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Finite-cutoff JT gravity and self-avoiding loops

Douglas Stanford, Zhenbin Yang

TL;DR

This work presents a finite-cutoff formulation of JT gravity by mapping the disk path integral to the statistical mechanics of a self-avoiding loop in hyperbolic space with positive pressure $p$ and fixed renormalized length $\upbeta$. In the semiclassical limit, three regimes across loop size are solved: a flat-space RGJ regime for small loops, an intermediate large-$p$ regime described by an entropic-tension effective theory for moderately large loops, and a Schwarzian regime for very large loops, with explicit expressions for the density of states and the partition function in each regime and smooth matching between them. The flat-space analysis uses the RGJ conjecture to yield an exact density of states in terms of Airy functions, while the Schwarzian regime reproduces the known Schwarzian density of states in the appropriate limit, clarifying how JT gravity emerges as an effective IR theory. The results illuminate the finite-cutoff structure of quantum JT gravity, expose differences from classical JT gravity, and clarify the nontrivial relation between UV (renormalized length) and IR (boundary length) notions of length in this setting.

Abstract

We study quantum JT gravity at finite cutoff using a mapping to the statistical mechanics of a self-avoiding loop in hyperbolic space, with positive pressure and fixed length. The semiclassical limit (small $G_N$) corresponds to large pressure, and we solve the problem in that limit in three overlapping regimes that apply for different loop sizes. For intermediate loop sizes, a semiclassical effective description is valid, but for very large or very small loops, fluctuations dominate. For large loops, this quantum regime is controlled by the Schwarzian theory. For small loops, the effective description fails altogether, but the problem is controlled using a conjecture from the theory of self-avoiding walks.

Finite-cutoff JT gravity and self-avoiding loops

TL;DR

This work presents a finite-cutoff formulation of JT gravity by mapping the disk path integral to the statistical mechanics of a self-avoiding loop in hyperbolic space with positive pressure and fixed renormalized length . In the semiclassical limit, three regimes across loop size are solved: a flat-space RGJ regime for small loops, an intermediate large- regime described by an entropic-tension effective theory for moderately large loops, and a Schwarzian regime for very large loops, with explicit expressions for the density of states and the partition function in each regime and smooth matching between them. The flat-space analysis uses the RGJ conjecture to yield an exact density of states in terms of Airy functions, while the Schwarzian regime reproduces the known Schwarzian density of states in the appropriate limit, clarifying how JT gravity emerges as an effective IR theory. The results illuminate the finite-cutoff structure of quantum JT gravity, expose differences from classical JT gravity, and clarify the nontrivial relation between UV (renormalized length) and IR (boundary length) notions of length in this setting.

Abstract

We study quantum JT gravity at finite cutoff using a mapping to the statistical mechanics of a self-avoiding loop in hyperbolic space, with positive pressure and fixed length. The semiclassical limit (small ) corresponds to large pressure, and we solve the problem in that limit in three overlapping regimes that apply for different loop sizes. For intermediate loop sizes, a semiclassical effective description is valid, but for very large or very small loops, fluctuations dominate. For large loops, this quantum regime is controlled by the Schwarzian theory. For small loops, the effective description fails altogether, but the problem is controlled using a conjecture from the theory of self-avoiding walks.

Paper Structure

This paper contains 19 sections, 112 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Brief review of JT gravity
  3. JT gravity and the self-avoiding loop measure
  4. Flat space JT gravity
  5. Ordinary (hyperbolic) JT gravity
  6. The small beta limit
  7. The flat space regime
  8. The intermediate regime
  9. Classical computations
  10. One-loop computations
  11. The Schwarzian regime
  12. Discussion
  13. A free particle analogy
  14. JT gravity as an effective description of JT gravity
  15. Deriving the density of states from the RGJ formula
  16. ...and 4 more sections

Figures (7)

  • Figure 1: The three approximations used in this paper are valid well inside the respective shaded regions. The horizontal axis is the renormalized length of the loop, and the vertical axis is the pressure (or equivalently the boundary value of the JT gravity dilaton).
  • Figure 2: Cutouts from the hyperbolic disk, bounded by a simple closed curve $\gamma$. Typical curves will be very wiggly at short distance scales, as shown at right.
  • Figure 3: Random triangulations give a discrete regularization of the path integral over 2d metrics. All triangles are taken to be equilateral, and curvature is localized at the vertices. Positive curvature means fewer than six edges meeting at a vertex, and negative curvature means more than six. A discretization of flat-space JT gravity could be defined as a sum over triangulations in which each vertex meets exactly six edges. This reduces to a sum over boundary shapes on the triangular lattice (right).
  • Figure 4: An ordinary random walk (left) and a self-avoiding walk (right) of $10^4$ steps each.
  • Figure 5: A plot of the function $[\text{Ai}(-E)^2 + \text{Bi}(-E)^2]^{-1}$. The region shown connects the behavior $\sqrt{E}$ for large positive $E$ to the behavior $e^{-\frac{4}{3}|E|^{3/2}}$ for large negative $E$.
  • ...and 2 more figures