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Uniform Lipschitz functions
Ioan Manolescu
joint work with:Alexander Glazman
University of Fribourg
22nd February 2019– Statistical Mechanics in les Diablerets –
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 1 / 10
What are Lipschitz functions?Integer valued function on the faces of the hexagonal lattice H, with values atadjacent faces di↵ering by at most 1.
0 0 0 0 0
1 1 2 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1
0 0 0
0
0
0
0
0
0
0
0
0 0 0 0 0 0 0 0 0 0 0 0 0
0
0
0
0
0
0
0
0
0
1 2 3 2 1
1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 0 1 1 1 1 1 1 �1
1 1
2 2
2 2 2
2
2
2
222
22
2
3 3
3
33
�1
�1
�1
�1
0
0 0
0
0
0 0 0 0
0
0 0
0
0
0
0
00
0 0
0
0
0
0
0
00
0
00
0
00
00
00000
0 0
0
0
0
0
�1�1
�1
�1�1
�1�1
�1
�1�2
0
�D uniform sample on finite domain D, with value 0 on boundary faces.
Main question: How does �D behave when D is large?
Option 1: �D(0) is tight, with exponential tails �! Localization
Option 2: �D(0) has logarithmic variance in the size of D!Log-delocalization
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 2 / 10
What are Lipschitz functions?Integer valued function on the faces of the hexagonal lattice H, with values atadjacent faces di↵ering by at most 1.
0 0 0 0 0
1 1 2 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1
0 0 0
0
0
0
0
0
0
0
0
0 0 0 0 0 0 0 0 0 0 0 0 0
0
0
0
0
0
0
0
0
0
1 2 3 2 1
1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 0 1 1 1 1 1 1 �1
1 1
2 2
2 2 2
2
2
2
222
22
2
3 3
3
33
�1
�1
�1
�1
0
0 0
0
0
0 0 0 0
0
0 0
0
0
0
0
00
0 0
0
0
0
0
0
00
0
00
0
00
00
00000
0 0
0
0
0
0
�1�1
�1
�1�1
�1�1
�1
�1�2
0
�D uniform sample on finite domain D, with value 0 on boundary faces.
Main question: How does �D behave when D is large?
Option 1: �D(0) is tight, with exponential tails �! Localization
Option 2: �D(0) has logarithmic variance in the size of D!Log-delocalization
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 2 / 10
What are Lipschitz functions?Integer valued function on the faces of the hexagonal lattice H, with values atadjacent faces di↵ering by at most 1.
0 0 0 0 0
1 1 2 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1
0 0 0
0
0
0
0
0
0
0
0
0 0 0 0 0 0 0 0 0 0 0 0 0
0
0
0
0
0
0
0
0
0
1 2 3 2 1
1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 0 1 1 1 1 1 1 �1
1 1
2 2
2 2 2
2
2
2
222
22
2
3 3
3
33
�1
�1
�1
�1
0
0 0
0
0
0 0 0 0
0
0 0
0
0
0
0
00
0 0
0
0
0
0
0
00
0
00
0
00
00
00000
0 0
0
0
0
0
�1�1
�1
�1�1
�1�1
�1
�1�2
0
�D uniform sample on finite domain D, with value 0 on boundary faces.
Main question: How does �D behave when D is large?
Option 1: �D(0) is tight, with exponential tails �! Localization
Option 2: �D(0) has logarithmic variance in the size of D!Log-delocalization
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 2 / 10
What are Lipschitz functions?Integer valued function on the faces of the hexagonal lattice H, with values atadjacent faces di↵ering by at most 1.
0 0 0 0 0
1 1 2 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1
0 0 0
0
0
0
0
0
0
0
0
0 0 0 0 0 0 0 0 0 0 0 0 0
0
0
0
0
0
0
0
0
0
1 2 3 2 1
1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 0 1 1 1 1 1 1 �1
1 1
2 2
2 2 2
2
2
2
222
22
2
3 3
3
33
�1
�1
�1
�1
0
0 0
0
0
0 0 0 0
0
0 0
0
0
0
0
00
0 0
0
0
0
0
0
00
0
00
0
00
00
00000
0 0
0
0
0
0
�1�1
�1
�1�1
�1�1
�1
�1�2
0
�D uniform sample on finite domain D, with value 0 on boundary faces.
Main question: How does �D behave when D is large?
