Period integrals from wall structures via tropical cycles, canonical coordinates in mirror symmetry and analyticity of toric degenerations

We give a simple expression for the integral of the canonical holomorphic volume form in degenerating families of varieties constructed from wall structures and with central fiber a union of toric varieties. The cycles to integrate over are constructed from tropical 1-cycles in the intersection complex of the central fiber. One application is a proof that the mirror map for the canonical formal families of Calabi-Yau varieties constructed by Gross and the second author is trivial. We also show that these families are the completion of an analytic family, without reparametrization, and that they are formally versal as deformations of logarithmic schemes. Other applications include canonical one-parameter type III degenerations of K3 surfaces with prescribed Picard groups. As a technical result of independent interest we develop a theory of period integrals with logarithmic poles on finite order deformations of normal crossing analytic spaces.