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THE SCALING LIMIT OF
RANDOM OUTERPLANAR MAPS
ALESSANDRA CARACENI
Scuola Normale Superiore
Université Paris Sud
THE SCALING LIMIT OF
RANDOM OUTERPLANAR MAPS
ALESSANDRA CARACENI
Scuola Normale Superiore
Université Paris Sud
Journées Aléa
19 mars 2015
OUTERPLANAR MAPS
OUTERPLANAR MAPS
outerface
• All of the vertices
belong to a single face
OUTERPLANAR MAPS
outerface
• All of the vertices
belong to a single face
• The map is simple
OUTERPLANAR MAPS
outerface
• All of the vertices
belong to a single face
• The map is simple
• The map is rooted
OUTERPLANAR MAPS
Let Mn be a (uniform)
random outerplanar
map with n vertices.
n = 24
OUTERPLANAR MAPS
Let Mn be a (uniform)
random outerplanar
map with n vertices.
What is the scaling
limit for Mn? What is
the appropriate
“scaling” factor?
n = 24
OUTERPLANAR MAPS
Let Mn be a (uniform)
random outerplanar
map with n vertices.
What is the scaling
limit for Mn? What is
the appropriate
“scaling” factor?
THE SCALING LIMIT OF PLANE TREES
root
THE SCALING LIMIT OF PLANE TREES
root
A plane tree is a
(rooted) map with
only one face.
6 vertices
5 edges
THE SCALING LIMIT OF PLANE TREES
t6
Let tn be a (uniform)
random plane tree
with n vertices.
THE SCALING LIMIT OF PLANE TREES
t6
Let tn be a (uniform)
random plane tree
with n vertices.
We can see (tn, d/n1/2) as a random
metric space (a measure on the
space of compact m. s. (X, dGH)).
THE SCALING LIMIT OF PLANE TREES
...
t2
t3
t4
We can see (tn, d/n1/2) as a random
metric space (a measure on the
space of compact m. s. (X, dGH)).
t5
t6
What is the weak limit of
the sequence (tn, d/n1/2) if
we let n go to infinity?
2
THE SCALING LIMIT OF PLANE TREES
...
t3
t4
t5
We can see (tn, d/n1/2) as a random
metric space (a measure on the
space of compact m. s. (X, dGH)).
t6
t20
What is the weak limit of
the sequence (tn, d/n1/2) if
we let n go to infinity?
THE SCALING LIMIT OF PLANE TREES
...
t4
t5
t6
We can see (tn, d/n1/2) as a random
metric space (a measure on the
space of compact m. s. (X, dGH)).
...
t20
t200
What is the weak limit of
the sequence (tn, d/n1/2) if
we let n go to infinity?
THE SCALING LIMIT OF PLANE TREES
...
t4
t5
t6
...
t20
t200
THE CRT
continuum random tree
We can see (tn, d/n1/2) as a random
metric space (a measure on the
space of compact m. s. (X, dGH)).
What is the weak limit of
the sequence (tn, d/n1/2) if
we let n go to infinity?
THE SCALING LIMIT OF PLANE TREES
THE CRT
continuum random tree
We can see (tn, d/n1/2) as a random
metric space (a measure on the
space of compact m. s. (X, dGH)).
What is the weak limit of
the sequence (tn, d/n1/2) if
we let n go to infinity?
THE BGH BIJECTION
THE BGH BIJECTION
A spanning tree of a
graph is a sub-graph which
is a tree and spans all of its
vertices.
THE BGH BIJECTION
A spanning tree of a
graph is a sub-graph which
is a tree and spans all of its
vertices.
Build a spanning tree by recursively
adding edges (start with those
around the root, proceed depth-first
and clockwise) unless they would
make a cycle.
