Construction and Borel summability of infrared ? 4 4 by a phase space expansion

The Nearest-Neighbor Self-Avoiding Walk with Complex Killing Rates

“…These latter models have been extensively studied with rigorous renormalization group techniques in four dimensions [11,13,19]. Another related model is the SAW on a Sierpiǹski gasket, which was studied in [20], again using renormalization group ideas.…”

From Constructive Field Theory to Fractional Stochastic Calculus. (I) An introduction: Rough Path Theory and Perturbative Heuristics

“…An interesting case is when one may show that the contribution of the renormalized vertex λ p P p (φ, ∇φ) goes to zero at distances which are large with respect to the bare scale; then the theory is said to be asymptotically free at large distances. The bestknown examples of this behaviour are maybe the weakly avoiding path or the φ 4 -theory, both in dimension D = 4 ; see [46,22,17] for rigorous results 17 . Our model is original for it combines in some sense features of models of both high-energy physics and statistical physics: namely, the bare scale is O(M −ρ ), but the theory is asymptotically free at large distances.…”

“…In this work we use the multi-scale cluster expansion developed in [17] more than twenty years ago which seems to us the most appropriate for these probabilistic models; it reduces to the minimum the use of abstract combinatorial identities and algebra, to the benefit of a very intuitive and visual (though sometimes heavy) tree expansion.…”

“…The multiscale expansion was devised in the eighties to provide a rigorous version of Wilson's renormalization group e.g. including the flow of the effective parameters: see [22] for the block-spin approach, and [17] for the continuous space multi-scale cluster expansion.…”

Exact Renormalization Group Equations and the Field Theoretical Approach to Critical Phenomena

“…[39] In fact, this removal is nothing but the construction of the perfect action (initialization of the functional form of the momentum-scale dependence). Indeed it has been shown that even in the absence of an ultraviolet stable fixed point, the perturbative series of the φ 4 4 field theory are Borel summable [40] provided that the RG flow be initialized somewhere and identifiable at any smaller cutoff scale 25 (infra-red φ 4 4 model 26 ). In the continuum limit, the problem of dealing with the infinite number of degrees of freedom is completely involved in the initial condition of the ideal scale dependence provided we have first picked out the slowest flow.…”