Using probabilistic limit theorems and concentration of measure, we prove that the infinite-volume limit of the spherical model on a d -dimensional torus is a massive GFF in the high-temperature regime, a standard GFF at the critical temperature, and a standard GFF plus a Rademacher random constant in the low-temperature regime. The proof at the critical temperature appears to be new and relies on a fine analysis of the zero-average Green’s function on the torus. We study the spin O ( N ) model in the double limit of spin dimensionality and torus size. Sending N → ∞ first, and then the torus size to infinity, we show that the different spin coordinates become i.i.d. fields, distributed as a massive GFF in the high-temperature regime, a standard GFF at the critical temperature, and a standard GFF plus a Gaussian random constant in the low-temperature regime.
Using probabilistic limit theorems and concentration of measure, we prove that the infinite-volume limit of the spherical model on a d -dimensional torus is a massive GFF in the high-temperature regime, a standard GFF at the critical temperature, and a standard GFF plus a Rademacher random constant in the low-temperature regime. The proof at the critical temperature appears to be new and relies on a fine analysis of the zero-average Green’s function on the torus. We study the spin O ( N ) model in the double limit of spin dimensionality and torus size. Sending N → ∞ first, and then the torus size to infinity, we show that the different spin coordinates become i.i.d. fields, distributed as a massive GFF in the high-temperature regime, a standard GFF at the critical temperature, and a standard GFF plus a Gaussian random constant in the low-temperature regime.
Suppose we have two finitely supported, admissible, probability measures on a hyperbolic group Γ . In this article we prove that the corresponding two Green metrics satisfy a counting central limit theorem when we order the elements of Γ according to one of the metrics. Our results also apply to various other metrics including length functions associated to Anosov representations and to group actions on hyperbolic metric spaces.
The now classical convergence in distribution theorem for well normalized sums of stationary martingale increments has been extended to multi-indexed martingale increments. In the present article we make progress in the identification of the limit law. In dimension one, as soon as the stationary martingale increments form an ergodic process, the limit law is normal, and it is still the case for multi-indexed martingale increments when one of the processes defined by one coordinate of the multidimensional time is ergodic. In the general case, the limit may be non normal. In the present paper we establish links between the dynamical properties of the ℤ d -measure preserving action associated to the stationary random field (like the positivity of the entropy of some factors) and the existence of a non normal limit law. The identification of a natural factor on which the ℤ d -action is of product type is a crucial step in this approach.
Many results of smooth hypoellipticity are available for scalar equations. Much remains to be done for systems and/or at different levels of regularity and in particular for L 1 -hypoellipticity. In this article we provide some examples and counter-examples.
