arXiv subjects
Publications and source records attributed to Takashi Hara.
We announce our recent proof that, for independent bond percolation in high dimensions, the scaling limits of the incipient infinite cluster's two-point and three-point functions are those of integrated super-Brownian excursion (ISE). The proof uses an extension of the lace expansion for percolation.
We present a general framework for understanding and analyzing critical behaviour in gravitational collapse. We adopt the method of renormalization group, which has the following advantages. (1) It provides a natural explanation for various types of universality and scaling observed in numerical studies. In particular, universality in initial data space and universality for different models are understood in a unified way. (2) It enables us to perform a detailed analysis of time evolution beyond linear perturbation, by providing rigorous controls on nonlinear terms. Under physically reasonable assumptions we prove: (1) Uniqueness of the relevant mode around a fixed point implies universality in initial data space. (2) The critical exponent $β_{BH}$ and the unique positive eigenvalue $κ$ of the relevant mode is exactly related by $β_{BH} = β/κ$, where $β$ is a scaling exponent. (3) The above (1) and (2) hold also for discretely self-similar case (replacing ``fixed point'' with ``limit cycle''). (4) Universality for diffent models holds under a certain condition. According to the framework, we carry out a rather complete (though not mathematically rigorous) analysis for perfect fluids with pressure proportional to density, in a wide range of the adiabatic index $γ$. The uniqueness of the relevant mode around a fixed point is established by Lyapunov analyses. This shows that the critical phenomena occurs not only for the radiation fluid but also for perfect fluids with $1 < γ\lesssim 1.88$. The accurate values of critical exponents are calculated for the models.
A scenario is presented, based on renormalization group (linear perturbation) ideas, which can explain the self-similarity and scaling observed in a numerical study of gravitational collapse of radiation fluid. In particular, it is shown that the critical exponent $β$ and the largest Lyapunov exponent ${\rm Re\, } κ$ of the perturbation is related by $β= ({\rm Re\, } κ) ^{-1}$. We find the relevant perturbation mode numerically, and obtain a fairly accurate value of the critical exponent $β\simeq 0.3558019$, also in agreement with that obtained in numerical simulation.
We give an elementary new method for obtaining rigorous lower bounds on the connective constant for self-avoiding walks on the hypercubic lattice $Z^d$. The method is based on loop erasure and restoration, and does not require exact enumeration data. Our bounds are best for high $d$, and in fact agree with the first four terms of the $1/d$ expansion for the connective constant. The bounds are the best to date for dimensions $d \geq 3$, but do not produce good results in two dimensions. For $d=3,4,5,6$, respectively, our lower bound is within 2.4\%, 0.43\%, 0.12\%, 0.044\% of the value estimated by series extrapolation.