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$ \def \nat {{\natural}} \def \F {{\mathbb F}} \def \I {{\mathbb I}} \def \C {{\mathbb C}} \def \R {{\mathbb R}} \def \reals {{\mathbb R}} \def \Z {{\mathbb Z}} \def \N {{\mathbb N}} \def \Q {{\mathbb Q}} \def \TT {{\mathbb T}} \def \T {{\mathbb T}} \def \CP {{\mathbb C}{\mathbb P}} \def \RP {{\mathbb R}{\mathbb P}} \def \ra {{\rightarrow}} \def \lra {{\longrightarrow}} \def \haf {{\frac{1}{2}}} \def \12 {{\frac{1}{2}}} \def \bD {\bar{\Delta}} \def \bG {\bar{\Gamma}} \def \p {\partial} \def \codim {\operatorname{codim}} \def \length {\operatorname{length}} \def \area {\operatorname{area}} \def \ev {\operatorname{\mathit{ev}}} \def \Area {\operatorname{Area}} \def \rk {\operatorname{rk}} \def \cl {\operatorname{cl}} \def \im {\operatorname{im}} \def \tr {\operatorname{tr}} \def \pr {\operatorname{pr}} \def \SB {\operatorname{SB}} \def \SH {\operatorname{SH}} \def \Sp {\operatorname{Sp}} \def \TSp {\widetilde{\operatorname{Sp}}} \def \TSpn {\widetilde{\operatorname{Sp}}{}^*} \def \U {\operatorname{U}} \def \GL {\operatorname{GL}} \def \LD {\underline{\operatorname{D}}} \def \HF {\operatorname{HF}} \def \HC {\operatorname{HC}} \def \CC {\operatorname{CC}} \def \CCM {\operatorname{CM}} \def \HQ {\operatorname{HQ}} \def \GW {\operatorname{GW}} \def \Gr {\operatorname{Gr}} \def \H {\operatorname{H}} \def \Tor {\operatorname{Tor}} \def \Tors {\operatorname{Tors}} \def \HFL {\operatorname{HF}^{\mathit{loc}}_*} \def \HML {\operatorname{HM}^{\mathit{loc}}_*} \def \HFLN {\operatorname{HF}^{\mathit{loc}}_{n}} \def \HMLM {\operatorname{HM}^{\mathit{loc}}_{m}} \def \HM {\operatorname{HM}} \def \CZ {\operatorname{CZ}} \def \CF {\operatorname{CF}} \def \CL {\operatorname{CL}} \def \SB {\operatorname{SB}} \def \Ham {\operatorname{Ham}} \def \Fix {\operatorname{Fix}} \def \vol {\operatorname{vol}} \def \PD {\operatorname{PD}} \def \pr {\preceq} \def \bPP {\bar{\mathcal{P}}} \def \barf {\bar{f}} \def \bx {\bar{x}} \def \by {\bar{y}} \def \bz {\bar{z}} \def \bg {\bar{\gamma}} \def \va {\vec{a}} \def \vb {\vec{b}} \def \bPhi {\bar{\Phi}} \def \MUCZ {\operatorname{\mu_{\scriptscriptstyle{CZ}}}} \def \MURS {\operatorname{\mu_{\scriptscriptstyle{RS}}}} \def \hMUCZ {\operatorname{\hat{\mu}_{\scriptscriptstyle{CZ}}}} \def \MUM {\operatorname{\mu_{\scriptscriptstyle{M}}}} \def \mum {\operatorname{\mu_{\scriptscriptstyle{M}}}} \def \s {\operatorname{c}} \def \hs {\hat{\operatorname{c}}} \def \sls {\operatorname{c}^{\scriptscriptstyle{LS}}} \def \odd {\scriptscriptstyle{odd}} \def \hn {\scriptscriptstyle{H}} \newcommand\congto{\ {\buildrel \cong \over \to}\ } \newcommand\wt{\widetilde} \newcommand\wh{\widehat} \def \ssminus {\smallsetminus} \def \chom {\operatorname{c_{hom}}} \def \CHZ {\operatorname{c_{\scriptscriptstyle{HZ}}}} \def \CGR {\operatorname{c_{\scriptscriptstyle{Gr}}}} \def \cf {\operatorname{c}} \def \Hof {\operatorname{\scriptscriptstyle{H}}} \def \CHHZ {{\mathcal H}_{\operatorname{\scriptscriptstyle{HZ}}}} \def \CBPS {\operatorname{c_{\scriptscriptstyle{BPS}}}} \newcommand{\TCH}{\tilde{\mathcal H}}$

Received: 16 March 2016 / Revised: 9 December 2016 / Accepted: 23 December 2016

Random Chain
Complexes

We study random, finite-dimensional, ungraded chain complexes over a finite
field and show that for a uniformly distributed differential a complex has the
smallest possible homology with the highest probability: either zero or one-
dimensional homology depending on the parity of the dimension of the complex.
We prove that as the order of the field goes to infinity the probability distribution
concentrates in the smallest possible dimension of the homology. On the other
hand, the limit probability distribution, as the dimension of the complex goes
to infinity, is a super-exponentially decreasing, but strictly positive, function of
the dimension of the homology.