[PPT] - 3 pictures to keep in mind 1 Tree to Excursion Trace around the PowerPoint Presentation

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3 pictures to keep in mind

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Tree to Excursion

Trace around the tree starting from root. Go up if seeing new edge, go down if seeing edge that’s already been seen.

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Tree to Excursion

Trace around the tree starting from root. Go up if seeing new edge, go down if seeing edge that’s already been seen.

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Tree to Excursion

Trace around the tree starting from root. Go up if seeing new edge, go down if seeing edge that’s already been seen.

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Tree to Excursion

Trace around the tree starting from root. Go up if seeing new edge, go down if seeing edge that’s already been seen.

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Tree to Excursion

Trace around the tree starting from root. Go up if seeing new edge, go down if seeing edge that’s already been seen.

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Tree to Excursion

Trace around the tree starting from root. Go up if seeing new edge, go down if seeing edge that’s already been seen.

2

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Tree to Excursion

Trace around the tree starting from root. Go up if seeing new edge, go down if seeing edge that’s already been seen.

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Tree to Excursion

Trace around the tree starting from root. Go up if seeing new edge, go down if seeing edge that’s already been seen.

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Tree to Excursion

Trace around the tree starting from root. Go up if seeing new edge, go down if seeing edge that’s already been seen.

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Tree to Excursion

Trace around the tree starting from root. Go up if seeing new edge, go down if seeing edge that’s already been seen.

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Tree to Excursion

Trace around the tree starting from root. Go up if seeing new edge, go down if seeing edge that’s already been seen.

2

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Tree to Excursion

Trace around the tree starting from root. Go up if seeing new edge, go down if seeing edge that’s already been seen.

2

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Tree to Excursion

Trace around the tree starting from root. Go up if seeing new edge, go down if seeing edge that’s already been seen.

2

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Tree to Excursion

Trace around the tree starting from root. Go up if seeing new edge, go down if seeing edge that’s already been seen.

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Tree to Excursion

Trace around the tree starting from root. Go up if seeing new edge, go down if seeing edge that’s already been seen.

2

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Tree to Excursion

Trace around the tree starting from root. Go up if seeing new edge, go down if seeing edge that’s already been seen.

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Tree to Excursion

Trace around the tree starting from root. Go up if seeing new edge, go down if seeing edge that’s already been seen.

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Excursion to Tree

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Excursion to Tree

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Excursion to Tree

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Excursion to Tree

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Excursion to Tree

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Excursion to Tree

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Excursion to Lamination

x ∼ y ⇐ ⇒ inf

[x,y] ❡ = ❡(x) = ❡(y) 4

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Excursion to Lamination

x ∼ y ⇐ ⇒ inf

[x,y] ❡ = ❡(x) = ❡(y) 4

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Excursion to Lamination

x ∼ y ⇐ ⇒ inf

[x,y] ❡ = ❡(x) = ❡(y) 4

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Theorem Statement

Let fn : D∗ → C be the solution to the welding problem for a uniformly random plane tree. Then as n → ∞, fn converges in distribution (w.r.t uniform convergence on ∂D) to a random map f .

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Proof Sketch

Convergence ←Equicontinuity/Tightness ←Estimate diam(f (arc)) ≈Show that each edge is small ←Find lots of thick annuli seperating edge from infinity

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Proof Sketch

Need to find lots of thick annuli to bound diamater of f (arc). Create annuli by joining points by geodesics (red).

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Proof Sketch

Need to find lots of thick annuli to bound diamater of f (arc). Create annuli by joining points by geodesics (red).

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Proof Sketch

Need to find lots of thick annuli to bound diamater of f (arc). Create annuli by joining points by geodesics (red).

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Proof Sketch

Need to find lots of thick annuli to bound diamater of f (arc). Create annuli by joining points by geodesics (red).

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Wishlist to make thick annuli

Conditions to ensure annulus is thick after welding?

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Wishlist to make thick annuli

1. Need bounded number of rectangles.

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Wishlist to make thick annuli

2. Need bounded geometry for rectangles.

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Wishlist to make thick annuli

3. Need to know something about the welding map.

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How to ensure that rectangle is still thick after welding

3. Need to know something about the equivalence relation.

If we can control the quality of the equivalence relation on a large subset of the edge then we get control on the modulus.

