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Uniform Lipschitz functions
Ioan Manolescu
joint work with:Alexander Glazman
University of Fribourg
19th June 2019Probability and quantum field theory:
discrete models, CFT, SLE and constructive aspects(Porquerolles)
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 1 / 12
What are Lipschitz functions?Integer valued function on the faces of the hexagonal lattice H, with values atadjacent faces differing 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
ΓD uniform sample on finite domain D, with value 0 on boundary faces.
Main question: How does ΓD behave when D is large?
Option 1: ΓD(0) is tight, with exponential tails −→ LocalizationOption 2: ΓD(0) has logarithmic variance in the size of D→Log-delocalization
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 2 / 12
What are Lipschitz functions?Integer valued function on the faces of the hexagonal lattice H, with values atadjacent faces differing 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
ΓD uniform sample on finite domain D, with value 0 on boundary faces.
Main question: How does ΓD behave when D is large?
Option 1: ΓD(0) is tight, with exponential tails −→ LocalizationOption 2: ΓD(0) has logarithmic variance in the size of D→Log-delocalization
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 2 / 12
What are Lipschitz functions?Integer valued function on the faces of the hexagonal lattice H, with values atadjacent faces differing 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
ΓD uniform sample on finite domain D, with value 0 on boundary faces.
Main question: How does ΓD behave when D is large?
Option 1: ΓD(0) is tight, with exponential tails −→ LocalizationOption 2: ΓD(0) has logarithmic variance in the size of D→Log-delocalization
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 2 / 12
What are Lipschitz functions?Integer valued function on the faces of the hexagonal lattice H, with values atadjacent faces differing 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
ΓD uniform sample on finite domain D, with value 0 on boundary faces.
Main question: How does ΓD behave when D is large?
Option 1: ΓD(0) is tight, with exponential tails −→ LocalizationOption 2: ΓD(0) has logarithmic variance in the size of D→Log-delocalization
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 2 / 12
What are Lipschitz functions?Integer valued function on the faces of the hexagonal lattice H, with values atadjacent faces differing by at most 1.
ΓD uniform sample on finite domain D, with value 0 on boundary faces.
Main question: How does ΓD behave when D is large?
Option 1: ΓD(0) is tight, with exponential tails −→ LocalizationOption 2: ΓD(0) has logarithmic variance in the size of D→Log-delocalization
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 2 / 12
What are Lipschitz functions?Integer valued function on the faces of the hexagonal lattice H, with values atadjacent faces differing by at most 1.
ΓD uniform sample on finite domain D, with value 0 on boundary faces.
Main question: How does ΓD behave when D is large?
Option 1: ΓD(0) is tight, with exponential tails −→ LocalizationOption 2: ΓD(0) has logarithmic variance in the size of D→Log-delocalization
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 2 / 12
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(ΓD(0)) ≤ C log r .
Moreover, ΓD(.)− ΓD(0) converges in law as D increases to H.
Observations:
Strong result: quantitative delocalisation; not just Var(ΓD(0))→∞ as Dincreases.
Covariances between points also diverge as log of distance between points.
Coherent with conjectured convergence of Γ 1n Λn
to the Gaussian Free Field.
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 3 / 12
Link to loop model:
Lipschitz function1 to 1←−−→ oriented loop configuration
many to 1−−−−−−→ loop configuration.
Conversely: a loop configuration corresponds to 2#loops oriented loop configs:
P(loop configuration) ∝ 2#loops.
Var(ΓD(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 19th June 2019 4 / 12
Link to loop model:
Lipschitz function1 to 1←−−→ oriented loop configuration
many to 1−−−−−−→ loop configuration.
Conversely: a loop configuration corresponds to 2#loops oriented loop configs:
P(loop configuration) ∝ 2#loops.
Var(ΓD(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 19th June 2019 4 / 12
Link to loop model:
Lipschitz function1 to 1←−−→ oriented loop configuration
many to 1−−−−−−→ loop configuration.
Conversely: a loop configuration corresponds to 2#loops oriented loop configs:
P(loop configuration) ∝ 2#loops.
Var(ΓD(0)) = ED,n,x(#loops surrounding 0)
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 4 / 12
Link to loop model:
Lipschitz function1 to 1←−−→ oriented loop configuration
many to 1−−−−−−→ loop configuration.
Conversely: a loop configuration corresponds to 2#loops oriented loop configs:
P(loop configuration) ∝ 2#loops.
Var(ΓD(0)) = ED,n,x(#loops surrounding 0)
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 4 / 12
Link to loop model:
Lipschitz function1 to 1←−−→ oriented loop configuration
many to 1−−−−−−→ loop configuration.
Conversely: a loop configuration corresponds to 2#loops oriented loop configs:
P(loop configuration) ∝ 2#loops.
Var(ΓD(0)) = ED,n,x(#loops surrounding 0)
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 4 / 12
Link to loop model:
Lipschitz function1 to 1←−−→ oriented loop configuration
many to 1−−−−−−→ loop configuration.
