scholarly journals Algebraic and combinatorial expansion in random simplicial complexes

2021 ◽  
Author(s):  
Nikolaos Fountoulakis ◽  
Michał Przykucki
Keyword(s):  
Simplicial Complexes ◽  
Random Simplicial Complexes

scholarly journals Large random simplicial complexes, I

2016 ◽  
Vol 08 (03) ◽  
pp. 399-429 ◽  
Author(s):  
A. Costa ◽  
M. Farber
Keyword(s):  
Probability Measure ◽  
Simplicial Complex ◽  
Parameter Space ◽  
Simplicial Complexes ◽  
High Dimensional ◽  
Topological Properties ◽  
Convex Domains ◽  
Dimensional Parameter ◽  
Random Simplicial Complexes ◽  
Simply Connected

In this paper we introduce and develop the multi-parameter model of random simplicial complexes with randomness present in all dimensions. Various geometric and topological properties of such random simplicial complexes are characterised by convex domains in the high-dimensional parameter space (rather than by intervals, as in the usual one-parameter models). We find conditions under which a multi-parameter random simplicial complex is connected and simply connected. Besides, we give an intrinsic characterisation of the multi-parameter probability measure. We analyse links of simplexes and intersections of multi-parameter random simplicial complexes and show that they are also multi-parameter random simplicial complexes.

scholarly journals Random simplicial complexes, duality and the critical dimension

2019 ◽  
pp. 1-31
Author(s):  
Michael Farber ◽  
Lewis Mead ◽  
Tahl Nowik
Keyword(s):  
Simplicial Complex ◽  
Knot Theory ◽  
Betti Numbers ◽  
Critical Dimension ◽  
Simplicial Complexes ◽  
Alexander Duality ◽  
Random Simplicial Complexes

In this paper, we discuss two general models of random simplicial complexes which we call the lower and the upper models. We show that these models are dual to each other with respect to combinatorial Alexander duality. The behavior of the Betti numbers in the lower model is characterized by the notion of critical dimension, which was introduced by Costa and Farber in [Large random simplicial complexes III: The critical dimension, J. Knot Theory Ramifications 26 (2017) 1740010]: random simplicial complexes in the lower model are homologically approximated by a wedge of spheres of dimension equal the critical dimension. In this paper, we study the Betti numbers in the upper model and introduce new notions of critical dimension and spread. We prove that (under certain conditions) an upper random simplicial complex is homologically approximated by a wedge of spheres of the critical dimension.

COBOUNDARY EXPANDERS

2012 ◽  
Vol 04 (04) ◽  
pp. 499-514 ◽  
Author(s):  
DOMINIC DOTTERRER ◽  
MATTHEW KAHLE
Keyword(s):  
Isoperimetric Inequality ◽  
Random Graphs ◽  
High Probability ◽  
Simplicial Complexes ◽  
Random Simplicial Complexes ◽  
Higher Dimensional

We describe a higher-dimensional generalization of edge expansion from graphs to simplicial complexes, which we discuss as a type of co-isoperimetric inequality. The main point is to observe that sufficiently dense random simplicial complexes have this expander-like property with high probability, a higher-dimensional analogue of the fact that random graphs are expanders.

scholarly journals On the phase transition in random simplicial complexes

2016 ◽  
Vol 184 (3) ◽  
pp. 745-773 ◽  
Author(s):  
Nathan Linial ◽  
Yuval Peled
Keyword(s):  
Phase Transition ◽  
Simplicial Complexes ◽  
Random Simplicial Complexes

scholarly journals On topological minors in random simplicial complexes

2015 ◽  
Vol 144 (4) ◽  
pp. 1815-1828 ◽  
Author(s):  
Anna Gundert ◽  
Uli Wagner
Keyword(s):  
Simplicial Complexes ◽  
Random Simplicial Complexes
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