Abstracts of papers by Takashi Hara (Part II)

20. Takashi Hara and Gordon Slade.
    The self-avoiding-walk and percolation critical points in high dimensions.
    Combinatorics, Probability and Computing, 4 (1995), 197--215.

    ABSTRACT: We prove existence of an asymptotic expansion in the inverse dimension, to all orders, for the connective constant for self-avoiding walks on Z^d. For the critical point, defined to be the reciprocal of the connective constant, the coefficients of the expansion are computed through order d^-6, with a rigorous error bound of order d^-7. Our method for computing terms in the expansion also applies to percolation, and for nearest-neighbour independent Bernoulli bond percolation on Z^d gives the 1/d-expansion for the critical point through order d^-3, with a rigorous error bound of order d^-4. The method uses the lace expansion.
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19. Takashi Hara, Gordon Slade and Alan D. Sokal.
    New lower bounds on the self-avoiding-walk connective constant.
    J. Statist. Phys. 72 (1993) 479--517.

    ABSTRACT: 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 > 2, 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.
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18. Takashi Hara and Gordon Slade.
    The number and size of branched polymers in high dimensions.
    J. Statist. Phys. 67 (1992) 1009--1038.

    ABSTRACT: We consider two models of branched polymers (lattice trees) on the d-dimensional hypercubic lattice: (i) the nearest-neighbour model in sufficiently high dimensions, and (ii) a ``spread-out'' or long-range model for d > 8, in which trees are constructed from bonds of length less than or equal to a large parameter L. We prove that for either model the critical exponent ϑ for the number of branched polymers exists and equals 5/2, and that the critical exponent ν for the radius of gyration exists and equals 1/4. This improves our earlier results for the corresponding generating functions. The proof uses the lace expansion, together with an analysis involving fractional derivatives which has been applied previously to the self-avoiding walk in a similar context.
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17. Takashi Hara and Gordon Slade.
    The lace expansion for self-avoiding walk in five or more dimensions.
    Rev. Math. Phys. 4 (1992) 235--327.

    ABSTRACT: This paper is a continuation of our companion paper [16], in which it was proved that the standard model of self-avoiding walk in five or more dimensions has the same critical behaviour as the simple random walk, assuming convergence of the lace expansion. We prove the convergence of the lace expansion, an upper and lower infrared bound, and a number of other estimates that were used in the companion paper. The proof requires a good upper bound on the critical point (or equivalently a lower bound on the connective constant). In an appendix, new upper bounds on the critical point in dimensions higher than two are obtained, using elementary methods which are independent of the lace expansion. The proof of convergence of the lace expansion is computer assisted. Numerical aspects of the proof, including methods for the numerical evaluation of simple random walk quantities such as the two-point function (or lattice Green function), are treated in an appendix.
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16. Takashi Hara and Gordon Slade.
    Self-avoiding walk in five or more dimensions. I. The critical behaviour.
    Commun. Math. Phys. 147 (1992) 101--136.

    ABSTRACT: We use the lace expansion to study the standard self-avoiding walk in the d-dimensional hypercubic lattice, for d greater than or equal to 5. We prove that the number c_n of n-step self-avoiding walks satisfies c_n ∼Aμⁿ (i.e., γ = 1), and that the mean-square displacement is linear in the number of steps (ν = 1/2). A strong bound is obtained for the generating function for c_n(x), the number of n-step self-avoiding walks ending at x. An infrared bound is proved, and the critical two-point function is shown to decay at least as fast as |x|^-2. We also prove that the scaling limit is Gaussian, in the sense of convergence in distribution to Brownian motion. Our results can be used to construct the infinite self-avoiding walk in five or more dimensions. The proof of convergence of the lace expansion uses explicit numerical estimates for a number of simple random walk quantities, and provides good numerical upper bounds on the critical two-point function and various related quantities. Without using the lace expansion, we obtain lower bounds on the connective constant μ, for d > 2, which for d = 3 slightly improves the existing rigorous lower bound.
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15. Takashi Hara and Gordon Slade.
    Critical behaviour of self-avoiding walk in five or more dimensions.
    Bull. AMS. 25 (1991) 417--423.