Option 1: �D(0) is tight, with exponential tails �! Localization
Option 2: �D(0) has logarithmic variance in the size of D!Log-delocalization
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 2 / 10
What are Lipschitz functions?Integer valued function on the faces of the hexagonal lattice H, with values atadjacent faces di↵ering by at most 1.
0 0 0 0 0
1 1 2 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1
0 0 0
0
0
0
0
0
0
0
0
0 0 0 0 0 0 0 0 0 0 0 0 0
0
0
0
0
0
0
0
0
0
1 2 3 2 1
1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 0 1 1 1 1 1 1 �1
1 1
2 2
2 2 2
2
2
2
222
22
2
3 3
3
33
�1
�1
�1
�1
0
0 0
0
0
0 0 0 0
0
0 0
0
0
0
0
00
0 0
0
0
0
0
0
00
0
00
0
00
00
00000
0 0
0
0
0
0
�1�1
�1
�1�1
�1�1
�1
�1�2
0
�D uniform sample on finite domain D, with value 0 on boundary faces.
Main question: How does �D behave when D is large?
Option 1: �D(0) is tight, with exponential tails �! Localization
Option 2: �D(0) has logarithmic variance in the size of D!Log-delocalization
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 2 / 10
Main results: Uniform Lipschitz functions delocalize logarithmically!Convergence to infinite volume measure for gradient.
Theorem (Glazman, M. 18)
For a domain D containing 0 let r be the distance form 0 to Dc.
c log r Var(�D(0)) C log r .
Moreover, �D(.)� �D(0) converges in law as D increases to H.
Observations:
Strong result: quantitative delocalisation; not just Var(�D(0))!1 as Dincreases.
Covariances between points also diverge as log of distance between points.
Coherent with conjectured convergence of � 1n⇤n
to the Gaussian Free Field.
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 3 / 10
Link to loop model:
Lipschitz functionbijection ���! oriented loop configuration
surjection�����! loop configuration.
Conversely: a loop configuration corresponds to 2#loops oriented loop configs:
P(loop configuration) / 2#loops.
Var(�D(0)) = ED,n,x(#loops surrounding 0)
0 0 0 0 0
1 1 2 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1
0 0 0
0
0
0
0
0
0
0
0
0 0 0 0 0 0 0 0 0 0 0 0 0
0
0
0
0
0
0
0
0
0
1 2 3 2 1
1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 0 1 1 1 1 1 1 �1
1 1
2 2
2 2 2
2
2
2
222
22
2
3 3
3
33
�1
�1
�1
�1
0
0 0
0
0
0 0 0 0
0
0 0
0
0
0
0
00
0 0
0
0
0
0
0
00
0
00
0
00
00
00000
0 0
0
0
0
0
�1�1
�1
�1�1
�1�1
�1
�1�2
0
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 4 / 10
Link to loop model:
Lipschitz functionbijection ���! oriented loop configuration
surjection�����! loop configuration.
Conversely: a loop configuration corresponds to 2#loops oriented loop configs:
P(loop configuration) / 2#loops.
Var(�D(0)) = ED,n,x(#loops surrounding 0)
0 0 0 0 0
1 1 2 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1
0 0 0
0
0
0
0
0
0
0
0
0 0 0 0 0 0 0 0 0 0 0 0 0
0
0
0
0
0
0
0
0
0
1 2 3 2 1
1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 0 1 1 1 1 1 1 �1
1 1
2 2
2 2 2
2
2
2
222
22
2
3 3
3
33
�1
�1
�1
�1
0
0 0
0
0
0 0 0 0
0
0 0
0
0
0
0
00
0 0
0
0
0
0
0
00
0
00
0
00
00
00000
0 0
0
0
0
0
�1�1
�1
�1�1
�1�1
�1
�1�2
0
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 4 / 10
Link to loop model:
Lipschitz functionbijection ���! oriented loop configuration
surjection�����! loop configuration.
Conversely: a loop configuration corresponds to 2#loops oriented loop configs:
P(loop configuration) / 2#loops.
Var(�D(0)) = ED,n,x(#loops surrounding 0)
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 4 / 10
Link to loop model:
Lipschitz functionbijection ���! oriented loop configuration
surjection�����! loop configuration.
Conversely: a loop configuration corresponds to 2#loops oriented loop configs:
P(loop configuration) / 2#loops.
Var(�D(0)) = ED,n,x(#loops surrounding 0)
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 4 / 10
Link to loop model:
Lipschitz functionbijection ���! oriented loop configuration
surjection�����! loop configuration.
Conversely: a loop configuration corresponds to 2#loops oriented loop configs:
P(loop configuration) / 2#loops.
