  In this paper, we prove the convergence and uniqueness of a general
discrete-time nonlinear Markov chain with specific conditions. The results have
important applications in discrete differential geometry. First, on a general
finite weighted graph, we prove the discrete-time Ollivier Ricci curvature flow
$d_{n+1}\coloneqq(1-\alpha\kappa_{d_{n}})d_{n}$ converges to a constant
curvature metric. Then the author in \cite[Theorem 5.1]{M23} proved a Laplacian
separation principle on a locally finite graph with non-negative Ollivier
curvature. Here we prove the Laplacian separation flow converges to the
constant Laplacian solution and generalize the result to nonlinear $p$-Laplace
operators. Moreover, our results can also be applied to study the long-time
behavior in the nonlinear Dirichlet forms theory and nonlinear
Perron--Frobenius theory. At last, we define the Ollivier Ricci curvature of
nonlinear Markov chain which is consistent with the classical Ollivier Ricci
curvature, sectional curvature \cite{CMS24}, coarse Ricci curvature on
hypergraphs \cite{IKTU21} and the modified Ollivier Ricci curvature for
$p$-Laplace. And we prove the convergence results for the nonlinear Markov
chain with nonnegative Ollivier Ricci curvature.
