Abstract
We obtain rigorous upper bounds on the critical exponent for self-avoiding polygons using probabilistic methods based on renewal processes and random walks.
Supported by a University Research Fellowship from NSERC of Canada.
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References
J.T. Chayes and L. Chayes, Random tubes as a model of pair correlations, Contemporary Mathematics 41 (1985) 11–41.
J.T. Chayes and L. Chayes, Percolation and random media, in Critical Phenomena, Random Systems and Gauge Theories; Les Houches, Session XLIII, K. Osterwalder and R. Stora (eds.). Elsevier, Amsterdam, Oxford, New York, 1986.
J.T. Chayes and L. Chayes, Ornstein-Zernike behavior for self-avoiding walks at all non-critical temperatures, Commun. Math. Phys. 105 (1986), 221–238.
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I ( 3rd edition ), Wiley, New York, 1968.
A.J. Guttmann, On the critical behaviour of self-avoiding walks, J. Phys. A 20 (1987), 1839–18M.
J.M. Hammersley, Percolation processes. II. The connective constant, Proc. Carob. Phil. Soc. 53 (1957), 642–645.
J.M. Hammersley, The number of polygons on a lattice, Proc. Camb. Phil. Soc. 57 (1961), 516–523.
J.M. Hammersley and D.J.A. Welsh, Further results on the rate of convergence to the connective constant of the hypercubical lattice, Quart. J. Math. Oxford Ser. 2 13 (1962), 108–110.
H. Kesten, On the number of self-avoiding walks, J. Math. Phys. 4 (1963), 960–969.
H. Kesten, On the number of self-avoiding walks II, J. Math. Phys. 5 (1964), 1128–1137.
J.C. Le Guillou and J. Zinn-Justin, Accurate critical exponents from field theory, J. Phys. France 50 (1989), 1365–1370.
N. Madras and A.D. Sokal, The pivot algorithm: a highly efficient Monte Carlo algorithm for the self-avoiding walk, J. Stat. Phys. 50 (1988), 109–186.
B. Nienhuis, Exact critical exponents of 0(n) models in two dimensions, Phys. Rev. Lett. 49 (1982), 1062–1065.
B. Nienhuis, Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas,J. Stat. Phys. 34 (1984), 731–761.
E. Nummelin, General Irreducible Markov Chains and Non-negative Operators, Cambridge University Press, Cambridge, 1984.
G. Slade, The scaling limit of self-avoiding random walk in high dimensions, Ann. Prob. 17 (1989), 91–107.
F. Spitzer, Principles of Random Walk, Springer, New York, 1976.
A.J. Stam, Renewal theory in r dimensions, Comp. Math. 23 (1971), 1–13.
S.G. Whittington, Statistical mechanics of polymer solutions and polymer adsorption, Adv. Chem. Phys. 51 (1982), 1–48.
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Madras, N. (1991). Bounds on the Critical Exponent of Self-Avoiding Polygons. In: Durrett, R., Kesten, H. (eds) Random Walks, Brownian Motion, and Interacting Particle Systems. Progress in Probability, vol 28. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0459-6_20
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DOI: https://doi.org/10.1007/978-1-4612-0459-6_20
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