Var(�D(0)) = ED,n,x(#loops surrounding 0)
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 4 / 10
Link to loop model:
Lipschitz functionbijection ���! oriented loop configuration
surjection�����! loop configuration.
Conversely: a loop configuration corresponds to 2#loops oriented loop configs:
P(loop configuration) / 2#loops.
Var(�D(0)) = ED,n,x(#loops surrounding 0)
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 4 / 10
Definition (Loop O(n) model)
A loop configuration is an even subgraph of D.The loop O(n) measure with edge-parameter x > 0 is given by
PD,n,x(!) =1
Zloop(D, n, x)n#loops
x#edges
1!loop config.
Dichotomy:
Exponential decay of loop sizes:the size of the loop of any pointhas exponential tail, unif. in D.
Macroscopic loops: the size ofthe loop of any point has power-law decay up to the size of D.In D there are loops at every scaleup to the size of D.
Phase diagram:
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 5 / 10
Definition (Loop O(n) model)
A loop configuration is an even subgraph of D.The loop O(n) measure with edge-parameter x > 0 is given by
PD,n,x(!) =1
Zloop(D, n, x)n#loops
x#edges
1!loop config.
Dichotomy:
Exponential decay of loop sizes:the size of the loop of any pointhas exponential tail, unif. in D.
Macroscopic loops: the size ofthe loop of any point has power-law decay up to the size of D.In D there are loops at every scaleup to the size of D.
Phase diagram:
1p3
n
x
xc=
1p
2+p2�n
11p2
1p2+
p2
n = 2, x = 1
Macroscopicloops
1p3
Exponentialdecay
1
2
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 5 / 10
Definition (Loop O(n) model)
A loop configuration is an even subgraph of D.The loop O(n) measure with edge-parameter x > 0 is given by
PD,n,x(!) =1
Zloop(D, n, x)n#loops
x#edges
1!loop config.
Dichotomy:
Exponential decay of loop sizes:the size of the loop of any pointhas exponential tail, unif. in D.
Macroscopic loops: the size ofthe loop of any point has power-law decay up to the size of D.In D there are loops at every scaleup to the size of D.
Phase diagram:
1p3
n
x11p2
1p2+
p2
n = 2, x = 1
1p3
Ising model
Duminil-Copin, Peled,Samotij, Spinka ’17
1
2
Easy
(Peierlsarg
ument)
See:Duminil-C
opin,Smirnov
12Duminil-C
opin,Kozma,
Yad
in14
Tag
gi18
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 5 / 10
Definition (Loop O(n) model)
A loop configuration is an even subgraph of D.The loop O(n) measure with edge-parameter x > 0 is given by
PD,n,x(!) =1
Zloop(D, n, x)n#loops
x#edges
1!loop config.
Theorem (Glazman, M. 18)
There exists a infinite volume Gibbs measure PH,2,1 for the loop O(2)model with x = 1.
PH,2,1 = limPD,2,1 as D ! H.
It is translation invariant, ergodic, formed entirely of loops.
The origin is surrounded PH,2,1-a.s. by infinitely many loops.
Order log n of these are in ⇤n ) “macroscopic loops”.
PH,2,1 is the unique infinite volume Gibbs measure for the loop O(2) model.
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 5 / 10
Case study: the Ising model (n = 1 and x 1).
PD,x(!) =1
Zloop(D,1,x) x#edges
1! loop config.
Loop configurationbijection ���! -spin configuration on faces (w on boundary).
For spin configuration �(w. on boundary),
PD,x(�) =1Z x
#
Ising model on faces with� = � 1
2 log x � 0.
Properties:
FKG: P(A\B) � P(A)P(B) if A,B "
Spatial Markov property
Maximal/minimal b.c..
Duality between and
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 6 / 10
Case study: the Ising model (n = 1 and x 1).
PD,x(!) =1
Zloop(D,1,x) x#edges
1! loop config.
Loop configurationbijection ���! -spin configuration on faces (w on boundary).
For spin configuration �(w. on boundary),
PD,x(�) =1Z x
#
Ising model on faces with� = � 1
2 log x � 0.
Properties:
FKG: P(A\B) � P(A)P(B) if A,B "
Spatial Markov property
Maximal/minimal b.c..
Duality between and
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 6 / 10
Case study: the Ising model (n = 1 and x 1).
PD,x(!) =1
Zloop(D,1,x) x#edges
1! loop config.
Loop configurationbijection ���! -spin configuration on faces (w on boundary).