THE BGH BIJECTION
A spanning tree of a
graph is a sub-graph which
is a tree and spans all of its
vertices.
x
x
x
x
x
x
Build a spanning tree by recursively
adding edges (start with those
around the root, proceed depth-first
and clockwise) unless they would
make a cycle.
x
x
x
THE BGH BIJECTION
A spanning tree of a
graph is a sub-graph which
is a tree and spans all of its
vertices.
x
x
x
x
Colour the “left” vertex of
each deleted edge.
x
x
Build a spanning tree by recursively
adding edges (start with those
around the root, proceed depth-first
and clockwise) unless they would
make a cycle.
x
x
x
THE BGH BIJECTION
A spanning tree of a
graph is a sub-graph which
is a tree and spans all of its
vertices.
x
x
x
x
Colour the “left” vertex of
each deleted edge.
x
x
Build a spanning tree by recursively
adding edges (start with those
around the root, proceed depth-first
and clockwise) unless they would
make a cycle.
x
x
x
Notice that the last
branch (around the root,
clockwise) cannot have
any coloured vertices.
THE BGH BIJECTION
A spanning tree of a
graph is a sub-graph which
is a tree and spans all of its
vertices.
Build a spanning tree by recursively
adding edges (start with those
around the root, proceed depth-first
and clockwise) unless they would
make a cycle.
Colour the “left” vertex of
each deleted edge.
Notice that the last
branch (around the root,
clockwise) cannot have
any coloured vertices.
THE BGH BIJECTION
THE BGH BIJECTION
THE BGH BIJECTION
THE BGH BIJECTION
Join each coloured
vertex v to the first
vertex unrelated to
v to be met after v
in the clockwise
contour of the
tree.
v
THE BGH BIJECTION
Join each coloured
vertex v to the first
vertex unrelated to
v to be met after v
in the clockwise
contour of the
tree.
v
THE BGH BIJECTION
Join each coloured
vertex v to the first
vertex unrelated to
v to be met after v
in the clockwise
contour of the
tree.
v
Forget the
colouring and the
outerplanar map is
retrieved.
THE BGH BIJECTION
v
THE BGH BIJECTION
(tcn, dc/n1/2)
(Mn, d/n1/2)
v
[Bonichon, Gavoille, Hanusse, ‘05] There is a
bijection between outerplanar maps with n vertices
and “well bicoloured” trees with n vertices.
(tcn, dc/n1/2)
(Mn, d/n1/2)
v
(Mn, d/n1/2)
(tcn, dc/n1/2)
THE MISSING LINK
(Mn, d/n1/2)
(tcn, dc/n1/2)
THE MISSING LINK
(Mn
, d/n1/2)
(tcn, dc/n1/2)
?
(tn, d/n1/2)
THE MISSING LINK
(Mn
, d/n1/2)
(tcn, dc/n1/2)
?
(tn, d/n1/2)
THE MISSING LINK
(Mn
, d/n1/2)
(tcn, dc/n1/2)
?
(tn, d/n1/2)
• Forgetting the colouring of a random bicoloured tree does not yield tn
(a well bicoloured tree tends to have a shorter last branch).
THE MISSING LINK
(Mn
, d/n1/2)
(tcn, dc/n1/2)
?
(tn, d/n1/2)
• Forgetting the colouring of a random bicoloured tree does not yield tn
(a well bicoloured tree tends to have a shorter last branch).
• The metric dc on tcn is not at all the same as the tree metric d (there
are “shortcuts” between coloured vertices and their “targets”).
THE MISSING LINK
(tcn, dc/n1/2)
?
(tn, d/n1/2)
w
v
ndom bicoloured tree does not yield tn
o have a shorter last branch).
l the same as the tree metric d (there
ed vertices and their “targets”).
d(v,w)=5
THE MISSING LINK
(tcn, dc/n1/2)
?
(tn, d/n1/2)
w
v
ndom bicoloured tree does not yield tn
o have a shorter last branch).
l the same as the tree metric d (there
ed vertices and their “targets”).
d(v,w)=5
dc(v,w)=3
THE MISSING LINK
(Mn
, d/n1/2)
(tcn, dc/n1/2)
?
(tn, d/n1/2)
• Forgetting the colouring of a random bicoloured tree does not yield tn
(a well bicoloured tree tends to have a shorter last branch).
• The metric dc on tcn is not at all the same as the tree metric d (there
are “shortcuts” between coloured vertices and their “targets”).
THE MISSING LINK
(Mn
, d/n1/2)
(tcn, dc/n1/2)
?
