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Liouville Brownian motion and quantum cones in dimension $d > 2$

Federico Bertacco, Ewain Gwynne

Abstract

For $d > 2$ and $γ\in (0, \sqrt{2d})$, we study the Liouville Brownian motion associated with the whole-space log-correlated Gaussian field in $\mathbb{R}^d$. We compute its spectral dimension, i.e., the short-time asymptotics of the heat kernel along the diagonal, which, in contrast to the two-dimensional case, depends on both $γ$ and on the thickness of the starting point. Furthermore, for even dimensions $d > 2$, we show that the spherical average process of the whole-space log-correlated Gaussian field in $\mathbb{R}^d$ can be identified with the integral of a stationary Gaussian Markov process of order $(d-2)/2$. Exploiting this representation, we construct the higher-dimensional analogue of the $β$-quantum cone for $β\in (-\infty, Q)$, with $Q = d/γ+ γ/2$. Lastly, for $α= Q - \sqrt{Q^2-4}$, we prove that the law of the $d$-dimensional $α$-quantum cone is invariant under shifts along the trajectories of the associated Liouville Brownian motion.

Liouville Brownian motion and quantum cones in dimension $d > 2$

Abstract

For and , we study the Liouville Brownian motion associated with the whole-space log-correlated Gaussian field in . We compute its spectral dimension, i.e., the short-time asymptotics of the heat kernel along the diagonal, which, in contrast to the two-dimensional case, depends on both and on the thickness of the starting point. Furthermore, for even dimensions , we show that the spherical average process of the whole-space log-correlated Gaussian field in can be identified with the integral of a stationary Gaussian Markov process of order . Exploiting this representation, we construct the higher-dimensional analogue of the -quantum cone for , with . Lastly, for , we prove that the law of the -dimensional -quantum cone is invariant under shifts along the trajectories of the associated Liouville Brownian motion.

Paper Structure

This paper contains 36 sections, 44 theorems, 275 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Identification of the spherical average in arbitrary even dimension
  4. Spectral dimension of the LBM in arbitrary dimension
  5. Quantum cones in arbitrary even dimension
  6. Outline and main differences with the two-dimensional case
  7. Some open problems
  8. Scaling limit of random walk on Voronoi tessellations
  9. Scaling exponent inside the exponential of the LHK
  10. Identification of the the spherical average process for the quantum cone
  11. Odd dimensions
  12. Background and preliminaries
  13. The whole-space log-correlated Gaussian field
  14. The spherical average
  15. Some properties of GMC measures
  16. ...and 21 more sections

Key Result

Theorem A

For $d > 2$ even and any $x \in \mathbb{R}^d$, the spherical average process $(h_r(x))_{r > 0}$ is $\mathfrak{c}_d$-times differentiable, where we recall eq:defcdd. Consider the processes $(\mathrm{S}_t)_{t \in \mathbb{R}}$ and $(\mathbf{S}_t)_{t \in \mathbb{R}}$ as defined in eq:defVecDer. Then the where $(\mathbf{B}_t)_{t \in R}$ denotes a $\mathfrak{c}_d$-dimensional two-sided Brownian motion,

Figures (2)

  • Figure 1: The blue (resp. green) curve represents a simulation of the process ${(\mathrm{S}_t)_{t \in \mathbb{R}}}$ in dimension $d = 4$ (resp. $d=10$), generated according to the representation given in \ref{['eq:defIntOU']}. The black curve corresponds to the driving two-sided Brownian motion $(B_t)_{t \in \mathbb{R}}$.
  • Figure 2: Subdivision of the ball $B(0, r)$ into four different regions. The first region consists in the blue ball $B(0, \abs{x}/2)$. The second region consists in the pink ball $B(x, \abs{x}/2)$. The third region consists in the green set $B(\mathop{\mathrm{mid}}\nolimits(0, x), 2 \abs{x}) \setminus B(0, \abs{x}/2) \setminus B(x, \abs{x}/2)$. The fourth region consists in the grey set $B(0, r) \setminus B(\mathop{\mathrm{mid}}\nolimits(0, x), 2 \abs{x})$.

Theorems & Definitions (105)

  • Definition 2.1
  • Theorem A: Identification of the spherical average process
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem B: Spectral dimension
  • ...and 95 more