Graded persistence diagrams and persistence landscapes

We introduce a refinement of the persistence diagram, the graded persistence diagram. It is the Mobius inversion of the graded rank function, which is obtained from the rank function using the unary numeral system. Both persistence diagrams and graded persistence diagrams are integer-valued functions on the Cartesian plane. Whereas the persistence diagram takes non-negative values, the graded persistence diagram takes values of 0, 1, or -1. The sum of the graded persistence diagrams is the persistence diagram. We show that the positive and negative points in the k-th graded persistence diagram correspond to the local maxima and minima, respectively, of the k-th persistence landscape. We prove a stability theorem for graded persistence diagrams: the 1-Wasserstein distance between k-th graded persistence diagrams is bounded by twice the 1-Wasserstein distance between the corresponding persistence diagrams, and this bound is attained. In the other direction, the 1-Wasserstein distance is a lower bound for the sum of the 1-Wasserstein distances between the k-th graded persistence diagrams. In fact, the 1-Wasserstein distance for graded persistence diagrams is more discriminative than the 1-Wasserstein distance for the corresponding persistence diagrams.Comment: accepted for publication in Discrete and Computational Geometr

Block persistence

We define a block persistence probability $p_l(t)$ as the probability that the order parameter integrated on a block of linear size $l$ has never changed sign since the initial time in a phase ordering process at finite temperature T<T_c. We argue that p_l(t)\sim l^{-z\theta_0}f(t/l^z) in the scaling limit of large blocks, where \theta_0 is the global (magnetization) persistence exponent and f(x) decays with the local (single spin) exponent \theta for large x. This scaling is demonstrated at zero temperature for the diffusion equation and the large n model, and generically it can be used to determine easily \theta_0 from simulations of coarsening models. We also argue that \theta_0 and the scaling function do not depend on temperature, leading to a definition of \theta at finite temperature, whereas the local persistence probability decays exponentially due to thermal fluctuations. We also discuss conserved models for which different scaling are shown to arise depending on the value of the autocorrelation exponent \lambda. We illustrate our discussion by extensive numerical results. We also comment on the relation between this method and an alternative definition of \theta at finite temperature recently introduced by Derrida [Phys. Rev. E 55, 3705 (1997)].Comment: Revtex, 18 pages (multicol.sty), 15 eps figures (uses epsfig), submitted to Eur. Phys. J.