    ABSTRACT: We use the lace expansion to prove that in five or more dimensions the standard self-avoiding walk on the hypercubic lattice behaves in many respects like the simple random walk. In particular, it is shown that the leading asymptotic behaviour of the number of n-step self-avoiding walks is purely exponential, that the mean-square displacement is asymptotically linear in the number of steps, and that the scaling limit is Gaussian, in the sense of convergence in distribution to Brownian motion. A number of related results are also proved. These results are optimal, according to the widely believed conjecture that the self-avoiding walk behaves unlike the simple random walk, in four or fewer dimensions.
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14. Takashi Hara and Gordon Slade.
    On the upper critical dimension of lattice trees and lattice animals.
    J. Statist. Phys. 59 (1990) 1469--1510.

    ABSTRACT: We give a rigorous proof of mean-field critical behaviour for the susceptibility (γ=1/2) and the correlation length (ν=1/4) for models of lattice trees and lattice animals in two cases: (i) for the usual model with trees or animals constructed from nearest-neighbour bonds, in sufficiently high dimensions, and (ii) for a class of "spread-out" or long-range models in which trees and animals are construced from bonds of various lengths, above eight dimensions. This provides further evidence that for these models the upper critical dimension is equal to eight. The proof involves obtaining an infrared bound and showing that a certain "square diagram" is finite at the critical point, and uses an expansion related to the lace expansion for the self-avoiding walk.
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13. Takashi Hara.
    Mean-field critical behaviour for correlation length for percolation in high dimensions.
    Prob. Th. Rel. Fields. 86 (1990) 337--385.

    ABSTRACT: Extending the method of [12], we prove that the correlation length ξ of independent bond percolation models exhibits mean-field type critical behaviour (i.e. ξ(p) ≈(p_c-p)^-1/2 as p ↑ p_c) in two situations: i) for nearest-neighbour independent bond percolation models on a d-dimensional hypercubic lattice Z^d, with d sufficiently large, and ii) for a class of "spread-out" independent bond percolation models, which are believed to belong to the same universality class as the nearest-neighbour model, in more than six dimensions. The proof is based on, and extends, a method used in [12], where it was used to prove the triangle condition and hence mean-field behaviour of the critical exponents γ, β, δ, Δ and ν₂ for the above two cases.
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12. Takashi Hara and Gordon Slade.
    Mean-field critical behaviour for percolation in high dimensions.
    Commun. Math. Phys. 128 (1990) 333--391.

    ABSTRACT: The triangle condition for percolation states that ∑_x,y τ(0,x) τ(x,y) τ(y,0) is finite at the critical point, where τ(x,y) is the probability that the sites x and y are connected. We use an expansion related to the lace expansion for a self-avoiding walk to prove that the triangle condition is satisfied in two situations: (i) for nearest-neighbour independent bond percolation on the d-dimensional hypercubic lattice, if d is sufficiently large, and (ii) in more than six dimensions for a class of `spread-out' models of independent bond percolation which are believed to be in the same universality class as the nearest-neighbour model. The class of models in (ii) includes the case where the bond occupation probability is constant for bonds of length less than some large number and is zero otherwise. In the course of the proof an infrared bound is obtained. The triangle condition is known to imply that various critical exponents take their mean-field (Bethe lattice) values (γ = γ = 1, δ = Δ_t = 2, t ≥ 2) and that the percolation density is continuous at the critical point. We also prove that ν₂ = 1/2 in (i) and (ii), where ν₂ is the critical exponent for the correlation length.
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11. Takashi Hara and Gordon Slade.
    The triangle condition for percolation.
    Bull. AMS. 21 (1989) 269--273.

    ABSTRACT: Aizenman and Newman introduced an unverified condition, the triangle condition, which has been shown to imply that a number of percolation critical exponents take their mean field values, and which is expected to hold above six dimensions for nearest neighbour percolation. We prove that the triangle condition is satisfied in sufficiently high dimensions for the nearest neighbour model, and above six dimensions for a class of "spread-out" models. The proof uses an expansion which is related to the lace expansion for self-avoiding walk.
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