(tn, d/n1/2)
• Forgetting the colouring of a random bicoloured tree does not yield tn
(a well bicoloured tree tends to have a shorter last branch).
• The metric dc on tcn is not at all the same as the tree metric d (there
are “shortcuts” between coloured vertices and their “targets”).
THE MISSING LINK
(Mn
, d/n1/2)
(tcn, dc/n1/2)
?
(tn, d/n1/2)
• The probability that the last branch of tcn has more than ɛn1/2 vertices is
infinitesimal. It is not hard to prove that (tcn, d/n1/2) and (tn, d/n1/2) admit the
same limit.
• We only need to control dc in terms of d up to errors of order ɛn1/2,
asymptotically almost surely.
THE MISSING LINK
(Mn
, d/n1/2)
(tcn, dc/n1/2)
?
(tn, d/n1/2)
on tcn, dc/n1/2~7d/9n1/2
• The probability that the last branch of tcn has more than ɛn1/2 vertices is
infinitesimal. It is not hard to prove that (tcn, d/n1/2) and (tn, d/n1/2) admit the
same limit.
• We only need to control dc in terms of d up to errors of order ɛn1/2,
asymptotically almost surely.
THE CORE ALGORITHM
Consider geodesics for dc between a
vertex u and the root.
THE CORE ALGORITHM
root
u
Consider geodesics for dc between a
vertex u and the root.
THE CORE ALGORITHM
root
u
Consider geodesics for dc between a
vertex u and the root.
We will build a geodesic via n
steps of an exploration
process on the tree; at each
step we wish to use only
“local” information in order
to determine the next one
(→Markov property).
THE CORE ALGORITHM
p(u)
u
root
Consider geodesics for dc between a
vertex u and the root.
We will build a geodesic via n
steps of an exploration
process on the tree; at each
step we wish to use only
“local” information in order
to determine the next one
(→Markov property).
The geodesic moves
•from u to p(u), or
THE CORE ALGORITHM
p(u)
u
root
t(u)
Consider geodesics for dc between a
vertex u and the root.
We will build a geodesic via n
steps of an exploration
process on the tree; at each
step we wish to use only
“local” information in order
to determine the next one
(→Markov property).
The geodesic moves
•from u to p(u), or
•from u to t(u), or
THE CORE ALGORITHM
p(u)
u
root
r(u)
t(u)
Consider geodesics for dc between a
vertex u and the root.
We will build a geodesic via n
steps of an exploration
process on the tree; at each
step we wish to use only
“local” information in order
to determine the next one
(→Markov property).
The geodesic moves
•from u to p(u), or
•from u to t(u), or
•from u to r(u).
THE CORE ALGORITHM
Consider geodesics for dc between a
vertex u and the root.
We will build a geodesic via n
steps of an exploration
process on the tree; at each
step we wish to use only
“local” information in order
to determine the next one
(→Markov property).
The geodesic moves
•from u to p(u), or
•from u to t(u), or
•from u to r(u).
THE CORE ALGORITHM
Consider geodesics for dc between a
vertex u and the root.
We will build a geodesic via n
steps of an exploration
process on the tree; at each
step we wish to use only
“local” information in order
to determine the next one
(→Markov property).
The geodesic moves
•from u to p(u), or
•from u to t(u), or
•from u to r(u).
