Abstracts of papers by Takashi Hara (Part III)

30. Sheldon Goldstein, Takashi Hara, and Hal Tasaki.
    Extremely quick thermalization in a macroscopic quantum system for a typical nonequilibrium subspace.
    New Journal of Physics 17 (2015) 045002.
    ABSTRACT)

    ABSTRACT: The fact that macroscopic systems approach thermal equilibrium may seem puzzling, for example, because it may seem to conflict with the time-reversibility of the microscopic dynamics. We here prove that in a macroscopic quantum system for a typical choice of 'nonequilibrium subspace', any initial state indeed thermalizes, and in fact does so very quickly, on the order of the Boltzmann time \tau_B = h/(k_B T). Therefore what needs to be explained is, not that macroscopic systems approach thermal equilibrium, but that they do so slowly.
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29. Sheldon Goldstein, Takashi Hara, and Hal Tasaki.
    Time Scales in the Approach to Equilibrium of Macroscopic Quantum Systems.
    Phys. Rev. Lett. 111 (2013) 140401.
    ABSTRACT)

    ABSTRACT: We prove two theorems concerning the time evolution in general isolated quantum systems. The theorems are relevant to the issue of the time scale in the approach to equilibrium. The first theorem shows that there can be pathological situations in which the relaxation takes an extraordinarily long time, while the second theorem shows that one can always choose an equilibrium subspace, the relaxation to which requires only a short time for any initial state.
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28. Takashi Hara.
    Decay of Correlations in Nearest-Neighbour Self-Avoiding Walk, Percolation, Lattice Trees and Animals.
    Ann. Prob. 36 (2008) 530--593.
    ABSTRACT, and PDF file (640 KB) of the final version published in Annals of Probability.
    (Older versions in In Los Alamos as /math-ph/0504021)

    ABSTRACT: We consider nearest-neighbour self-avoiding walk, bond percolation, lattice trees, and bond lattice animals on d-dimensional hypercubic lattice. The two-point functions of these models are respectively the generating function for self-avoiding walks from the origin to x, the probability of a connection from the origin to x, and the generating functions for lattice trees or lattice animals containing the origin and x. Using the lace expansion, we prove that the two-point function at the critical point is asymptotic to |x|^{2-d} as |x| goes to infinity, for d \geq 5 for self-avoiding walk, for d \geq 19 for percolation, and for sufficiently large d for lattice trees and animals. These results are complementary to those of Hara, Hofstad and Slace, where spread-out models were considered. In the course of the proof, we also provide a sufficient (and rather sharp if d > 4) condition under which the two-point function of a random walk is asymptotic to |x|^{2-d} as |x| goes to infinity.
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27. Takashi Hara, Remco van der Hofstad, and Gordon Slade.
    Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models.
    Ann. Prob. 31 (2003) 349--408.
    ABSTRACT, and PDF file (520 KB) of the final version published in Annals of Probability.
    (Older versions in Los Alamos as /math-ph/0011046, in Texas mp_arc, paper# 00-468.)

    ABSTRACT: We consider spread-out models of self-avoiding walk, bond percolation, lattice trees and bond lattice animals on Z^d, having long finite-range connections, above their upper critical dimensions d = 4 (self-avoiding walk), d = 6 (percolation) and d = 8 (trees and animals). The two-point functions for these models are respectively the generating function for self-avoiding walks from the origin to x in Z^d, the probability of a connection from 0 to x, and the generating function for lattice trees or lattice animals containing 0 and x. We use the lace expansion to prove that for sufficiently spread-out models above the upper critical dimension, the two-point function of each model decays, at the critical point, as a multiple of |x|^2-d as x goes to infinity. We use a new unified method to prove convergence of the lace expansion. The method is based on x-space methods rather than the Fourier transform. Our results also yield unified and simplified proofs of the bubble condition for self-avoiding walk, the triangle condition for percolation, and the square condition for lattice trees and lattice animals, for sufficiently spread-out models above the upper critical dimension.
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26. Takashi Hara, Tetsuya Hattori, and Hiroshi Watanabe.
    Triviality of hierarchical Ising model in four dimensions.
    Commmun. Math. Phys. 220 (2001) 13--40.
    PDF file (204 KB). The original publication is available on LINK (Springer's internet service, http://link.springer.de) . (In Texas mp_arc, paper# 00-397.)