For spin configuration �(w. on boundary),
PD,x(�) =1Z x
#
Ising model on faces with� = � 1
2 log x � 0.
Properties:
FKG: P(A\B) � P(A)P(B) if A,B "
Spatial Markov property
Maximal/minimal b.c..
Duality between and
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 6 / 10
Case study: the Ising model (n = 1 and x 1).
PD,x(!) =1
Zloop(D,1,x) x#edges
1! loop config.
Loop configurationbijection ���! -spin configuration on faces (w on boundary).
For spin configuration �(w. on boundary),
PD,x(�) =1Z x
#
Ising model on faces with� = � 1
2 log x � 0.
Properties:
FKG: P(A\B) � P(A)P(B) if A,B "
Spatial Markov property
Maximal/minimal b.c..
Duality between and
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 6 / 10
Case study: the Ising model (n = 1 and x 1).
PD,x(!) =1
Zloop(D,1,x) x#edges
1! loop config.
Loop configurationbijection ���! -spin configuration on faces (w on boundary).
For spin configuration �(w. on boundary),
PD,x(�) =1Z x
#
Ising model on faces with� = � 1
2 log x � 0.
Properties:
FKG: P(A\B) � P(A)P(B) if A,B "
Spatial Markov property
Maximal/minimal b.c..
Duality between and
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 6 / 10
Case study: the Ising model (n = 1 and x 1).
PD,x(!) =1
Zloop(D,1,x) x#edges
1! loop config.
Loop configurationbijection ���! -spin configuration on faces (w on boundary).
For spin configuration �(w. on boundary),
PD,x(�) =1Z x
#
Ising model on faces with� = � 1
2 log x � 0.
Properties:
FKG: P(A\B) � P(A)P(B) if A,B "Spatial Markov property
Maximal/minimal b.c..
Duality between and
?
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 6 / 10
Case study: the Ising model (n = 1 and x 1).
PD,x(!) =1
Zloop(D,1,x) x#edges
1! loop config.
Loop configurationbijection ���! -spin configuration on faces (w on boundary).
For spin configuration �(w. on boundary),
PD,x(�) =1Z x
#
Ising model on faces with� = � 1
2 log x � 0.
Properties:
FKG: P(A\B) � P(A)P(B) if A,B "Spatial Markov property
Maximal/minimal b.c..
Duality between and
?
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 6 / 10
Case study: the Ising model (n = 1 and x 1).
PD,x(!) =1
Zloop(D,1,x) x#edges
1! loop config.
Loop configurationbijection ���! -spin configuration on faces (w on boundary).
For spin configuration �(w. on boundary),
PD,x(�) =1Z x
#
Ising model on faces with� = � 1
2 log x � 0.
Properties:
FKG: P(A\B) � P(A)P(B) if A,B "Spatial Markov property
Maximal/minimal b.c..
Duality between and
max
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 6 / 10
Case study: the Ising model (n = 1 and x 1).
PD,x(!) =1
Zloop(D,1,x) x#edges
1! loop config.
Loop configurationbijection ���! -spin configuration on faces (w on boundary).
For spin configuration �(w. on boundary),
PD,x(�) =1Z x
#
Ising model on faces with� = � 1
2 log x � 0.
Properties:
FKG: P(A\B) � P(A)P(B) if A,B "Spatial Markov property
Maximal/minimal b.c..
Duality between and
min
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 6 / 10
Case study: the Ising model (n = 1 and x 1).
PD,x(!) =1
Zloop(D,1,x) x#edges
1! loop config.
Loop configurationbijection ���! -spin configuration on faces (w on boundary).
For spin configuration �(w. on boundary),
PD,x(�) =1Z x
#
Ising model on faces with� = � 1
2 log x � 0.
Properties:
FKG: P(A\B) � P(A)P(B) if A,B "Spatial Markov property
Maximal/minimal b.c..
Duality between and
a
b c
d
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 6 / 10
Case study: the Ising model (n = 1 and x 1).
PD,x(!) =1
Zloop(D,1,x) x#edges
1! loop config.
Loop configurationbijection ���! -spin configuration on faces (w on boundary).
For spin configuration �(w. on boundary),
PD,x(�) =1Z x
#
Ising model on faces with� = � 1
2 log x � 0.
Properties:
FKG: P(A\B) � P(A)P(B) if A,B "Spatial Markov property
Maximal/minimal b.c..
Duality between and
a
b c
d
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 6 / 10
Case study: the Ising model (n = 1 and x 1).
