from LQG and SLE viewpoints
Minjae Park
(University of Chicago)
Based on the joint work with Jacopo Borga and Ewain Gwynne
on various domains (sphere, disk, ...)
SLE: Schramm-Lowener evolution
LQG: Liouville quantum gravity
CLE: conformal loop ensemble
spanning tree: subgraph without any cycle spanning all the vertices
Uniform spanning tree peano curve
by R. Lyons
Critical percolation interfaces
by S. Watson
\((M_n, P_n, Γ_n)\) converges under an appropriate scaling limit to an independent triple consisting of a \(\sqrt{2}\)-LQG sphere, a whole-plane SLE\(_8\) from \(\infty\) to \(\infty\), and a whole-plane CLE\(_6\).
\(P_n\): Hamiltonian path on \(2n\) vertices
\(\Gamma_n\): Loops from two uniform arc diagrams
\(M_n\): Underlying planar map with \(2n\) vertices
\((M_n, P_n)\) is completely encoded by two independent uniform walk excursions, which converges to an independent pair of \(\sqrt{2}\)-LQG and SLE\(_8\) in the peanosphere sense
Tutte embedded Hamiltonian path on 500K points
Tutte embedded loops
on 20K and 500K points
For example, top-to-bottom open crossing in a box has probability 1/2, analogous to the critical percolation
Uniform meandric system on 2M points with the first 300 largest loops colored
Kargin conjectured the largest loop size grows like \(n^{0.8}\) purely based on simulation
\(\Delta_0\): Dimension w.r.t. Euclidean metric
\(\Delta_\gamma\): Dimension w.r.t. \(\gamma\)-LQG metric
Beffara computed the Euclidean dimension of SLE\(_6\) which equals to 7/4. Therefore, the \(\sqrt{2}\)-LQG dimension of CLE\(_6\) should be
For each fixed \(k\in\mathbb N\), \(\# \{\text{vertices of $k$th largest loop}\} = n^{\alpha + o(1)}\) with probability tending to \(1\) as \(n\to \infty\).
There is a.s. no infinite path in the uniform infinite meandric system.
Consider an infinite volume version of the uniform meandric system via Benjamini-Schramm local limit, studied in [CKST], [FT]
[CKST] proved that there is at most one infinite path in the uniform infinite meandric system.
\((M, P, Γ)\) converges under an appropriate scaling limit to an independent triple consisting of a \(\sqrt{2}\)-LQG cone, a whole-plane SLE\(_8\) from \(\infty\) to \(\infty\), and a whole-plane CLE\(_6\).
It is clear that this conjecture implies the non-existence of infinite path in the uniform infinite meandric system.
\(\sqrt{2}\)-LQG cone
Recall that [DMS] provides a natural way to define a quantum surface with half-plane topology by cutting a whole-plane quatum surface with an independent SLE
✂️ whole-plane SLE\(_2\)
from 0 to \(\infty\)
\(\sqrt{2}\)-LQG wedge
There exists a.s. no infinite path in the uniform infinite half-plane meandric system. Furthermore, \(\frac12\) is diconnected from \(+\infty\) by infinitely many loops.
Proof is based on the Burton-Keane type argument.
Define the uniform (infinite) half-plane meandric system by cutting the bottom arcs of uniform meandric system with the ray \(\{\frac{1}{2}\}\times (-\infty,0)\)
The graph distance diameter of \(M_n\) grows like \(n^{\frac{1}{d}-o(1)}\) as \(n\to \infty\).
\(d = \) Hausdorff dimension of \(\sqrt{2}\)-LQG satisfying \(3.55\le d\le 3.63\), so \(1/d\approx 0.28\)
The graph distance diameter of the largest loop grows like \(n^{\frac{1}{d}-o(1)}\) as \(n\to \infty\) which is of the same order with the entire map.
Therefore, the size of largest loop is at least \(n^{1/d-o(1)}\), not optimal at all compared to \(n^\alpha\), but much better than the trivial lower bound \(O(\log n)\) stated in Kargin.
Analogous to [CGHP] and [DCGPS]
Macroscopic loops in \(O(n)\) loop model
For each \(\alpha > 0\), there exists \(A(\alpha) > 0\) and a coupling of \((L,R)\) with \((\mathcal L , \mathcal R)\) such that the following is true with probability \(1-O(n^{-\alpha})\):
\[A^{-1} (\log n)^{-3} \text{dist}_{\mathcal G}(x , y) \leq \text{dist}_{\mathcal M}(x , y) \leq A (\log n)^3 \text{dist}_{\mathcal G}(x , y).\]
\((L,R)\): two independent random walks encoding the uniform meandric system \(\mathcal M\)
\((\mathcal{L}, \mathcal{R})\): two independent Brownian motions encoding the mated-CRT map \(\mathcal{G}\)
Then, compare the mated-CRT map with LQG, where plenty of tools and standard LQG/SLE techniques are available.
Concatenation of boundary loops on 18M points,
whose scaling limit should be SLE\(_6\)