Abstract
As was mentioned in Section 1.1, there is still no rigorous proof of the finiteness of the critical exponent γ for the number of self-avoiding walks [see Equation (1.1.4)] in dimensions two, three and four. The best rigorous upper bounds on c N /µ N are essentially of the form exp(O(N p)) for some constant 0 > p > 1. It is a major open problem to replace this bound by a polynomial in N. We remark that subadditivity (Section 1.2) by itself gives no information about such subexponential behaviour.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1996 Birkhäuser Boston
About this chapter
Cite this chapter
Madras, N., Slade, G. (1996). Some combinatorial bounds. In: The Self-Avoiding Walk. Probability and Its Applications. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4132-4_3
Download citation
DOI: https://doi.org/10.1007/978-1-4612-4132-4_3
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-3891-7
Online ISBN: 978-1-4612-4132-4
eBook Packages: Springer Book Archive