1D Ising models, compound geometric distributions and selfdecomposability (Q5933507)

Consider \(N\) sites on a line with spin variables \(\sigma_1,\sigma_2,\dots ,\sigma_N\) and assume the nearest-neighbour, pairwise interactions. The partition function of the 1D Ising model has the form NEWLINE\[NEWLINE Z_N(K,h)=\sum_{\sigma_1,\sigma_2,\dots ,\sigma_N} \exp\left[K\sum_{j=1}^{N-1}\sigma_j\sigma_{j+1}+ h\sum_{j=1}^{N}\sigma_j\right], NEWLINE\]NEWLINE where \(\sigma_i=\pm 1,\;i=1,2,\dots ,N\), and the sum is taken over all possible \(2^N\) configurations of \(\{\pm 1\}\). It is shown that NEWLINE\[NEWLINE Z_N(K,h)=e^{NK}2^{-1} \left({N\over {2[N/2]}}\right)^{-1} \cosh^N h\prod_{j=1}^{[N/2]}\left(\frac{2}{1+\cos\theta_{j,N}}-\frac{1-e^{-4K}}{\cosh^2h}\right) NEWLINE\]NEWLINE where \(\theta_{j,N}=(2j-1)\pi/N, \;j=1,2,\dots , [N/2]\).NEWLINENEWLINENEWLINEAs a consequence of this factorization it is proved that the inverse of the partition function, as a function of the external field, is a product of characteristic functions of compound geometric distributions. Moreover, these products are characteristic functions of selfdecomposable (or Levy class \(L\)) probability distributions. In the paper it is also given an interpretation of the ratio of the partition function in terms of a randomly stopped random walk. This point of view will provide new approaches to complex physical models since the class \(L\) is quite rich and well understood in probability theory.