Talk:Bethe lattice
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[edit]I'm not an expert in the field, but I happened to read a related article (Pys.Rev.E 75(2007)026105 on the Ising model on networks with Cayley-tree-like structure. In it, it was mentioned the difference of behaviors when the model is applied on a Cayley tree, vs. the model applied on a Bethe lattice. So the two must be different - while here I read they're the same thing... could you elaborate on this??? Thanks Alexxx m 15:45, 14 March 2007 (UTC)
- The main difference between the two is that a Cayley tree contains the boundaries whereas the Bethe lattice does not. Since the total number of nodes on the tree is
![{\displaystyle T_{n}={\frac {z\left[(z-1)^{n}-1\right]}{z-2}}+1}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/7be7cda765ec94f6227eb255f6ff0f08efae7b10)
if there are a total of n shells. Then, in the thermodynamic limit, the number of sites on the outer shell Nn does not vanish compared to this number (they both grow exponentially at the same rate). Therefore it is not possible to apply the usual approximation in the thermodynamic limit that the boundary conditions can be ignored. The Bethe lattice, is then defined as a sub-lattice of the total tree that is infinitely far away from the boundaries and where all sites have exactly the same coordination number. Both problems have quite different behaviors. Poudro (talk) 12:59, 23 March 2009 (UTC)
Baxter, Rodney J. (1982), Exactly solved models in statistical mechanics, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 978-0-12-083180-7, MR 0690578
I'm not an expert in the field, but it seems like a Bethe lattice is simply an n-regular tree, not a "tree-like structure". Can somebody please confirm this? 201.246.104.171 (talk) 03:10, 6 January 2012 (UTC)
What exactly is meant by "a Cayley tree contains the boundaries" (Poudro, above) ??? I can guess, but would rather just be told. 2601:200:C000:1A0:21A7:B740:95E6:BA8D (talk) 20:47, 5 September 2021 (UTC)
Mistake
[edit]After the first paragraph in the introductory section, this sentence appears:
"(Note that the Bethe lattice is actually an unrooted tree, since any vertex will serve equally well as a root.) "
Although it is true that any vertex will serve equally as a root, that is not the reason that the Bethe lattice is an unrooted tree.
The reason that it is an unrooted tree is that it is defined to be an unrooted tree.
I hope someone knowledgeable about the subject will fix this error. 2601:200:C000:1A0:21A7:B740:95E6:BA8D (talk) 20:41, 5 September 2021 (UTC)