2.754 ... c 0 2.433 ... 2logn pn p = c/n [Newman, Andrew Freeness of the random fundamental group. J. Topol. Anal. 12 (2020), no. 1, 29–35.] ⇡1(Y )notfree 2.754 ... c 0 2.433 ... 2logn pn p = c/n Hoffman, Christopher. Kahle, Matthew, Paquette, Elliot Spectral graps of random graphs and applications. IMRN. (2019), rnz077.] ⇡1(Y )isKazhdan 2.754 ... c 0 2.433 ... 2logn pn p = c/n Babson, Eric; Hoffman, Christopher; Kahle, Matthew The fundamental group of random 2-complexes. J. Amer. Math. Soc. 24 (2011), no. 1, 1–28. ⇡1(Y )=0 2.754 ... c 0 2.433 ... 2logn pn If 2 log n/n p 1/pn, are there any ⌧ ⌧ nontrivial finite quotients of ⇡1(Y (n, p))? • No maps to any abelian groups. Łuczak, Tomasz; Peled, Yuval Integral homology of random simplicial complexes. Discrete Comput. Geom. 59 (2018), no. 1, 131–142. • No maps to any fixed finite targets. Meshulam, Roy. Bounded quotients of the fundamental group of a random 2-complex. arXiv:1308.3769 • Kazhdan’s Property (T)––not many unitary representations. Hoffman, Christopher. Kahle, Matthew, Paquette, Elliot Spectral graps of random graphs and applications. IMRN. (2019), rnz077.] Recent work Kahle, Matthew; Paquette, Elliot; Roldán, Érika Topology and geometry of random 2-dimensional cubical complexes. arXiv:2001.07812 Recent work Kahle, Matthew and Newman, Andrew. Topology and geometry of random 2-dimensional hypertrees. arXiv:2004.13572 Thanks for your time and attention! Acknowledgments:.