Spectral analysis of the disordered stochastic 1-D Ising model (Q1809279)

The particle structure of the generator of the Glauber dynamics in a one-dimensional Ising model with random i.i.d. nearest-neighbour interactions \(\omega_{x,x+1}\), \(x\in\mathbb{Z}\), is investigated. The distribution of \(\omega_{x,x+1}\) is supposed absolutely continuous and with support \(0<\gamma_1\leq \omega_{x,x+ 1}\leq\gamma_2< \infty\). For every inverse temperature \(\beta\) and every realization \(\omega\) there exists a unique Gibbs measure \(\mu^\beta_\omega\) on the spin configuration space \(X\), which is a Markov chain, and \({\mathcal L}_2(X, \mu^\beta_\omega)\) decomposes in a direct sum of \(k\)-particle invariant subspaces of the corresponding Glauber dynamics generator \(L^\beta(\omega)\), unitary equivalent with a free Fermion theory with one-particle Hamiltonian \(\widetilde L^\beta_1(\omega)\) on \(\ell_2(\mathbb{Z})\). In particular, the spectrum in the \(k\)-particle subspace is the sum of \(k\) one-particle spectra. The main result is that \(\omega\)-a.e., \(\widetilde L^\beta_1(\omega)\) has pure point spectrum, its eigenvalues are a dense subset of a non-random interval \([-1-c, -1+c]\) and its eigenfunctions have exponential localization. Moreover, for any realization \(\omega , \sigma(\widetilde L^\beta_1(\omega))\subset [-1-c, -1+c]\).