Conversely: a loop configuration corresponds to 2#loops oriented loop configs:
P(loop configuration) ∝ 2#loops.
Var(ΓD(0)) = ED,n,x(#loops surrounding 0)
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 4 / 12
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(ω) =1
Zloop(D, n, x)n#loopsx#edges 1ωloop 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 19th June 2019 5 / 12
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(ω) =1
Zloop(D, n, x)n#loopsx#edges 1ωloop 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:
1√3
n
x
xc=
1√
2+√2−n
11√2
1√2+√2
n = 2, x = 1
Macroscopicloops
1√3
Exponentialdecay
1
2
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 5 / 12
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(ω) =1
Zloop(D, n, x)n#loopsx#edges 1ωloop 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:
1√3
n
x11√2
1√2+√2
n = 2, x = 1
1√3
Ising model
Duminil-Copin, Peled,Samotij, Spinka ’17
1
2
Easy
(Peierlsarg
ument)
See:Duminil-C
opin,Smirnov
12
Duminil-C
opin,Kozm
a,Yadin
14
Taggi18
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 5 / 12
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(ω) =1
Zloop(D, n, x)n#loopsx#edges 1ωloop 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 Λn ⇒ “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 19th June 2019 5 / 12
Case study: the Ising model (n = 1 and x ≤ 1).
PD,x(ω) = 1Zloop(D,1,x) x
#edges 1ω loop config.
Loop configurationbijection←−−−→ -spin configuration on faces (w on boundary).
For spin configuration σ(w. on boundary),
PD,x(σ) = 1Z x
#
Ising model on faces withβ = − 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 19th June 2019 6 / 12
Case study: the Ising model (n = 1 and x ≤ 1).
PD,x(ω) = 1Zloop(D,1,x) x
#edges 1ω loop config.
Loop configurationbijection←−−−→ -spin configuration on faces (w on boundary).
For spin configuration σ(w. on boundary),
PD,x(σ) = 1Z x
#
Ising model on faces withβ = − 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 19th June 2019 6 / 12
Case study: the Ising model (n = 1 and x ≤ 1).
PD,x(ω) = 1Zloop(D,1,x) x
#edges 1ω loop config.
Loop configurationbijection←−−−→ -spin configuration on faces (w on boundary).
For spin configuration σ(w. on boundary),
PD,x(σ) = 1Z x
#
Ising model on faces withβ = − 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 19th June 2019 6 / 12
Case study: the Ising model (n = 1 and x ≤ 1).
PD,x(ω) = 1Zloop(D,1,x) x
#edges 1ω loop config.
Loop configurationbijection←−−−→ -spin configuration on faces (w on boundary).
For spin configuration σ(w. on boundary),
PD,x(σ) = 1Z x
#
Ising model on faces withβ = − 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 19th June 2019 6 / 12
Case study: the Ising model (n = 1 and x ≤ 1).
PD,x(ω) = 1Zloop(D,1,x) x
#edges 1ω loop config.
Loop configurationbijection←−−−→ -spin configuration on faces (w on boundary).
For spin configuration σ(w. on boundary),
PD,x(σ) = 1Z x
#
Ising model on faces withβ = − 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 19th June 2019 6 / 12
Case study: the Ising model (n = 1 and x ≤ 1).
PD,x(ω) = 1Zloop(D,1,x) x
#edges 1ω loop config.
Loop configurationbijection←−−−→ -spin configuration on faces (w on boundary).
For spin configuration σ(w. on boundary),
PD,x(σ) = 1Z x
#
Ising model on faces withβ = − 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 19th June 2019 6 / 12
Case study: the Ising model (n = 1 and x ≤ 1).
PD,x(ω) = 1Zloop(D,1,x) x
#edges 1ω loop config.
Loop configurationbijection←−−−→ -spin configuration on faces (w on boundary).
For spin configuration σ(w. on boundary),
PD,x(σ) = 1Z x
#
Ising model on faces withβ = − 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 19th June 2019 6 / 12
Case study: the Ising model (n = 1 and x ≤ 1).
PD,x(ω) = 1Zloop(D,1,x) x
#edges 1ω loop config.
Loop configurationbijection←−−−→ -spin configuration on faces (w on boundary).
For spin configuration σ(w. on boundary),
PD,x(σ) = 1Z x
#
Ising model on faces withβ = − 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 19th June 2019 6 / 12
Case study: the Ising model (n = 1 and x ≤ 1).
PD,x(ω) = 1Zloop(D,1,x) x
#edges 1ω loop config.
Loop configurationbijection←−−−→ -spin configuration on faces (w on boundary).
For spin configuration σ(w. on boundary),
PD,x(σ) = 1Z x
#
Ising model on faces withβ = − 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 19th June 2019 6 / 12
Case study: the Ising model (n = 1 and x ≤ 1).
PD,x(ω) = 1Zloop(D,1,x) x
#edges 1ω loop config.
