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3 Citations
Résumé:

In (J. Stat. Phys. 115:415-449, 2004) Brydges, Guadagni and Mitter proved the existence of multiscale expansions of a class of lattice Green's functions as sums of positive definite finite range functions (called fluctuation covariances). The lattice Green's functions in the class considered are integral kernels of inverses of second order positive self-adjoint elliptic operators with constant coefficients and fractional powers thereof. The rescaled fluctuation covariance in the nth term of the expansion lives on a lattice with spacing L −n and satisfies uniform bounds. Our main result in this note is that the sequence of these terms converges in appropriate norms at a rate L −n/2 to a smooth, positive definite, finite range continuum function.

Commentaires: 14 pages

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In this talk I will show that a large class of Gaussian Random Fields in the continuum or the lattice can be written as a sum of independent Gaussian random fields called fluctuation fields which enjoy the following properties: their covariances have finite range (compact support) and the fields are almost surely smooth. The fluctuation covariances satisfy very strong uniform bounds . After suitable rescaling the sequence of fluctuation fields converges in distribution to a a smooth continuum Gaussian random field whose covariance has finite range. This finite range multiscale expansion is the basis of a new mathematical form of Wilson's Renormalization Group where non local effects are minimized and estimates rendered simpler. In particular, on the lattice, this gives an alternative to the Kadanoff-Wilson renormalization group based on the block spin transformation. The talk is based on my joint work with D. Brydges and G. Guadagni (J. Stat.Phys. 115,415-449 (2004)) and a further paper with D. Brydges (2011, in preparation).

Commentaires: International Workshop at the Erwin Schroedinger International Institute for Mathematical Physics, University of Vienna.

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I give a review of some rigorous results on self avoiding walks in cubic and hypercubic lattices, including the case of self-avoiding Levy walks. I then explain the connection with supersymmetric field theory and go on to give an exposition of rigorous renormalization group analysis of the supersymmetric measure in the particular case of self avoiding Levy walks below the critical dimension. The talk is based on my joint work with D. Brydges and G. Guadagni (J.Stat.Phys. 115, 415-449 (2004), further work with D.Brydges (2011, in preparation) on finite range multiscale expansions. The renormalization group analysis of the supersymmetric measure based on the above expansions is carried out in joint work with B. Scoppola (J.Stat.Phys. 133, 921-1011 (2008)).

Commentaires: International workshop held at the Centro de Ciencias de Benasque Pedro Pascual, Benasque, Espagne

Ref HAL: hal-00417638_v1
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The Green's function of a weakly self-avoiding Levy walk with long range jumps in a large but finite cube in Z^3 can be expressed as the two point correlation function in a supersymmetric field theory. We have proved the global existence of the renormalization group trajectory of the underlying supersymmetric measure at all renormalization group scales . We establish the existence of the critical (stable) manifold and prove that the interactions are bounded away from zero on all scales. This is a step in a program to study rigorously the critical exponents of a self- avoiding Levy walk. Based on joint work with Benedetto Scoppola published in J.Stat Phys (2008) 133:921-1011.

Ref HAL: hal-00378134_v1
DOI: 10.1007/s10955-008-9626-8
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19 Citations
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We consider an Euclidean supersymmetric field theory in $\math{Z}^{3}$ given by a supersymmetric $\Phi^{4}$ perturbation of an underlying massless Gaussian measure on scalar bosonic and Grassmann fields with covariance the Green's function of a (stable) L\'evy random walk in $\math{Z}^{3}$. The Green's function depends on the L\'evy-Khintchine parameter $\a={3+\e\over 2}$ with $0<\a<2$. For $\a ={3\over 2}$ the $\Phi^{4}$ interaction is marginal. We prove for $\a-{3\over 2}={\e\over 2}>0$ sufficiently small and initial parameters held in an appropriate domain the existence of a global renormalization group trajectory uniformly bounded on all renormalization group scales and therefore on lattices which become arbitrarily fine. At the same time we establish the existence of the critical (stable) manifold. The interactions are uniformly bounded away from zero on all scales and therefore we are constructing a non-Gaussian supersymmetric field theory on all scales. The interest of this theory comes from the easily established fact that the Green's function of a (weakly) self-avoiding L\'evy walk in $\math{Z}^{3}$ is a second moment (two point correlation function) of the supersymmetric measure governing this model. The rigorous control of the critical renormalization group trajectory is a preparation for the study of the critical exponents of the (weakly) self-avoiding L\'evy walk in $\math{Z}^{3}$.

Ref HAL: hal-00286465_v1
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We consider an Euclidean supersymmetric field theory in $\math{Z}^{3}$ given by a supersymmetric $\Phi^{4}$ perturbation of an underlying massless Gaussian measure on scalar bosonic and Grassmann fields with covariance the Green's function of a (stable) L\'evy random walk in $\math{Z}^{3}$. The Green's function depends on the L\'evy-Khintchine parameter $\a={3+\e\over 2}$ with $0<\a<2$. For $\a ={3\over 2}$ the $\Phi^{4}$ interaction is marginal. We prove for $\a-{3\over 2}={\e\over 2}>0$ sufficiently small and initial parameters held in an appropriate domain the existence of a global renormalization group trajectory uniformly bounded on all renormalization group scales and therefore on lattices which become arbitrarily fine. At the same time we establish the existence of the critical (stable) manifold. The interactions are uniformly bounded away from zero on all scales and therefore we are constructing a non-Gaussian supersymmetric field theory on all scales. The interest of this theory comes from the easily established fact that the Green's function of a (weakly) self-avoiding L\'evy walk in $\math{Z}^{3}$ is a second moment (two point correlation function) of the supersymmetric measure governing this model. The rigorous control of the critical renormalization group trajectory is a preparation for the study of the critical exponents of the (weakly) self-avoiding L\'evy walk in $\math{Z}^{3}$.

Ref HAL: hal-00286461_v1
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We report on the rigorous construction of an analogue of the Wilson-Fisher fixed point in three dimensions. The model corresponds to a perturbation by a $\phi^4$ interaction of a Gaussian measure on scalar fields with a covariance depending on a real parameter $\epsilon$ in the range $0\le \epsilon \le 1$. For $\epsilon =1$ one recovers the covariance of a massless scalar field in ${\bf R}^3$. For $\epsilon =0$, $\phi^{4}$ is a marginal interaction. For $0\le \epsilon < 1$ the covariance continues to be Osterwalder-Schrader and pointwise positive. We consider the infinite volume critical theory with a fixed ultraviolet cutoff at the unit length scale and we prove that for $\epsilon > 0$, sufficiently small, there exists a non-gaussian fixed point (with one unstable direction) of the Renormalization Group iterations. We construct the stable critical manifold near this fixed point and prove that under Renormalization Group iterations the critical theories converge to the fixed point.