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We compare the marked length spectra of some pairs of proper and cocompactcubical actions of a non-virtually cyclic group on $\text{CAT}(0)$ cubecomplexes. The cubulations are required to be virtually co-special, have thesame sets of convex-cocompact subgroups, and admit a contracting element. Thereare many groups for which these conditions are always fulfilled for any pair ofcubulations, including non-elementary cubulable hyperbolic groups, manycubulable relatively hyperbolic groups, and many right-angled Artin and Coxetergroups.  For these pairs of cubulations, we study the Manhattan curve associated totheir combinatorial metrics. We prove that this curve is analytic and convex,and a straight line if and only if the marked length spectra are homothetic.The same result holds if we consider invariant combinatorial metrics in whichthe lengths of the edges are not necessarily one. In addition, for theirstandard combinatorial metrics, we prove a large deviation theorem withshrinking intervals for their marked length spectra. We deduce the same resultfor pairs of word metrics on hyperbolic groups.  The main tool is the construction of a finite-state automaton thatsimultaneously encodes the marked length spectra of both cubulations in acoherent way, in analogy with results about (bi)combable functions onhyperbolic groups by Calegari and Fujiwara. The existence of this automatonallows us to apply the machinery of thermodynamic formalism for suspensionflows over subshifts of finite type, from which we deduce our results.

Rigidity phenomena and the statistical properties of group actions on $\text{CAT}(0)$ cube complexes