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How to ensure that rectangle is still thick after welding

If we can control the quality of the equivalence relation on a large subset of the edge then we get control on the modulus. It’s not enough that the sets on each side are large. The equivalence relation should also behave nicely.

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How to ensure that rectangle is still thick after welding

If we can control the quality of the equivalence relation on a large subset of the edge then we get control on the modulus. It’s not enough that the sets on each side are large. The equivalence relation should also behave nicely.

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How to ensure that rectangle is still thick after welding

If we can control the quality of the equivalence relation on a large subset of the edge then we get control on the modulus. It’s not enough that the sets on each side are large. The equivalence relation should also behave nicely.

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Modulus and Gluing of rectangles

Lemma We have L inf

µ−,µ+ E(µ−) + E(µ+).

µ− ranges over probability measures on I − ∩ supp(∼)
µ+ ranges over probability measures on I + ∩ supp(∼)
µ− and µ+ must be be equivalent with respect to ∼.
Energy:

E(µ) =

log

1 |x − y|dµ(x)dµ(y).

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How to use this lemma for our problem:

Consider a toy model where we glue two squares together using the equivalence relation from a random excursion. What should we take for µ− and µ+?

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How to use this lemma for our problem:

Consider a toy model where we glue two squares together using the equivalence relation from a random excursion. What should we take for µ− and µ+? Notice that the support of ∼ is exactly the images of the left sided and right sided inverse map of the excursion ❡ on [0, ❡(1/2)].

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How to use this lemma for our problem:

Consider a toy model where we glue two squares together using the equivalence relation from a random excursion. What should we take for µ− and µ+? Notice that the support of ∼ is exactly the images of the left-sided and right-sided inverse map of the excursion ❡ on [0, ❡(1/2)].

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How to use this lemma for our problem:

Thus we should take µ−, µ+ to be pullback of Lebesgue measure

n [0, ❡(1/2)] via ❡.

This demonstrates that the modulus L is controlled by the H¨

lder

regularity of ❡.

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Wishlist to make thick annuli

To get thick annuli, want

1. Bounded number of rectangles
2. Bounded geometry rectangles
3. Control over ∼ on interface ( = regularity of excursion)

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Finding lots of thick annuli

Now want to find lots of configurations that satisfy the conditions

f our list.

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How to construct the annuli

Fix λ > 1. Start with large finite tree T with the graph distance. Let Xk be the points on T which are distance λk from the root. Join these points with geodesics in the complement of the tree.

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How to construct the annuli

Fix λ > 1. Start with large finite tree T with the graph distance. Let Xk be the points on T which are distance λk from the root. Join these points with geodesics in the complement of the tree.

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How to construct the annuli

Fix λ > 1. Start with large finite tree T with the graph distance. Let Xk be the points on T which are distance λk from the root. Join these points with geodesics in the complement of the tree.

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How to construct the annuli

Fix λ > 1. Start with large finite tree T with the graph distance. Let Xk be the points on T which are distance λk from the root. Join these points with geodesics in the complement of the tree.

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How to construct the annuli

Fix λ > 1. Start with large finite tree T with the graph distance. Let Xk be the points on T which are distance λk from the root. Join these points with geodesics in the complement of the tree.

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How to construct the annuli

Fix λ > 1. Start with large finite tree T with the graph distance. Let Xk be the points on T which are distance λk from the root. Join these points with geodesics in the complement of the tree.

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How to construct the annuli

Fix λ > 1. Start with large finite tree T with the graph distance. Let Xk be the points on T which are distance λk from the root. Join these points with geodesics in the complement of the tree.

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How to construct the annuli

Fix λ > 1. Start with large finite tree T with the graph distance. Let Xk be the points on T which are distance λk from the root. Join these points with geodesics in the complement of the tree.

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Rest of the proof

Analyze this exploration process via excursion picture, show

that w.h.p. on most scales k the list of conditions is satisfied.

Borel-Cantelli + Union bound =

⇒ H¨

lder equicontinuity of

welding map

Uniqueness of limit from Jones-Smirnov removability theorem.

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