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Decay of quantum correlations on a lattice by heat kernel methods. (English) Zbl 1165.82007
In the recent years, many authors have investigated the estimates of the correlation of two local observables for classical lattice models at high or at low temperature. In this elegant paper the authors study some estimates of the correlation of two local observables in quantum lattice models at high temperature. They give another estimate with a different hypothesis on the interaction. They obtain an upper bound for the correlation of two local observables supported in two disjoint sets. Also, defining a decomposition of any function \(f\) on \((\mathbb{R}_p)^{\mathbb{Z}^{d}}\) as a sum of functions \(T_{\mathbb{Q}}f\) associated to the boxes of \(\mathbb{Z}^{d}\), they apply this decomposition to the obtained function and prove estimates for each term of the decomposition. They obtain the cluster type decomposition of the heat kernel and describe the heat kernel of the Hamiltonian for finite subsets of the lattice, which may tend to the whole lattice.

MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
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