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The ant on loops: Alexander-Orbach conjecture for the critical level set of the Gaussian free field

Shirshendu Ganguly, Kyeongsik Nam

Abstract

Alexander and Orbach (AO) in 1982 conjectured that the simple random walk on critical percolation clusters (also known as the ant in the labyrinth) in Euclidean lattices exhibit mean field behavior; for instance, its spectral dimension is $4/3$. While false in low dimensions, this is expected to be true above the upper critical dimension of six. First rigorous results in this direction go back to Kesten who verified this on the tree. After many developments, in a breakthrough work, Kozma and Nachmias [KN] established the AO conjecture for bond percolation on $\mathbb{Z}^d$ for $d>19$ and $d>6$ for the spread out lattice. We investigate the validity of the AO conjecture for the critical level set of the Gaussian Free Field (GFF), a canonical dependent percolation model of central importance. In an influential work, Lupu proved that for the cable graph of $\mathbb{Z}^d$ (which is obtained by also including the edges), the signed clusters of the associated GFF are given by the corresponding clusters in a Poisson loop soup, thus reducing the analysis to the study of the latter. In 2021, Werner put forth an evocative picture for the critical behavior in this setting drawing an analogy with the usual bond case. Building on this, Cai and Ding established the universality of the extrinsic one arm exponent for all $d > 6.$ In this article, we carry this program further, and consider the random walk on sub-sequential limits of the cluster of the origin conditioned to contain far away points. These form candidates for the Incipient Infinite Cluster (IIC), first introduced by Kesten in the planar case. Inspired by the program of [KN], introducing several novel ideas to tackle the long range nature of this model and its effect on the intrinsic geometry of the percolation cluster, we establish that the AO conjecture indeed holds for any sub-sequential IIC, for all large enough dimensions.

The ant on loops: Alexander-Orbach conjecture for the critical level set of the Gaussian free field

Abstract

Alexander and Orbach (AO) in 1982 conjectured that the simple random walk on critical percolation clusters (also known as the ant in the labyrinth) in Euclidean lattices exhibit mean field behavior; for instance, its spectral dimension is . While false in low dimensions, this is expected to be true above the upper critical dimension of six. First rigorous results in this direction go back to Kesten who verified this on the tree. After many developments, in a breakthrough work, Kozma and Nachmias [KN] established the AO conjecture for bond percolation on for and for the spread out lattice. We investigate the validity of the AO conjecture for the critical level set of the Gaussian Free Field (GFF), a canonical dependent percolation model of central importance. In an influential work, Lupu proved that for the cable graph of (which is obtained by also including the edges), the signed clusters of the associated GFF are given by the corresponding clusters in a Poisson loop soup, thus reducing the analysis to the study of the latter. In 2021, Werner put forth an evocative picture for the critical behavior in this setting drawing an analogy with the usual bond case. Building on this, Cai and Ding established the universality of the extrinsic one arm exponent for all In this article, we carry this program further, and consider the random walk on sub-sequential limits of the cluster of the origin conditioned to contain far away points. These form candidates for the Incipient Infinite Cluster (IIC), first introduced by Kesten in the planar case. Inspired by the program of [KN], introducing several novel ideas to tackle the long range nature of this model and its effect on the intrinsic geometry of the percolation cluster, we establish that the AO conjecture indeed holds for any sub-sequential IIC, for all large enough dimensions.
Paper Structure (37 sections, 42 theorems, 323 equations, 15 figures)

This paper contains 37 sections, 42 theorems, 323 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Models
  3. Main result
  4. Idea of proofs
  5. Organization of the article
  6. Notations
  7. Acknowledgement
  8. Preliminaries
  9. Loops on ${\mathbb{Z}}^d$ and on $\widetilde{\mathbb{Z}}^d$
  10. Discrete-time loops on $\mathbb{Z}^d$
  11. Continuous-time loops on $\mathbb{Z}^d$
  12. Continuous-time loops on $\widetilde{\mathbb{Z}}^d$
  13. Loop soup
  14. Isomorphism theorem
  15. BKR inequality
  16. ...and 22 more sections

Key Result

Theorem 1

Let $d>20$ and consider any sub-sequential limit $\mathbb{P}_{\textup{IIC}}$ of the measures $\nu^x$ defined above in limitiic and a graph $\mathcal{G}$ sampled from it. Let $\{X_i\}_{i\ge 1}$ be the discrete time simple random walk on $\mathcal{G}$. Then $\mathbb{P}_{\textup{IIC}}$-a.s., Above, $\tau_r$ is the hitting time of (intrinsic) distance $r$ from the origin (the expectation ${\mathbb{E}

Figures (15)

  • Figure 1: The two paths $0\leftrightarrow z$ and $0 \leftrightarrow x$ last meet at $w$ leading to $0\overset{r}\leftrightarrow w \circ w\overset{r}\leftrightarrow z \circ {w\leftrightarrow x}.$
  • Figure 2: The chain of loops $0\leftrightarrow z$ and $0\leftrightarrow x$ last intersect at $\Gamma$ from which emanate three disjoint arms none of which use $\Gamma$ leading to $0\leftrightarrow w_1 \circ w_2 \leftrightarrow z \circ w_3\leftrightarrow x \circ \{\Gamma \in \mathcal{L}\}$
  • Figure 3: While $\Gamma$ is the last loop on the path $0\leftrightarrow x$ which intersects the path $0\leftrightarrow z,$ and hence the arm $w_3 \leftrightarrow x$ is disjoint from everything else, including $\Gamma,$ owing to the intrinsic constraint of the type $0\overset{r}\leftrightarrow z$, the path certifying the latter might use edges from another loop $\Gamma'$ both before and after intersecting $\Gamma$ making the corresponding arms not disjoint.
  • Figure 4: On the left, the figure shows the $\textup{BLE}$$(\Gamma_1, \Gamma_2, \Gamma_3, u_1,u_2,u_3)$ along with the points $v_1,v_2,v_3$ from which emanate disjoint arms to $x,0,z$ respectively. On the right, the connections between the $\textup{BLE}$ loops are tree expanded extrinsically around $\Gamma$ leading to the points $w_1,w_2, w_3$ wich are connected to $u_1,u_2, u_3$ disjointly without using any of $\Gamma_1,\Gamma_2,\Gamma_3$ or $\Gamma.$
  • Figure 5: The brown curve denotes the boundary of $B(0,r),$ and there could be large loops (denoted by black) which extends from within the interior of $B(0,r)$ to much further.
  • ...and 10 more figures

Theorems & Definitions (88)

  • Theorem 1
  • Proposition 2.1: Proposition 2.1 in lupu
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4: lupu
  • Lemma 2.5: fkg2
  • Lemma 2.6: BKR inequality
  • Lemma 2.7
  • Proposition 2.8: Proposition 6.6 in cd
  • ...and 78 more