PD,x(!) =1
Zloop(D,1,x) x#edges
1! loop config.
Loop configurationbijection ���! -spin configuration on faces (w on boundary).
For spin configuration �(w. on boundary),
PD,x(�) =1Z x
#
Ising model on faces with� = � 1
2 log x � 0.
Properties:
FKG: P(A\B) � P(A)P(B) if A,B "Spatial Markov property
Maximal/minimal b.c..
Duality between and
a
b c
d
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 6 / 10
Spin as percolation models: Same spin representation holds for any n and x
Theorem (Duminil-Copin, Glazman,Peled, Spinka 17)
For n � 1 and x < 1/pn the spin
model has FKG!
+ Spatial Markov property+
Theorem (Dichotomy theorem)
Either:
(A) exponential decay of inside -bc,
or
(B) RSW of inside , hence clusters
of any size of any spin
1p3
n
x11p2
1p2+
p2
n = 2, x = 1
1p3
Ising model1
2
Dum
inil-Copin,Glazm
an,
Peled, S
pinka18
FKG
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 7 / 10
Spin as percolation models: Same spin representation holds for any n and x
Theorem (Duminil-Copin, Glazman,Peled, Spinka 17)
For n � 1 and x < 1/pn the spin
model has FKG!
+ Spatial Markov property+
Theorem (Dichotomy theorem)
Either:
(A) exponential decay of inside -bc,
or
(B) RSW of inside , hence clusters
of any size of any spin
1p3
n
x11p2
1p2+
p2
n = 2, x = 1
1p3
Ising model1
2
Dum
inil-Copin,Glazm
an,
Peled, S
pinka18
FKG
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 7 / 10
Spin as percolation models: Same spin representation holds for any n and x
Theorem (Duminil-Copin, Glazman,Peled, Spinka 17)
For n � 1 and x < 1/pn the spin
model has FKG!
+ Spatial Markov property+
Theorem (Dichotomy theorem)
Either:
(A) exponential decay of inside -bc,
or
(B) RSW of inside , hence clusters
of any size of any spin
1p3
n
x11p2
1p2+
p2
n = 2, x = 1
1p3
Ising model1
2
Dum
inil-Copin,Glazm
an,
Peled, S
pinka18
FKG
D
P[ ]0
⇤n
< e�cn
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 7 / 10
Spin as percolation models: Same spin representation holds for any n and x
Theorem (Duminil-Copin, Glazman,Peled, Spinka 17)
For n � 1 and x < 1/pn the spin
model has FKG!
+ Spatial Markov property+
Theorem (Dichotomy theorem)
Either:
(A) exponential decay of inside -bc,
or
(B) RSW of inside , hence clusters
of any size of any spin
1p3
n
x11p2
1p2+
p2
n = 2, x = 1
1p3
Ising model1
2
Dum
inil-Copin,Glazm
an,
Peled, S
pinka18
FKG
0
D
⇤n/2
P[ ]� �⇤n
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 7 / 10
Spin as percolation models: Same spin representation holds for any n and x
Theorem (Duminil-Copin, Glazman,Peled, Spinka 17)
For n � 1 and x < 1/pn the spin
model has FKG!
+ Spatial Markov property+
Theorem (Dichotomy theorem)
Either:
(A) exponential decay of inside -bc,
or
(B) RSW of inside , hence clusters
of any size of any spin
1p3
n
x11p2
1p2+
p2
n = 2, x = 1
1p3
Ising model1
2
Dum
inil-Copin,Glazm
an,
Peled, S
pinka18
FKG
D
n
2n
P[ ]� �
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 7 / 10
Spin as percolation models: Same spin representation holds for any n and x
Theorem (Duminil-Copin, Glazman,Peled, Spinka 17)
For n � 1 and x < 1/pn the spin
model has FKG!
+ Spatial Markov property+
Theorem (Dichotomy theorem)
Either:
(A) exponential decay of inside -bc,
or
(B) RSW of inside , hence clusters
of any size of any spin
1p3
n
x11p2
1p2+
p2
n = 2, x = 1
1p3
Ising model1
2
Dum
inil-Copin,Glazm
an,
Peled, S
pinka18
FKG
??
D
n
2n
P[ ]� �
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 7 / 10
Back to n = 2 = 1 + 1, x = 1: P(!) / 2#loops
Coloured loop measure: uniform on pairs (!r ,!b) of non-intersecting loops.
