[Paper Review] Random Complexes and l^2-Betti Numbers

Russell Lyons|arXiv (Cornell University)|Nov 18, 2008

Stochastic processes and statistical mechanics27 references3 citations

TL;DR

This paper introduces a higher-dimensional analogue of uniform spanning trees on finite and infinite CW-complexes using determinantal probability measures, linking them to ℓ²-Betti numbers. It establishes a uniform isoperimetric inequality and provides an enumeration formula for k-dimensional subcomplexes, generalizing classical results on spanning trees and extending them to higher homology via ℓ²-topological invariants.

ABSTRACT

Uniform spanning trees on finite graphs and their analogues on infinite graphs are a well-studied area. On a Cayley graph of a group, we show that they are related to the first $\ell^2$-Betti number of the group. Our main aim, however, is to present the basic elements of a higher-dimensional analogue on finite and infinite CW-complexes, which relate to the higher $\ell^2$-Betti numbers. One consequence is a uniform isoperimetric inequality extending work of Lyons, Pichot, and Vassout. We also present an enumeration similar to recent work of Duval, Klivans, and Martin.

Motivation & Objective

• To develop a higher-dimensional analogue of uniform spanning trees on finite and infinite CW-complexes.
• To relate the resulting probability measures to higher ℓ²-Betti numbers of the complex.
• To establish a uniform isoperimetric inequality for ℓ²-homology in the context of random subcomplexes.
• To provide an enumeration formula for k-dimensional subcomplexes analogous to classical results on spanning trees.
• To extend the theory of determinantal measures to infinite complexes with free and wired measures, distinguishing them via ℓ²-homology.

Proposed method

• Uses determinantal probability measures arising from orthogonal projections onto row spaces of incidence matrices of CW-complexes.
• Defines two probability measures on k-dimensional subcomplexes—free and wired—based on the k-th Betti number and ℓ²-homology.
• Applies the Cauchy-Binet formula to relate determinants of boundary matrices to torsion in homology groups.
• Introduces the quotient group $ Q_k(S) $ to measure homological obstructions in subcomplexes and relates it to the order of homology groups.
• Derives a key identity (Lemma 6.1) linking the determinant of the boundary matrix $ ext{det} \partial_{S,T} $ to the product of torsion invariants $ t_k(T), t_k'(S), t_{k-1}(S^c) $.
• Uses the structure of the homology exact sequence of pairs to prove that the determinant equals the order of a certain homology group, enabling the enumeration result.

Experimental results

Research questions

• RQ1 How can the concept of uniform spanning trees be generalized to higher-dimensional CW-complexes using probabilistic and algebraic tools?
• RQ2 What is the relationship between the resulting random subcomplexes and the ℓ²-Betti numbers of the complex?
• RQ3 How do free and wired measures on infinite complexes differ, and what topological invariant captures this difference?
• RQ4 Can an enumeration formula for k-dimensional subcomplexes be derived that generalizes Cayley’s formula and Kalai’s result?
• RQ5 What is the role of torsion in homology in determining the weights of subcomplexes in the probability measure?

Key findings

• The determinant of the boundary matrix $ \partial_{S,T} $ equals the product of the orders of certain quotient groups: $ |\det \partial_{S,T}| = t_{d-1}(T) t_{d-2}(S^c) t_{d-1}'(S) / t_{d-2}(X) $, establishing a precise link between combinatorics and homology.
• The product of the non-zero eigenvalues of $ \partial_d \partial_d^* $ is given by $ \frac{h_{d-1}(X) h'_{d-2}(X)}{t_{d-2}(X)^2} $, where $ h_k'(X) $ is a sum over cobases of squared torsion invariants.
• The four measures (free and wired on finite and infinite complexes) coincide if and only if the reduced $ \ell^2 $-homology group vanishes, providing a topological criterion for measure equivalence.
• The theory yields a uniform isoperimetric inequality that extends previous results by Lyons, Pichot, and Vassout, with the difference in measures quantified by the $ k $-th $ \ell^2 $-Betti number.
• The probability measures are determinantal, hence satisfy strong probabilistic properties such as negative association, and are proportional to weights depending on the torsion of homology groups.
• The enumeration of k-dimensional subcomplexes generalizes classical results: for a simplex, the number of such subcomplexes is $ n^{n-2 \choose k} $, extending Kalai’s result.