Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Upper bounds on the fluctuations for a class of degenerate convex $\nabla φ$-interface models

Paul Dario

Abstract

We derive upper bounds on the fluctuations of a class of random surfaces of the $\nabla φ$-type with convex interaction potentials. The Brascamp-Lieb concentration inequality provides an upper bound on these fluctuations for uniformly convex potentials. We extend these results to twice continuously differentiable convex potentials whose second derivative grows asymptotically like a polynomial and may vanish on an (arbitrarily large) interval. Specifically, we prove that, when the underlying graph is the $d$-dimensional torus of side length $L$, the variance of the height is smaller than $C \ln L$ in two dimensions and remains bounded in dimension $d \geq 3$. The proof makes use of the Helffer-Sjöstrand representation formula (originally introduced by Helffer and Sjöstrand (1994) and used by Naddaf and Spencer (1997) and Giacomin, Olla Spohn (2001) to identify the scaling limit of the model), the anchored Nash inequality (and the corresponding on-diagonal heat kernel upper bound) established by Mourrat and Otto (2016) and Efron's monotonicity theorem for log-concave measures (Efron (1965)).

Upper bounds on the fluctuations for a class of degenerate convex $\nabla φ$-interface models

Abstract

We derive upper bounds on the fluctuations of a class of random surfaces of the -type with convex interaction potentials. The Brascamp-Lieb concentration inequality provides an upper bound on these fluctuations for uniformly convex potentials. We extend these results to twice continuously differentiable convex potentials whose second derivative grows asymptotically like a polynomial and may vanish on an (arbitrarily large) interval. Specifically, we prove that, when the underlying graph is the -dimensional torus of side length , the variance of the height is smaller than in two dimensions and remains bounded in dimension . The proof makes use of the Helffer-Sjöstrand representation formula (originally introduced by Helffer and Sjöstrand (1994) and used by Naddaf and Spencer (1997) and Giacomin, Olla Spohn (2001) to identify the scaling limit of the model), the anchored Nash inequality (and the corresponding on-diagonal heat kernel upper bound) established by Mourrat and Otto (2016) and Efron's monotonicity theorem for log-concave measures (Efron (1965)).
Paper Structure (38 sections, 17 theorems, 257 equations)

This paper contains 38 sections, 17 theorems, 257 equations.

Table of Contents

  1. Introduction
  2. Outline of the proof
  3. The Helffer-Sjöstrand representation formula
  4. The on-diagonal heat kernel upper bound in degenerate environment of Mourrat and Otto
  5. A fluctuation estimate for the Langevin dynamic and stochastic integrability of the moderated environment
  6. Extension of the argument to the potentials satisfying Assumption \ref{['assumption1']}
  7. Discussion and background
  8. Random surfaces
  9. Parabolic equations with degenerate random coefficients and the random conductance model
  10. Further comments and perspective
  11. Organization of the paper
  12. Convention for constants and exponents
  13. Notation and preliminary results
  14. General notation
  15. Functions
  16. ...and 23 more sections

Key Result

Theorem 1.2

Under Assumption assumption1, there exists a constant $C := C(d , V) < \infty$ such that, for any $L \geq 2$,

Theorems & Definitions (39)

  • Theorem 1.2: Localization and Delocalization
  • Remark 1.3
  • Remark 1.4
  • Remark 2.1
  • Remark 2.2
  • Definition 2.3: Langevin dynamic in the torus
  • Proposition 2.4: Helffer-Sjöstrand representation formula on the torus
  • Theorem 2.5: Efron's monotonicity theorem Efron1965
  • Proposition 2.6: discrete Gagliardo-Nirenberg-Sobolev inequality on the torus
  • Remark 2.7
  • ...and 29 more