Combinatorics and Complexity of Partition Functions by Alexander Barvinok

Author:Alexander Barvinok
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

and the convergence is uniform over . Moreover, for any fixed , the convergence is also uniform over all .

Proof

Let us fix and an . As is discussed in Sect. 5.2.3, there exists such that up to an error , the ratio depends only on the -neighborhood of in . If , the neighborhood of looks like the k-regular tree with root at , uniform weight 1 on every edge and m levels, see Fig. 5.3. Hence it follows that the limit in question, say y, indeed exists.

Fig. 5.3A 3-regular tree with root at 0 and 3 levels

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