    ABSTRACT: Existence of critical renormalization group trajectory for a hierarchical Ising model in 4 dimensions is shown. After 70 iterations of renormalization group transformations, the critical Ising model is mapped into a vicinity of the Gaussian fixed point. Convergence of the subsequent trajectory to the Gaussian fixed point is shown by power decay of the effective coupling constant. The analysis in the strong coupling regime is computer-aided and Newman's inequalities on truncated correlations are used to give mathematical rigor to the numerical bounds. In order to obtain a criterion for convergence to the Gaussian fixed point, characteristic functions and Newman's inequalities are systematically used.
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25. Takashi Hara and Gordon Slade.
    The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion.
    J. Math. Phys. 41 (2000) 1244--1293.
    PDF file (648 KB) (In Los Alamos as /math-ph/9903043.)

    ABSTRACT: For independent nearest-neighbour bond percolation on Z^d with d >> 6, we prove that the incipient infinite cluster's two-point function and three-point function converge to those of integrated super-Brownian excursion (ISE) in the scaling limit. The proof is based on an extension of the new expansion for percolation derived in a previous paper, and involves treating the magnetic field as a complex variable. A special case of our result for the two-point function implies that the probability that the cluster of the origin consists of n sites, at the critical point, is given by a multiple of n^-3/2, plus an error term of order n^-3/2-ε with ε > 0. This is a strong version of the statement that the critical exponent δ is given by δ = 2.
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24. Takashi Hara and Gordon Slade.
    The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents.
    J. Statist. Phys. 99 (2000) 1075--1168.
    PDF file (744 KB) (In Los Alamos as /math-ph/9903042.)

    ABSTRACT: This is the first of two papers on the critical behaviour of bond percolation models in high dimensions. In this paper, we obtain strong joint control of the critical exponents η and δ, for the nearest-neighbour model in very high dimensions d >> 6 and for sufficiently spread-out models in all dimensions d > 6. The exponent η describes the low frequency behaviour of the Fourier transform of the critical two-point connectivity function, while δ describes the behaviour of the magnetization at the critical point. Our main result is an asymptotic relation showing that, in a joint sense, η = 0 and δ = 2. The proof uses a major extension of our earlier expansion method for percolation. This result provides evidence that the scaling limit of the incipient infinite cluster is the random probability measure on R^d known as integrated super-Brownian excursion (ISE), in dimensions above 6. In the sequel to this paper[25], we extend our methods to prove that the scaling limits of the incipient infinite cluster's two-point and three-point functions are those of ISE for the nearest-neighbour model in dimensions d >> 6.
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23. Tatsuhiko Koike, Takashi Hara and Satoshi Adachi
    Critical behavior in gravitational collapse of a perfect fluid.
    Phys. Rev. D59 (1999) 104008.

    ABSTRACT:

22. Takashi Hara and Gordon Slade.
    The incipient infinite cluster in high-dimensional percolation.
    Elec. Research Announcements of AMS, 4 (1998) 48--55.
    (In Los Alamos as /math-ph/9805023.)

    ABSTRACT: 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.
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21. Tatsuhiko Koike, Takashi Hara and Satoshi Adachi.
    Critical behaviour in gravitational collapse of radiation fluid --- A renormalization group (linear perturbation) approach.
    (In Los Alamos as gr-qc/9503007.)
    Phys. Rev. Lett. 74 (1995) 5170--5173.

    ABSTRACT: 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 Re κ of the perturbation is related by β = (Re κ) ^-1. We find the relevant perturbation mode numerically, and obtain a fairly accurate value of the critical exponent β ≅ 0.3558019, also in agreement with that obtained in numerical simulation.
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