Spin measure: µD uniform on red/blue spin configurations { , }F ⇥ { , }Fwith no simultaneous disagreement and , on outer layer.
Red spin marginal: ⌫D(�r ) =1Z
P�b
1{�r?�b} = 1Z 2
#free faces. Has FKG!!!
Spatial Markov: Generally NO!
Yes for ! ⌫D and ! ⌫D - maximal
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 8 / 10
Back to n = 2 = 1 + 1, x = 1: P(!) / 2#loops
Coloured loop measure: uniform on pairs (!r ,!b) of non-intersecting loops.
Spin measure: µD uniform on red/blue spin configurations { , }F ⇥ { , }Fwith no simultaneous disagreement and , on outer layer.
Red spin marginal: ⌫D(�r ) =1Z
P�b
1{�r?�b} = 1Z 2
#free faces. Has FKG!!!
Spatial Markov: Generally NO!
Yes for ! ⌫D and ! ⌫D - maximal
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 8 / 10
Back to n = 2 = 1 + 1, x = 1: P(!) / 2#loops
Coloured loop measure: uniform on pairs (!r ,!b) of non-intersecting loops.
Spin measure: µD uniform on red/blue spin configurations { , }F ⇥ { , }Fwith no simultaneous disagreement and , on outer layer.
Red spin marginal: ⌫D(�r ) =1Z
P�b
1{�r?�b} = 1Z 2
#free faces. Has FKG!!!
Spatial Markov: Generally NO!
Yes for ! ⌫D and ! ⌫D - maximal
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 8 / 10
Back to n = 2 = 1 + 1, x = 1: P(!) / 2#loops
Coloured loop measure: uniform on pairs (!r ,!b) of non-intersecting loops.
Spin measure: µD uniform on red/blue spin configurations { , }F ⇥ { , }Fwith no simultaneous disagreement and , on outer layer.
Red spin marginal: ⌫D(�r ) =1Z
P�b
1{�r?�b} = 1Z 2
#free faces. Has FKG!!!
Spatial Markov: Generally NO!
Yes for ! ⌫D and ! ⌫D - maximal
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 8 / 10
Back to n = 2 = 1 + 1, x = 1: P(!) / 2#loops
Coloured loop measure: uniform on pairs (!r ,!b) of non-intersecting loops.
Spin measure: µD uniform on red/blue spin configurations { , }F ⇥ { , }Fwith no simultaneous disagreement and , on outer layer.
Red spin marginal: ⌫D(�r ) =1Z
P�b
1{�r?�b} = 1Z 2
#free faces. Has FKG!!!
Spatial Markov: Generally NO!
Yes for ! ⌫D and ! ⌫D - maximal
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 8 / 10
Back to n = 2 = 1 + 1, x = 1: P(!) / 2#loops
Coloured loop measure: uniform on pairs (!r ,!b) of non-intersecting loops.
Spin measure: µD uniform on red/blue spin configurations { , }F ⇥ { , }Fwith no simultaneous disagreement and , on outer layer.
Red spin marginal: ⌫D(�r ) =1Z
P�b
1{�r?�b} = 1Z 2
#free faces. Has FKG!!!
Spatial Markov: Generally NO!
Yes for ! ⌫D and ! ⌫D - maximal
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 8 / 10
Back to n = 2 = 1 + 1, x = 1: P(!) / 2#loops
Coloured loop measure: uniform on pairs (!r ,!b) of non-intersecting loops.
Spin measure: µD uniform on red/blue spin configurations { , }F ⇥ { , }Fwith no simultaneous disagreement and , on outer layer.
Red spin marginal: ⌫D(�r ) =1Z
P�b
1{�r?�b} = 1Z 2
#free faces. Has FKG!!!
Spatial Markov: Generally NO!
Yes for ! ⌫D and ! ⌫D - maximal
?
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 8 / 10
Back to n = 2 = 1 + 1, x = 1: P(!) / 2#loops
Coloured loop measure: uniform on pairs (!r ,!b) of non-intersecting loops.
Spin measure: µD uniform on red/blue spin configurations { , }F ⇥ { , }Fwith no simultaneous disagreement and , on outer layer.
Red spin marginal: ⌫D(�r ) =1Z
P�b
1{�r?�b} = 1Z 2
#free faces. Has FKG!!!
Spatial Markov: Generally NO!
Yes for ! ⌫D and ! ⌫D - maximal
?
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 8 / 10
Back to n = 2 = 1 + 1, x = 1: P(!) / 2#loops
Coloured loop measure: uniform on pairs (!r ,!b) of non-intersecting loops.
