We study random, finite-dimensional, ungraded chain complexes over a finite field and we are interested in the probability that such a complex has homology of a given dimension. We show that for a uniformly distributed differential the complex has the smallest possible homology with the highest probability.

To be more specific, consider an $ n$-dimensional vector space $ V$ over a finite field $ \F=\F_q$ of order $ q$ and let $ D$ be a differential on $ V$, i.e., a linear operator $ D\colon V\to V$ with $ D^2=0$. We are interested in the probability $ p_r(q,n)$ with which a chain complex $ (V,D)$ has homology $ \ker D/\im D$ of a given dimension $ r$ for fixed $ n$ and $ q$. The differential $ D$ is uniformly distributed and $ p_r(q,n)$ is simply the ratio $ c_r(q,n)/c(q,n)$, where $ c_r(q,n)$ is the number of complexes with $ r$-dimensional homology and $ c(q,n)$ is the number of all complexes. The numbers $ c_r(q,n)$ are explicitly calculated by [Kovacs1987]; see also Theorem 2.2.

We mainly focus on large complexes, i.e., on the limits as $ q$ or $ n$ go to infinity. Clearly, $ r$ and $ n$ must have the same parity and we separately analyze the asymptotic behavior of the sequence $ p_0(q,n), p_2(q,n),\ldots $, where $ n$ is even, and the sequence $ p_1(q,n), p_3(q,n),\ldots $ for $ n$ odd.

As $ q\to\infty$ with $ n$ fixed, the probability concentrates in the lowest possible dimension, i.e., $ p_0(q,n)\to 1$ or $ p_1(q,n)\to 1$ depending on the parity of $ n$, while $ p_r(q,n)\to 0$ for $ r>1$. This is consistent with the observation that over $ \C$ and even $ \R$ (see Lemma 3.1) a generic complex has $ 0$- or $ 1$-dimensional homology, i.e., that such complexes form the highest dimensional stratum in the variety of all $ n$-dimensional complexes. Indeed, one can expect the probability distributions for large $ q$ to approximate the generic situation in zero characteristic. We do not know, however, if the density functions converge in any sense as $ q\to\infty$ to some probability density on the variety of $ n$-dimensional complexes over, e.g., $ \R$.

When $ q$ and $ r$ are fixed and $ n\to\infty$ through either even or odd integers depending on the parity of $ r$, the situation is more subtle. In this case, all limit probabilities $ p_r(q)=\lim_{n\to\infty}p_r(q,n)$ are positive. However, the sequences $ p_0,p_2,\ldots$ and $ p_1,p_3,\ldots$ are super-exponentially decreasing and for a large $ q$ all terms in these sequences but the first one are very close to zero while the first is then, of course, close to 1. When $ q=2$ and $ r$ is even, we have $ p_0\approx 0.6$, $ p_2\approx 0.4$, $ p_4\approx 0.0075$ and other terms are very small. We explicitly calculate the ratios $ p_r(q)/p_0(q)$ and $ p_r(q)/p_1(q)$ and $ p_0$ and $ p_1$ in Theorem 2.1.

The proofs of these facts are elementary and quite simple. However, we have not been able to find in the literature any probability calculations in this basic case where random chain complexes are stripped of all additional structures including a grading. (The combinatorial part of our proof, Theorem 2.2, is contained in [[Kovacs1987], Lemma 5].) In contrast, random complexes of geometrical origin and underlying random geometrical and topological objects have been studied extensively and from various perspectives. Among such random objects are, for instance, random simplicial complexes of various types [see ([Aronshtam et al.2013],[Bobrowski and Kahle2014],[Costa and Farber2014],[Costa and Farber2015],[Kahle2011],[Meshulam2013],[Meshulam and Wallach2009],[Pippenger and Schleich2006],[Yogeshwaran et al.2014]) and references therein] and random Morse functions [see, e.g., ([Arnold2006],[Arnold2007],[Collier et al.2017],[Nicolaescu2012])].

These works utilize several models of randomness all of which appear to be quite different from the one, admittedly rather naive, used here. This makes a direct comparison difficult. One way to interpret our result is that, for a large complex, sufficiently non-trivial homology is indicative of some structure, a constraint limiting randomness. Note that such a structure can be as simple as a $ \Z$-grading confined to a fixed range of degrees. A dimensional constraint of this type is usually inherent in geometrical complexes, and it would be interesting to analyze its effect (if any) on the probability distribution in our purely algebraic setting. Another consequence of the result is that the assertion that a complex has large homology carries more information than the assertion that it has small homology.

