Topology of random $2$-dimensional cubical complexes

Abstract We study a natural model of a random $2$-dimensional cubical complex which is a subcomplex of an n-dimensional cube, and where every possible square $2$-face is included independently with probability p. Our main result exhibits a sharp threshold $p=1/2$ for homology vanishing as $n \to \infty $. This is a $2$-dimensional analogue of the Burtin and Erdoős–Spencer theorems characterising the connectivity threshold for random graphs on the $1$-skeleton of the n-dimensional cube. Our main result can also be seen as a cubical counterpart to the Linial–Meshulam theorem for random $2$-dimensional simplicial complexes. However, the models exhibit strikingly different behaviours. We show that if $p> 1 - \sqrt {1/2} \approx 0.2929$, then with high probability the fundamental group is a free group with one generator for every maximal $1$-dimensional face. As a corollary, homology vanishing and simple connectivity have the same threshold, even in the strong ‘hitting time’ sense. This is in contrast with the simplicial case, where the thresholds are far apart. The proof depends on an iterative algorithm for contracting cycles – we show that with high probability, the algorithm rapidly and dramatically simplifies the fundamental group, converging after only a few steps.

[1]  Béla Bollobás.  The Evolution of the Cube , 1983 .

[2]  Elliot Paquette,et al.  The integer homology threshold in $Y_d(n, p)$ , 2018, 1808.10647.

[3]  Matthew Kahle,et al.  Spectral Gaps of Random Graphs and Applications , 2012, International Mathematics Research Notices.

[4]  Nathan Linial,et al.  Collapsibility and Vanishing of Top Homology in Random Simplicial Complexes , 2010, Discret. Comput. Geom..

[6]  M. Farber,et al.  Large Random Simplicial Complexes, II; the fundamental groups , 2015, 1509.04837.

[7]  Yasuaki Hiraoka,et al.  Limit Theorems for Random Cubical Homology , 2016, Discret. Comput. Geom..

[8]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[9]  Matthew Kahle,et al.  The fundamental group of random 2-complexes , 2007, 0711.2704.

[10]  M. Farber,et al.  Large Random Simplicial Complexes, III; The Critical Dimension , 2015, 1512.08714.

[11]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[12]  Matthew Kahle,et al.  Cohen–Lenstra Heuristics for Torsion in Homology of Random Complexes , 2017, Exp. Math..

[13]  Louis J. Billera,et al.  Neighborly cubical spheres and a cubical lower bound conjecture , 1997 .

[14]  Michael Davis,et al.  The geometry and topology of Coxeter groups , 2008 .

[15]  Michael Farber,et al.  Topological embeddings into random 2-complexes , 2021, Random Struct. Algorithms.

[16]  R. Meshulam,et al.  Homological connectivity of random k-dimensional complexes , 2009, Random Struct. Algorithms.

[17]  Nathan Linial,et al.  Homological Connectivity Of Random 2-Complexes , 2006, Comb..

[18]  Yasuaki Hiraoka,et al.  Tutte polynomials and random-cluster models in Bernoulli cell complexes , 2016, 1602.04561.

[19]  M. Farber,et al.  Large random simplicial complexes, I , 2015, 1503.06285.

[20]  Erik Lundberg,et al.  Homotopy types of random cubical complexes , 2019, J. Appl. Comput. Topol..

[21]  J. Spencer,et al.  EVOLUTION OF THE n-CUBE , 1979 .

[22]  N. Wallach,et al.  Homological connectivity of random k-dimensional complexes , 2009 .

[23]  B. Bollobás.  The evolution of random graphs , 1984 .

[24]  Daniel C. Cohen,et al.  Topology of random 2-complexes , 2010, 1006.4229.

[25]  Yuval Peled,et al.  On Simple Connectivity of Random 2-Complexes , 2018, Discret. Comput. Geom..

[26]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[27]  Nathan Linial,et al.  On the phase transition in random simplicial complexes , 2014, 1410.1281.

[28]  Tomasz Luczak,et al.  Integral Homology of Random Simplicial Complexes , 2016, Discret. Comput. Geom..

[29]  Sergiu Hart,et al.  A note on the edges of the n-cube , 1976, Discret. Math..

[30]  Alexander A. Sapozhenko,et al.  On Random Cubical Graphs , 1992 .

[31]  Michael Farber,et al.  The asphericity of random 2‐dimensional complexes , 2012, Random Struct. Algorithms.