Spin measure: µD uniform on red/blue spin configurations { , }F ⇥ { , }Fwith no simultaneous disagreement and , on outer layer.
Red spin marginal: ⌫D(�r ) =1Z
P�b
1{�r?�b} = 1Z 2
#free faces. Has FKG!!!
Spatial Markov: Generally NO! Yes for ! ⌫D
and ! ⌫D - maximal
?constant blue spin
constant blue spin
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 8 / 10
Back to n = 2 = 1 + 1, x = 1: P(!) / 2#loops
Coloured loop measure: uniform on pairs (!r ,!b) of non-intersecting loops.
Spin measure: µD uniform on red/blue spin configurations { , }F ⇥ { , }Fwith no simultaneous disagreement and , on outer layer.
Red spin marginal: ⌫D(�r ) =1Z
P�b
1{�r?�b} = 1Z 2
#free faces. Has FKG!!!
Spatial Markov: Generally NO! Yes for ! ⌫D and ! ⌫D - maximal
?any blue spins
any blue spins
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 8 / 10
A taste of the proof. Step 1: infinite vol. measure
Red marginal: ⌫H
= limD!H
# ⌫D
Blue marginal: i.i.d. colouring of red config. ) joint measure µH:
0 is surrounded by infinitely many circuits µH-a.s.
in particular µH is translation invariant and ergodic;
Weak RSW result for percolation:
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 9 / 10
A taste of the proof. Step 1: infinite vol. measure
Red marginal: ⌫H
= limD!H
# ⌫DBlue marginal: i.i.d. colouring of red config. ) joint measure µ
H:
0 is surrounded by infinitely many circuits µH-a.s.
in particular µH is translation invariant and ergodic;
Weak RSW result for percolation:
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 9 / 10
A taste of the proof. Step 2: ergodicity (via RSW)
Red marginal: ⌫H
= limD!H
# ⌫DBlue marginal: i.i.d. colouring of red config. ) joint measure µ
H:
0 is surrounded by infinitely many circuits µH-a.s.
in particular µH is translation invariant and ergodic;
Weak RSW result for percolation:
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 9 / 10
A taste of the proof. Step 2: ergodicity (via RSW)
Red marginal: ⌫H
= limD!H
# ⌫DBlue marginal: i.i.d. colouring of red config. ) joint measure µ
H:
0 is surrounded by infinitely many circuits µH-a.s.
in particular µH is translation invariant and ergodic;
Weak RSW result for percolation:
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 9 / 10
A taste of the proof. Step 2: ergodicity (via RSW)
Red marginal: ⌫H
= limD!H
# ⌫DBlue marginal: i.i.d. colouring of red config. ) joint measure µ
H:
0 is surrounded by infinitely many circuits µH-a.s.
in particular µH is translation invariant and ergodic;
Weak RSW result for percolation:
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 9 / 10
A taste of the proof. Step 2: ergodicity (via RSW)
Red marginal: ⌫H
= limD!H
# ⌫DBlue marginal: i.i.d. colouring of red config. ) joint measure µ
H:
0 is surrounded by infinitely many circuits µH-a.s.
in particular µH is translation invariant and ergodic;
Weak RSW result for percolation:
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 9 / 10
A taste of the proof. Step 2: ergodicity (via RSW)
Red marginal: ⌫H
= limD!H
# ⌫DBlue marginal: i.i.d. colouring of red config. ) joint measure µ
H:
0 is surrounded by infinitely many circuits µH-a.s.
in particular µH is translation invariant and ergodic;
Weak RSW result for percolation:
µH ( )� 1
µH ( )µH ( )µH ( )
+
+
+
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 9 / 10
A taste of the proof. Step 2: ergodicity (via RSW)
Red marginal: ⌫H
= limD!H
# ⌫DBlue marginal: i.i.d. colouring of red config. ) joint measure µ
H:
0 is surrounded by infinitely many circuits µH-a.s.
in particular µH is translation invariant and ergodic;
Weak RSW result for percolation:
µH ( )� 14
n
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 9 / 10
A taste of the proof. Step 2: ergodicity (via RSW)
Red marginal: ⌫H
= limD!H
# ⌫DBlue marginal: i.i.d. colouring of red config. ) joint measure µ
H:
0 is surrounded by infinitely many circuits µH-a.s.
