Abstract
We define the connectivity indexc for an infinite graph by the requirement that to disconnect a subset of at leastV points from the rest of the graph requires the deletion of a minimum ofS(V) bonds whereS(V) ∼V (c−1)/c for largeV. For ad-dimensional hypercubical lattice withd integral,c=d. We construct explicit examples of lattices with nonintegral connectivity indexc, 1<c<∞. It is argued that the connectivity index is an important parameter determining the critical behaviour of Hamiltonians on these lattices.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Belavin A A and Yurishchev M A 1973Zh. Eskp. Theor. Fiz. 64 407
Bollini C G and Giambiagi J J 1972Nuovo Cimento B12 20
Brezin E, Le Gillou J C, Zinn Justin J and Nickel B G 1974Phys. Rev. D9 1121
Collet P and Eckmann J P 1978A renormalization group analysis of the hierarchical model in statistical mechanics (Berlin: Springer-Verlag).
Dhar D 1977J. Math. Phys. 18 577
Dhar D 1978J. Math. Phys. 19 5
Fisher M E 1974Rev. Mod. Phys. 46 597
Gefen Y, Mandelbrot B B and Aharony A 1980Phys. Rev. Lett. 45 855
Harris A B, Lubensky T C and Chen J H 1976Phys. Rev. Lett. 36 415
Joyce G S 1972Phase transitions and critical phenomena eds. C Domb and M S Green (London, New York: Academic Press) Vol. 2
Mandelbrot B B 1977Fractals: form, chance and dimension (San Fransisco: Freeman)
Nelson D R and Fisher M E 1975Ann. Phys. 91 226
Obukhov S P 1980Physica A101 145
Wilson and Kogut 1974Phys. Rep. C 12 75
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dhar, D. On the connectivity index for lattices of nonintegral dimensionality. Pramana - J. Phys 15, 545–549 (1980). https://doi.org/10.1007/BF02848326
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02848326