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Suppose we have an undirected graph with integer valued nodes where node $i$ is connected to 0<|i-j|\le 2$ implies nodes $i+1$ i$ and $i+2$. j$ are connected. Let $c_n$ be the number of self-avoiding walks on this graph of length $n$ starting at origin. Define the connective constant as $$\mu = \lim_{n\to \infty} c_n^{\frac{1}{n}}$$ What is known about $\mu$? This quantity seems to be related to the transition temperature of an Ising model on such graph, has such model been studied? |
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Connective constant for self-avoiding walks on a skip-chainSuppose we have an undirected graph with integer valued nodes where node $i$ is connected to nodes $i+1$ and $i+2$. Let $c_n$ be the number of self-avoiding walks on this graph of length $n$ starting at origin. Define the connective constant as $$\mu = \lim_{n\to \infty} c_n^{\frac{1}{n}}$$ What is known about $\mu$? This quantity seems to be related to the transition temperature of an Ising model on such graph, has such model been studied?
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