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Discrete RG flow of a hierarchical singular SPDE

Leonard Ferdinand, Simon Gabriel

Abstract

This paper illustrates the Renormalisation Group (RG) approach to singular SPDEs following a framework introduced by Kupiainen \cite{Kupiainen2016}. We study a linear elliptic SPDE with a hierarchical Laplace operator and multiplicative noise, in two dimensions. Although this model is a significant simplification, it captures the core mechanisms of the RG method in a transparent setting. Particular emphasis is placed on the dynamical system governing the flow of the effective force coefficients, the central object of the RG method.

Discrete RG flow of a hierarchical singular SPDE

Abstract

This paper illustrates the Renormalisation Group (RG) approach to singular SPDEs following a framework introduced by Kupiainen \cite{Kupiainen2016}. We study a linear elliptic SPDE with a hierarchical Laplace operator and multiplicative noise, in two dimensions. Although this model is a significant simplification, it captures the core mechanisms of the RG method in a transparent setting. Particular emphasis is placed on the dynamical system governing the flow of the effective force coefficients, the central object of the RG method.
Paper Structure (1 section, 1 theorem, 5 equations)

This paper contains 1 section, 1 theorem, 5 equations.

Table of Contents

  1. Introduction

Key Result

Theorem 1.1

Let $d = 2$ and $r \geqslant 1$. There exists an odd $L \in \BF{N}$, sufficiently large and depending only on $r$, such that the following holds. For every $g \in (0,1]$, there exists an event $\Omega_g \subset \Omega$ such that $\BF{P}(\Omega_g) \geqslant 1 - C e^{-c g^{-2}}$. On $\Omega_g$, the SP The Hölder norm $\| \mathbin{ {$$} {$⋅$} \hbox{|_{\CB C^{1-\kappa}}$is defined in Definiti

Theorems & Definitions (1)

  • Theorem 1.1