THE CORE ALGORITHM
leaf
move to p(u), erase u
(t17, u17)
(t16, u16)
THE CORE ALGORITHM
leaf
move to p(u), erase u
(t17, u17)
(t16, u16)
THE CORE ALGORITHM
leaf
move to p(u), erase u
parent
tu
(t17, u17)
(t16, u16)
THE CORE ALGORITHM
leaf
move to p(u), erase u
parent
if u has a right sibling:
go to p(u), erase tu
tu
(t17, u17)
(t16, u16)
(t15, u15)
THE CORE ALGORITHM
leaf
move to p(u), erase u
parent
if u has a right sibling:
go to p(u), erase tu
(t17, u17)
(t16, u16)
(t15, u15)
THE CORE ALGORITHM
leaf
move to p(u), erase u
parent
if u has a right sibling:
go to p(u), erase tu
(t17, u17)
(t16, u16)
(t15, u15)
THE CORE ALGORITHM
leaf
move to p(u), erase u
parent
if u has a right sibling:
go to p(u), erase tu
else go to p(u), attach
tr(u) to p(u), erase tu
(t17, u17)
(t16, u16)
(t15, u15)
(t14, u14)
THE CORE ALGORITHM
leaf
move to p(u), erase u
parent
if u has a right sibling:
go to p(u), erase tu
else go to p(u), attach
tr(u) to p(u), erase tu
(t17, u17)
(t16, u16)
(t15, u15)
(t14, u14)
THE CORE ALGORITHM
leaf
move to p(u), erase u
parent
if u has a right sibling:
go to p(u), erase tu
else go to p(u), attach
tr(u) to p(u), erase tu
(t17, u17)
(t16, u16)
(t15, u15)
(t14, u14)
THE CORE ALGORITHM
leaf
move to p(u), erase u
parent
if u has a right sibling:
go to p(u), erase tu
else go to p(u), attach
tr(u) to p(u), erase tu
+
(t17, u17)
(t16, u16)
(t15, u15)
(t14, u14)
(t13, u13)
THE CORE ALGORITHM
leaf
move to p(u), erase u
parent
if u has a right sibling:
go to p(u), erase tu
else go to p(u), attach
tr(u) to p(u), erase tu
(t17, u17)
(t16, u16)
(t15, u15)
(t14, u14)
(t13, u13)
THE CORE ALGORITHM
leaf
move to p(u), erase u
parent
if u has a right sibling:
go to p(u), erase tu
else go to p(u), attach
tr(u) to p(u), erase tu
(t17, u17)
(t16, u16)
(t15, u15)
(t14, u14)
(t13, u13)
THE CORE ALGORITHM
leaf
move to p(u), erase u
parent
(t12, u12)
if u has a right sibling:
go to p(u), erase tu
else go to p(u), attach
tr(u) to p(u), erase tu
if u and p(u) have no
right siblings, JUMP!
go to p(u), erase tu
JUMP
(t17, u17)
(t16, u16)
(t15, u15)
(t14, u14)
(t13, u13)
THE CORE ALGORITHM
leaf
move to p(u), erase u
parent
(t12, u12)
(t11, u11)
if u has a right sibling:
go to p(u), erase tu
else go to p(u), attach
tr(u) to p(u), erase tu
if u and p(u) have no
right siblings, JUMP!
go to p(u), erase tu
JUMP
if u and p(u) have no
right siblings, you’re
still JUMPING!
go to p(u), erase tu
(t17, u17)
(t16, u16)
(t15, u15)
(t14, u14)
(t13, u13)
THE CORE ALGORITHM
leaf
move to p(u), erase u
parent
(t12, u12)
(t11, u11)
(t10, u10)
if u has a right sibling:
go to p(u), erase tu
else go to p(u), attach
tr(u) to p(u), erase tu
if u and p(u) have no
right siblings, JUMP!
go to p(u), erase tu
JUMP
if u and p(u) have no
right siblings, you’re
still JUMPING!
go to p(u), erase tu
else LAND:
go to t(u), erase tp(u)
(t17, u17)
(t16, u16)
(t15, u15)
(t14, u14)
(t13, u13)
THE CORE ALGORITHM
leaf
move to p(u), erase u
parent
(t12, u12)
(t11, u11)
(t10, u10)
if u has a right sibling:
go to p(u), erase tu
else go to p(u), attach
tr(u) to p(u), erase tu
if u and p(u) have no
right siblings, JUMP!
go to p(u), erase tu
JUMP
if u and p(u) have no
right siblings, you’re
still JUMPING!
go to p(u), erase tu
else LAND:
go to t(u), erase tp(u)
elseif u has no right
siblings and p(u) has one,
go to t(u), erase tp(u)
elseif u has one right
sibling, attach tt(u) to
p(u), erase tu
else go to p(u), erase tu
(t17, u17)
(t16, u16)
(t15, u15)
(t14, u14)
(t13, u13)
THE CORE ALGORITHM
leaf
move to p(u), erase u
parent
(t12, u12)
(t11, u11)
(t10, u10)
if u has a right sibling:
go to p(u), erase tu
else go to p(u), attach
tr(u) to p(u), erase tu
if u and p(u) have no
right siblings, JUMP!
