Witten Laplacian method for generalized models of Kac type

We first study the exponential decay of correlations for classical models of Kac type. Using recent results [J. Jon, “On spectral properties of Witten-Laplacians, their range of projections and Brascamp-Lieb inequality,” Integral Equ. Oper. Theory 36(3), 288–324 (2000)10.1007/BF01213926] about the Witten Laplacians, we improve the bounds obtained [A. Lo, “Witten Laplacian methods for the decay of correlations,” J. Stat. Phys. 132(2), 355–396 (2008)10.1007/s10955-008-9547-6] to justify the conjecture that the critical point in the (d+1)-dimensional Kac model is given by β c = 1/4d. We also provide a new approach for proving the exponential decay of the usual truncated correlations which is more direct than the existing ones in the sense that it does not use any cluster type expansions.