CURVATURE TENSOR OF A STATISTICAL MANIFOLD ASSOCIATED WITH A CORRELATED-WALK MODEL

The curvature tensor of a statistical manifold associated with a corre lated-walk model is investigated. Ln the model, a walker moves along a linear lattice right or left or stays, depending on the last steps. I n the case of symmetric walks, two jump probabilities and a stay proba bility (r(0)) specify the transition probabilities and constitute a th ree-dimensional (3D) space. The 3D space is foliated by the stay proba bility r(0). A Riemann curvature, defined by the method of information geometry, on each 2D leaf is not only a function of two jump paramete rs, but also a step time N. The dynamic evolution of the Riemann scala r curvature R and also the asymptotic properties of the R in N --> inf inity are investigated in detail. It is found that remarkable features appear in the dynamic process of the R. Such dynamic features of the R are able to be well understood in terms of the degree of correlation or the activity of stepping. In N --> infinity, each leaf is shown to approach the saddle surface of R = -1 except for the leaf of r(0) = 1 . The exceptional leaf approaches the spherical surface of R values of R are shown to have relation to regularity of paths or stability of s tochastic processes. The relation to stability is also discussed by co ntrast with the values of R of Fermi gases and Bose gases. It is shown that the R's of the leaves r(0) = 0 and 1 are almost equal to those o f Fermi gases and Bose gases, respectively, and that the R's of the tw o leaves reflect the difference of the stability of the two leaves. Th e asymptotic property of a one-parameter curvature, called the a curva ture, is also investigated. The a curvature at a = 1 is shown to appro ach zero for N -> infinity or approaching equilibrium states. This sug gests that the zero a = 1 curvature is a universal property in a broad er class of equilibrium systems including thermodynamic systems.