go to p(u), erase tu
JUMP
if u and p(u) have no
right siblings, you’re
still JUMPING!
go to p(u), erase tu
else LAND:
go to t(u), erase tp(u)
elseif u has no right
siblings and p(u) has one,
go to t(u), erase tp(u)
elseif u has one right
sibling, attach tt(u) to
p(u), erase tu
else go to p(u), erase tu
(t17, u17)
(t16, u16)
(t15, u15)
(t14, u14)
(t13, u13)
THE CORE ALGORITHM
leaf
move to p(u), erase u
parent
(t12, u12)
(t11, u11)
(t10, u10)
…
(t0, root)
(t7, u7)
(t6, u6)
JUMP
if u and p(u) have no
right siblings, you’re
still JUMPING!
go to p(u), erase tu
else LAND:
go to t(u), erase tp(u)
if u has a right sibling:
go to p(u), erase tu
else go to p(u), attach
tr(u) to p(u), erase tu
if u and p(u) have no
right siblings, JUMP!
go to p(u), erase tu
elseif u has no right
siblings and p(u) has one,
go to t(u), erase tp(u)
elseif u has one right
sibling, attach tt(u) to
p(u), erase tu
else go to p(u), erase tu
(t17, u17)
(t16, u16)
(t15, u15)
(t14, u14)
(t13, u13)
THE CORE ALGORITHM
leaf
move to p(u), erase u
parent
(t12, u12)
(t11, u11)
(t10, u10)
…
(t0, root)
(t7, u7)
(t6, u6)
c(u , root)=n-#JUMPs
d
n
JUMP
if u and p(u) have no
right siblings, you’re
still JUMPING!
go to p(u), erase tu
else LAND:
go to t(u), erase tp(u)
if u has a right sibling:
go to p(u), erase tu
else go to p(u), attach
tr(u) to p(u), erase tu
if u and p(u) have no
right siblings, JUMP!
go to p(u), erase tu
elseif u has no right
siblings and p(u) has one,
go to t(u), erase tp(u)
elseif u has one right
sibling, attach tt(u) to
p(u), erase tu
else go to p(u), erase tu
THE CORE ALGORITHM
on random trees
Running the algorithm on a
certain random pair Xn=(T,u),
where T is a well bicoloured
plane tree and u is a vertex of
depth n in T makes the
sequence of “states” into a
Markov chain.
1/4
leaf
3/32
1/4
1/8
1/16
5/16
parent
1/2
7/32
1/2
1/8
5/8
1/4
1/4
7/16
JUMP
THE CORE ALGORITHM
on random trees
Running the algorithm on a
certain random pair Xn=(T,u),
where T is a well bicoloured
plane tree and u is a vertex of
depth n in T makes the
sequence of “states” into a
Markov chain.
1/4
leaf
3/32
1/4
1/8
1/16
5/16
parent
1/2
7/32
1/2
1/8
5/8
1/4
1/4
size-biased geometric GW tree, geometric GW tree conditioned to survive…
7/16
JUMP
THE CORE ALGORITHM
on random trees
Running the algorithm on a
certain random pair Xn=(T,u),
where T is a well bicoloured
plane tree and u is a vertex of
depth n in T makes the
sequence of “states” into a
Markov chain.
1/4
leaf
3/32
1/4
1/8
1/16
5/16
parent
1/2
7/32
1/2
1/8
5/8
1/4
1/4
size-biased geometric GW tree, geometric GW tree conditioned to survive…
7/16
•Stationary distribution gives probability 2/9 to jump;
JUMP
THE CORE ALGORITHM
on random trees
Running the algorithm on a
certain random pair Xn=(T,u),
where T is a well bicoloured
plane tree and u is a vertex of
depth n in T makes the
sequence of “states” into a
Markov chain.
