Below we consider the most interesting case when renormalised coupling constant is assumed to be independent of N (or energy).
Singular statistics
In the other situations, by renormalising the random variables we always construct some process satisfying f (1) = 1.
Random walks in random environment on trees and multiplicative chaos
Introduce now the renormalised random variables ˜ηi = ηα i for this value of α ∈]0, 1[.
Random walks in random environment on trees and multiplicative chaos
We analyse the renormalisation group ﬂow at the order of one loop.
2d random Dirac fermions: large N approach
The ﬁrst step of our analysis is the renormalisation group ﬂow for the effective coupling constants.
2d random Dirac fermions: large N approach
The theory of percolation, , suggests a route towards understanding the geometry of the supercritical phase, namely by developing a rigorous block renormalisation argument.
The Random-Cluster Model
One may think of block arguments as a form of rigorous renormalisation.
The Random-Cluster Model
Sur le temps local d’intersection du mouvement brownien plan et la m´ethode de renormalisation de Varadhan.
Statistics of a vortex filament model
We also develop a self-similar renormalisation group approach based on thinning out the scatterers, this scheme is similar to that used with success for diffusion in Gaussian random potentials and agrees with known exact results.
Perturbation theory for the effective diffusion constant in a medium of random scatterer
The most successful to date is the self-similar renormalisation group (RG) scheme of .
Perturbation theory for the effective diffusion constant in a medium of random scatterer
The basic idea of self-similar renormalisation group schemes is to resum the original perturbation expansions available to us in a physically motivated way.
Perturbation theory for the effective diffusion constant in a medium of random scatterer
We remark that one may verify from the results on the weak disorder perturbation theory in section 2 that the Fourier space based renormalisation group method does not reproduce the exact results for scatterering generated potentials known in one and two dimensions.
Perturbation theory for the effective diffusion constant in a medium of random scatterer
The Fourier based renormalisation group does however reproduce known exact results in one and two dimensions for Gaussian potentials .
Perturbation theory for the effective diffusion constant in a medium of random scatterer
It was noted that in the case of diffusion in a Gaussian ﬁeld a so called t-slicing renormalisation group procedure gave identical results to the Fourier space renormalisation group.
Perturbation theory for the effective diffusion constant in a medium of random scatterer
In the case of anisotropic potentials these two renormalisation group schemes give different but numerically very close results.
Perturbation theory for the effective diffusion constant in a medium of random scatterer
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