in particular µH is translation invariant and ergodic;
Weak RSW result for percolation:
( )� cn
2n
µH
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 9 / 10
A taste of the proof. Step 2: ergodicity (via RSW)
Red marginal: ⌫H
= limD!H
# ⌫DBlue marginal: i.i.d. colouring of red config. ) joint measure µ
H:
0 is surrounded by infinitely many circuits µH-a.s.
in particular µH is translation invariant and ergodic;
Weak RSW result for percolation:
( )� c0⇤n
⇤n/2µH
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 9 / 10
A taste of the proof. Step 3: µH= µ
H
Red marginal: ⌫H
= limD!H
# ⌫DBlue marginal: i.i.d. colouring of red config. ) joint measure µ
H:
0 is surrounded by infinitely many circuits µH-a.s.
in particular µH is translation invariant and ergodic;
Burton-Keane applies ) zero or one infinite -cluster
Zhang’s trick ) no infinite -cluster and no infinite -cluster
Weak RSW result for percolation:
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 9 / 10
A taste of the proof. Step 3: µH= µ
H
Red marginal: ⌫H
= limD!H
# ⌫DBlue marginal: i.i.d. colouring of red config. ) joint measure µ
H:
0 is surrounded by infinitely many circuits µH-a.s.
in particular µH is translation invariant and ergodic;
Burton-Keane applies ) zero or one infinite -cluster
Zhang’s trick ) no infinite -cluster and no infinite -cluster
Weak RSW result for percolation:
1
11
1
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 9 / 10
A taste of the proof. Step 3: µH= µ
H
Red marginal: ⌫H
= limD!H
# ⌫DBlue marginal: i.i.d. colouring of red config. ) joint measure µ
H:
0 is surrounded by infinitely many circuits µH-a.s.
in particular µH is translation invariant and ergodic;
Burton-Keane applies ) zero or one infinite -cluster
Zhang’s trick ) no infinite -cluster and no infinite -cluster
Weak RSW result for percolation:
1
11
1
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 9 / 10
A taste of the proof. Step 3: µH= µ
H
Red marginal: ⌫H
= limD!H
# ⌫DBlue marginal: i.i.d. colouring of red config. ) joint measure µ
H:
0 is surrounded by infinitely many circuits µH-a.s.
in particular µH is translation invariant and ergodic;
Burton-Keane applies ) zero or one infinite -cluster
Zhang’s trick ) no infinite -cluster and no infinite -cluster
Infinitely many blue loops
Weak RSW result for percolation:
⇤n
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 9 / 10
A taste of the proof. Step 3: µH= µ
H
Red marginal: ⌫H
= limD!H
# ⌫DBlue marginal: i.i.d. colouring of red config. ) joint measure µ
H:
0 is surrounded by infinitely many circuits µH-a.s.
in particular µH is translation invariant and ergodic;
Burton-Keane applies ) zero or one infinite -cluster
Zhang’s trick ) no infinite -cluster and no infinite -cluster
Infinitely many blue loops and infinitely many red loops
Weak RSW result for percolation:
⇤n
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 9 / 10
A taste of the proof. Step 3: µH= µ
H
Red marginal: ⌫H
= limD!H
# ⌫DBlue marginal: i.i.d. colouring of red config. ) joint measure µ
H:
0 is surrounded by infinitely many circuits µH-a.s.
in particular µH is translation invariant and ergodic;
Burton-Keane applies ) zero or one infinite -cluster
Zhang’s trick ) no infinite -cluster and no infinite -cluster
Infinitely many blue loops and infinitely many red loops
Changing colours of loops + infinitely many circuits) µ
Hunique infinite-volume measure: µ
H= lim
D!H
µD = limD!H
µD
Weak RSW result for percolation:
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 9 / 10
A taste of the proof. Step 4: delocalization
Red marginal: ⌫H
= limD!H
# ⌫DBlue marginal: i.i.d. colouring of red config. ) joint measure µ
H:
0 is surrounded by infinitely many circuits µH-a.s.
in particular µH is translation invariant and ergodic;
Burton-Keane applies ) zero or one infinite -cluster
Zhang’s trick ) no infinite -cluster and no infinite -cluster
Infinitely many blue loops and infinitely many red loops
Changing colours of loops + infinitely many circuits) µ
Hunique infinite-volume measure: µ
H= lim
D!H
µD = limD!H
µD
infinitely many loops around 0 ) delocalisation for µH.
Var(�D(0))!1 as D increases to H ) Delocalisation in finite volume.
Weak RSW result for percolation:
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 9 / 10
Thank you!
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 22nd Feb. 2019 10 / 10
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