1/4
leaf
3/32
1/4
1/8
1/16
5/16
parent
1/2
7/32
1/2
1/8
5/8
1/4
1/4
size-biased geometric GW tree, geometric GW tree conditioned to survive…
7/16
dc(u,root) ~7|u|/9
•Stationary distribution gives probability 2/9 to jump;
•Large deviation estimates yield PXn(|dc(u, root)/n-7/9|>ɛ)<C1exp(-C2n);
JUMP
THE CORE ALGORITHM
on random trees
Running the algorithm on a
certain random pair Xn=(T,u),
where T is a well bicoloured
plane tree and u is a vertex of
depth n in T makes the
sequence of “states” into a
Markov chain.
1/4
leaf
3/32
1/4
1/8
1/16
5/16
parent
1/2
7/32
1/2
1/8
5/8
1/4
1/4
size-biased geometric GW tree, geometric GW tree conditioned to survive…
7/16
dc(u,root) ~7|u|/9
•Stationary distribution gives probability 2/9 to jump;
•Large deviation estimates yield PXn(|dc(u, root)/n-7/9|>ɛ)<C1exp(-C2n);
•Extension to generic pairs of vertices.
JUMP
THE MISSING LINK
on tcn, dc/n1/2~7d/9n1/2
THE MISSING LINK
on tcn, dc/n1/2~7d/9n1/2
C=7/9
•
The probability that there is a pair of vertices u,v in tcn such that
|dc(u, v)-Cd(u, v)|>ɛn1/2 is infinitesimal in n.
THE MISSING LINK
on tcn, dc/n1/2~7d/9n1/2
C=7/9
•
The probability that there is a pair of vertices u,v in tcn such that
|dc(u, v)-Cd(u, v)|>ɛn1/2 is infinitesimal in n.
dGH((tcn, dc/n1/2),(tcn, Cd/n1/2))≤maxu,v∈tcn |dc(u,v)/n1/2 - Cd(u,v)/n1/2|
THE MISSING LINK
on tcn, dc/n1/2~7d/9n1/2
C=7/9
•
The probability that there is a pair of vertices u,v in tcn such that
|dc(u, v)-Cd(u, v)|>ɛn1/2 is infinitesimal in n.
dGH((tcn, dc/n1/2),(tcn, Cd/n1/2))≤maxu,v∈tcn |dc(u,v)/n1/2 - Cd(u,v)/n1/2|
The probability P(dGH((tcn, dc/n1/2),(tcn, Cd/n1/2))>ɛ) is infinitesimal.
The probability P(dGH((tcn, dc/n1/2),(tcn, Cd/n1/2))>ɛ) is infinitesimal.
AND FINALLY…
The probability P(dGH((tcn, dc/n1/2),(tcn, Cd/n1/2))>ɛ) is infinitesimal.
(tcn, dc/n1/2)
(tcn, Cd/n1/2)
AND FINALLY…
The probability P(dGH((tcn, dc/n1/2),(tcn, Cd/n1/2))>ɛ) is infinitesimal.
GHJ bijection
(Mn, d/n1/2)
(tcn, dc/n1/2)
(tcn, Cd/n1/2)
AND FINALLY…
The probability P(dGH((tcn, dc/n1/2),(tcn, Cd/n1/2))>ɛ) is infinitesimal.
GHJ bijection
(Mn, d/n1/2)
(tcn, dc/n1/2)
the last branch “vanishes” in the limit
(tcn, Cd/n1/2)
(tn, Cd/n1/2)
AND FINALLY…
The probability P(dGH((tcn, dc/n1/2),(tcn, Cd/n1/2))>ɛ) is infinitesimal.
GHJ bijection
(Mn, d/n1/2)
the last branch “vanishes” in the limit
(tcn, dc/n1/2)
(tcn, Cd/n1/2)
C·CRT
(tn, Cd/n1/2)
AND FINALLY…
The probability P(dGH((tcn, dc/n1/2),(tcn, Cd/n1/2))>ɛ) is infinitesimal.
GHJ bijection
(Mn, d/n1/2)
the last branch “vanishes” in the limit
(tcn, dc/n1/2)
(tcn, Cd/n1/2)
(tn, Cd/n1/2)
C·CRT
[C., ‘14] Let Mn be a random outerplanar map with n vertices and
d its graph distance; then (in the sense of convergence in law for the
Gromov-Hausdorff metric)
limn→∞ (Mn, d/n1/2) = (CRT, 7d/9).
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