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An invariance principle for the 2d weakly self-repelling Brownian polymer

Giuseppe Cannizzaro, Harry Giles

Abstract

We investigate the large-scale behaviour of the Self-Repelling Brownian Polymer (SRBP) in the critical dimension $d=2$. The SRBP is a model of self-repelling motion, which is formally given by the solution a stochastic differential equation driven by a standard Brownian motion and with a drift given by the negative gradient of its own local time. As with its discrete counterpart, the "true" self-avoiding walk (TSAW) of [D.J. Amit, G. Parisi, & L. Peliti, Asymptotic behaviour of the "true" self-avoiding walk, Phys. Rev. B, 1983], it is conjectured to be logarithmically superdiffusive, i.e. to be such that its mean-square displacement grows as $t(\log t)^β$ for $t$ large and some currently unknown $β\in(0,1)$. The main result of the paper is an invariance principle for the SRBP under the weak coupling scaling, which corresponds to scaling the SRBP diffusively and simultaneously tuning down the strength of the self-interaction in a scale-dependent way. The diffusivity for the limiting Brownian motion is explicit and its expression provides compelling evidence that the $β$ above should be $1/2$. Further, we derive the scaling limit of the so-called environment seen by the particle process, which formally solves a non-linear singular stochastic PDE of transport-type, and prove this is given by the solution of a stochastic linear transport equation with enhanced diffusivity.

An invariance principle for the 2d weakly self-repelling Brownian polymer

Abstract

We investigate the large-scale behaviour of the Self-Repelling Brownian Polymer (SRBP) in the critical dimension . The SRBP is a model of self-repelling motion, which is formally given by the solution a stochastic differential equation driven by a standard Brownian motion and with a drift given by the negative gradient of its own local time. As with its discrete counterpart, the "true" self-avoiding walk (TSAW) of [D.J. Amit, G. Parisi, & L. Peliti, Asymptotic behaviour of the "true" self-avoiding walk, Phys. Rev. B, 1983], it is conjectured to be logarithmically superdiffusive, i.e. to be such that its mean-square displacement grows as for large and some currently unknown . The main result of the paper is an invariance principle for the SRBP under the weak coupling scaling, which corresponds to scaling the SRBP diffusively and simultaneously tuning down the strength of the self-interaction in a scale-dependent way. The diffusivity for the limiting Brownian motion is explicit and its expression provides compelling evidence that the above should be . Further, we derive the scaling limit of the so-called environment seen by the particle process, which formally solves a non-linear singular stochastic PDE of transport-type, and prove this is given by the solution of a stochastic linear transport equation with enhanced diffusivity.
Paper Structure (31 sections, 40 theorems, 94 equations)

This paper contains 31 sections, 40 theorems, 94 equations.

Table of Contents

  1. Introduction and main results
  2. Introduction and related works
  3. The model and main results
  4. Structure of the paper
  5. From the environment seen by the particle to the invariance principle
  6. The SRBP and the environment seen by the particle
  7. The Itô trick and tightness
  8. Beyond the classical conditions: Theorem \ref{['thm:3']}
  9. A good family of observables
  10. An (alternative) graded sector condition
  11. The replacement lemma and the family $\{v^{n,\lambda}\}_{n,\lambda}$
  12. Estimates for the approximate solutions
  13. Controlling the nuisance region $N$
  14. A priori estimates
  15. Proof of Proposition \ref{['p:MainEstimates']}
  16. ...and 16 more sections

Key Result

theorem 1

Under the weak coupling in eq:coup, the finite dimensional distributions of $(X^\varepsilon_t)_{t\ge0}$ converge in probability with respect to $\pi$ to those of an isotropic Brownian motion with diffusivity $\varsigma^2(\alpha)\eqdef 1+\sigma^2(\alpha)$, where $\sigma^2(\alpha)$ is given explicitly Moreover, for $T\ge0$, with respect to the annealed measure $\mathbf{P}$, we have an invariance pri

Theorems & Definitions (90)

  • theorem 1
  • remark 1
  • theorem 2
  • definition 1
  • lemma 1
  • proof
  • remark 2
  • lemma 2
  • proof
  • remark 3
  • ...and 80 more