The main motivation for our setting comes from Hamiltonian Floer theory for closed symplectic manifolds; see, e.g., [Salamon1999] and references therein. A Hamiltonian diffeomorphism is the time-one map of the isotopy generated by a time-dependent Hamiltonian. To such a diffeomorphism one can associate a certain complex, called the Floer complex, generated by its fixed points or, equivalently, the one-periodic orbits of the isotopy. Hence the dimension of the Floer homology gives a lower bound for the number of one-periodic orbits. The homology is independent of the Hamiltonian diffeomorphism. In addition, one can fix the free homotopy class of the orbits. (This construction is similar to Morse theory and, in fact, Floer theory is a version of Morse theory for the action functional.)

In many instances, e.g., often generically or for all symplectic manifolds with vanishing first Chern class such as tori, the dimension of the Floer complex grows with the order of iteration of the diffeomorphism; see [Ginzburg and G[U+00A8]urel2015]. In other words, the complex gets larger and larger as time in this discrete dynamical system grows. Moreover, the differential in the complex is usually impossible to describe explicitly, and hence it makes sense to compare the behavior of the complex and its homology with the generic or random situation. The Floer homology for contractible periodic orbits is isomorphic to the homology of the underlying manifold. Therefore, by our result, even though the Floer complex appears to be very "noisy" for large iterations and random on a bounded action scale, it has large homology groups and is actually very far from random. For non-contractible orbits, the dimension of the Floer complex is also known to grow in many settings; see [Ginzburg and G[U+00A8]urel2015],[G[U+00A8]urel2013]. However, in this case the Floer homology is zero and the complex may well be close to random. Note also that in some instances the Floer complex is $ \Z$-graded, but the grading is not supported within any specific interval of degrees. Moreover, in contrast with geometrical random complexes, the grading range of the Floer homology usually grows with the order of iteration ([Salamon and Zehnder1992]), and while it is not clear how to correctly account for an unbounded grading in a random model, such a grading is unlikely to affect the probability distribution.

One aspect of Floer theory which is completely ignored in our model is the action filtration. This filtration is extremely important and, in particular, it allows one to treat Floer theory in the context of persistent homology and topological data analysis; see [Carlsson2009],[Ghrist2008]. This connection has recently been explored in [Polterovich and Shelukhin2014],[Usher and Zhang2015]. However, it is not entirely clear how to meaningfully incorporate the action filtration into our model.

Let, as in the introduction, $ (V,D)$ be an ungraded $ n$-dimensional chain complex with differential $ D$ over a finite field $ \F=\F_q$ of order $ q$. In other words, $ V=\F^n$ and $ D$ is a linear operator on $ V$ with $ D^2=0$. We denote by $ c(q,n)$ the number of such complexes, i.e., the number of differentials $ D$. The dimension $ r$ of the homology $ \ker D/\im D$ has the same parity as $ n$ and we let $ c_r(q,n)$ be the number of complexes with homology of dimension $ r$. (In what follows, we always assume that $ r$ and $ n$ have the same parity.) Clearly, \begin{eqnarray*} c(q,n)=c_0(q,n)+c_2(q,n)+\cdots+c_{n}(q,n) \end{eqnarray*} when $ n$ is even and \begin{eqnarray*} c(q,n)=c_1(q,n)+c_3(q,n)+\cdots+c_{n}(q,n) \end{eqnarray*} when $ n$ is odd.

Furthermore, denote by \begin{eqnarray*} p_r(q,n)=\frac{c_r(q,n)}{c(q,n)} \end{eqnarray*} the probability (with respect to the uniform distribution) of a complex to have $ r$-dimensional homology. Our main result describes the behavior of $ p_r(q,n)$ as the size of the complex, i.e., $ q$ or $ n$, goes to infinity.

The proof of this theorem is based on an explicit calculation of $ c_r(q,n)$. To state the result, denote by $ \GL_k(q)$ the general linear group of $ k\times k$ invertible matrices over $ \F_q$ and recall that \begin{eqnarray*} |\GL_k(q)|=q^{k(k-1)/2}\prod_{j=1}^k(q^j-1). \end{eqnarray*} Then we have the following particular case of [[Kovacs1987], Lemma 5].

The proof of Theorem 2.2 is based on the observation that the differential in a finite-dimensional complex over any field $ \F$ can be brought to its Jordan normal form or, equivalently, a complex over $ \F$ can be decomposed into a sum of elementary complexes, i.e., into a sum of two-dimensional complexes with zero homology and one-dimensional complexes. To be more precise, we have the following elementary observation.

When $ \F$ is algebraically closed, this follows immediately from the Jordan normal form theorem. Hence, the emphasis here is on the fact that the field $ \F$ is immaterial. For the sake of completeness, we outline a proof of the lemma.

Proof.

Proof.

Proof of Theorem 2.1.