scholarly journals Law of large numbers for Betti numbers of homogeneous and spatially independent random simplicial complexes

2021 ◽  
Author(s):  
Shu Kanazawa
Keyword(s):  
Law Of Large Numbers ◽  
Betti Numbers ◽  
Simplicial Complexes ◽  
Large Numbers ◽  
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.

scholarly journals Strong Law of Large Numbers for Betti Numbers in the Thermodynamic Regime

2018 ◽  
Vol 174 (4) ◽  
pp. 865-892 ◽  
Author(s):  
Akshay Goel ◽  
Khanh Duy Trinh ◽  
Kenkichi Tsunoda
Keyword(s):  
Law Of Large Numbers ◽  
Betti Numbers ◽  
Strong Law ◽  
Large Numbers ◽  
Thermodynamic Regime

scholarly journals Large random simplicial complexes, III the critical dimension

2017 ◽  
Vol 26 (02) ◽  
pp. 1740010 ◽  
Author(s):  
A. Costa ◽  
M. Farber
Keyword(s):  
Fundamental Group ◽  
Random Graphs ◽  
Betti Numbers ◽  
Critical Dimension ◽  
Simplicial Complexes ◽  
Topological Properties ◽  
Open Questions ◽  
Random Simplicial Complexes ◽  
Special Cases

In this paper, we study the notion of critical dimension of random simplicial complexes in the general multi-parameter model described in [Random simplicial complexes, preprint (2014), arXiv:1412.5805 ; Large random simplicial complexes, I, preprint (2015), arXiv:1503.06285 ; Large random simplical complexes, II, preprint (2015), arXiv:1509.04837 ]. This model includes as special cases the Linial–Meshulam–Wallach model [Homological connectivity of random 2-complexes, Combinatorica 26 (2006) 475–487; Homological connectivity of random [Formula: see text]-complexes, Random Struct. Alogrithms 34 (2009) 408–417.] as well as the clique complexes of random graphs. We characterize the concept of critical dimension in terms of various geometric and topological properties of random simplicial complexes such as their Betti numbers, the fundamental group, the size of minimal cycles and the degrees of simplexes. We mention in the text a few interesting open questions.

Introduction

2017 ◽  
Author(s):  
Jochen Rau
Keyword(s):  
Probability Theory ◽  
Phase Transitions ◽  
Statistical Mechanics ◽  
Conservation Laws ◽  
Law Of Large Numbers ◽  
Macroscopic Scale ◽  
Classical Probability ◽  
Large Numbers ◽  
Microscopic World ◽  
New Phenomena

Statistical mechanics concerns the transition from the microscopic to the macroscopic realm. On a macroscopic scale new phenomena arise that have no counterpart in the microscopic world. For example, macroscopic systems have a temperature; they might undergo phase transitions; and their dynamics may involve dissipation. How can such phenomena be explained? This chapter discusses the characteristic differences between the microscopic and macroscopic realms and lays out the basic challenge of statistical mechanics. It suggests how, in principle, this challenge can be tackled with the help of conservation laws and statistics. The chapter reviews some basic notions of classical probability theory. In particular, it discusses the law of large numbers and illustrates how, despite the indeterminacy of individual events, statistics can make highly accurate predictions about totals and averages.

scholarly journals Limit theorems for multi-type general branching processes with population dependence

2020 ◽  
Vol 52 (4) ◽  
pp. 1127-1163
Author(s):  
Jie Yen Fan ◽  
Kais Hamza ◽  
Peter Jagers ◽  
Fima C. Klebaner
Keyword(s):  
Carrying Capacity ◽  
Population Model ◽  
General Framework ◽  
Limit Theorems ◽  
Mating Systems ◽  
Law Of Large Numbers ◽  
Branching Processes ◽  
Special Section ◽  
Large Numbers ◽  
Type Population

AbstractA general multi-type population model is considered, where individuals live and reproduce according to their age and type, but also under the influence of the size and composition of the entire population. We describe the dynamics of the population as a measure-valued process and obtain its asymptotics as the population grows with the environmental carrying capacity. Thus, a deterministic approximation is given, in the form of a law of large numbers, as well as a central limit theorem. This general framework is then adapted to model sexual reproduction, with a special section on serial monogamic mating systems.

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