Loop configurationbijection←−−−→ -spin configuration on faces (w on boundary).
For spin configuration σ(w. on boundary),
PD,x(σ) = 1Z x
#
Ising model on faces withβ = − 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 19th June 2019 6 / 12
Case study: the Ising model (n = 1 and x ≤ 1).
PD,x(ω) = 1Zloop(D,1,x) x
#edges 1ω loop config.
Loop configurationbijection←−−−→ -spin configuration on faces (w on boundary).
For spin configuration σ(w. on boundary),
PD,x(σ) = 1Z x
#
Ising model on faces withβ = − 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 19th June 2019 6 / 12
Case study: the Ising model (n = 1 and x ≤ 1).
PD,x(ω) = 1Zloop(D,1,x) x
#edges 1ω loop config.
Loop configurationbijection←−−−→ -spin configuration on faces (w on boundary).
For spin configuration σ(w. on boundary),
PD,x(σ) = 1Z x
#
Ising model on faces withβ = − 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 19th June 2019 6 / 12
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/√n the spin
model has FKG!
+ Spatial Markov property⇓
Theorem (Dichotomy theorem)
Either:(A) exponential decay of inside -bc,or(B) RSW of inside , hence clustersof any size of any spin
1√3
n
x11√2
1√2+√2
n = 2, x = 1
1√3
Ising model1
2
Dum
inil-Copin,Glazm
an,
Peled,Spinka
18
FKG
D
P[ ]0
Λn
< e−cn
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 7 / 12
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/√n the spin
model has FKG!
+ Spatial Markov property⇓
Theorem (Dichotomy theorem)
Either:(A) exponential decay of inside -bc,or(B) RSW of inside , hence clustersof any size of any spin
1√3
n
x11√2
1√2+√2
n = 2, x = 1
1√3
Ising model1
2
Dum
inil-Copin,Glazm
an,
Peled,Spinka
18
FKG
D
P[ ]0
Λn
< e−cn
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 7 / 12
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/√n the spin
model has FKG!
+ Spatial Markov property⇓
Theorem (Dichotomy theorem)
Either:(A) exponential decay of inside -bc,or(B) RSW of inside , hence clustersof any size of any spin
1√3
n
x11√2
1√2+√2
n = 2, x = 1
1√3
Ising model1
2
Dum
inil-Copin,Glazm
an,
Peled,Spinka
18
FKG
D
P[ ]0
Λn
< e−cn
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 7 / 12
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/√n the spin
model has FKG!
+ Spatial Markov property⇓
Theorem (Dichotomy theorem)
Either:(A) exponential decay of inside -bc,or(B) RSW of inside , hence clustersof any size of any spin
1√3
n
x11√2
1√2+√2
n = 2, x = 1
1√3
Ising model1
2
Dum
inil-Copin,Glazm
an,
Peled,Spinka
18
FKG
0
D
Λn/2
P[ ]≥ δΛn
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 7 / 12
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/√n the spin
model has FKG!
+ Spatial Markov property⇓
Theorem (Dichotomy theorem)
Either:(A) exponential decay of inside -bc,or(B) RSW of inside , hence clustersof any size of any spin
1√3
n
x11√2
1√2+√2
n = 2, x = 1
1√3
Ising model1
2
Dum
inil-Copin,Glazm
an,
Peled,Spinka
18
FKG
D
n
2n
P[ ]≥ δ
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 7 / 12
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/√n the spin
model has FKG!
+ Spatial Markov property⇓
Theorem (Dichotomy theorem)
Either:(A) exponential decay of inside -bc,or(B) RSW of inside , hence clustersof any size of any spin
1√3
n
x11√2
1√2+√2
n = 2, x = 1
1√3
Ising model1
2
Dum
inil-Copin,Glazm
an,
Peled,Spinka
18
FKG
??
D
n
2n
P[ ]≥ δ
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 7 / 12
Back to n = 2 = 1 + 1, x = 1: P(ω) ∝ 2#loops
Coloured loop measure: uniform on pairs (ωr , ωb) 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: νD(σr ) = 1Z
∑σb
1{σr⊥σb} = 1Z 2#free faces. Has FKG!!!
Spatial Markov: Generally NO!
Yes for → νD and → νD - maximal
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 8 / 12
Back to n = 2 = 1 + 1, x = 1: P(ω) ∝ 2#loops
Coloured loop measure: uniform on pairs (ωr , ωb) 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: νD(σr ) = 1Z
∑σb
1{σr⊥σb} = 1Z 2#free faces. Has FKG!!!
Spatial Markov: Generally NO!
Yes for → νD and → νD - maximal
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 8 / 12
Back to n = 2 = 1 + 1, x = 1: P(ω) ∝ 2#loops
Coloured loop measure: uniform on pairs (ωr , ωb) 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: νD(σr ) = 1Z
∑σb
1{σr⊥σb} = 1Z 2#free faces. Has FKG!!!
Spatial Markov: Generally NO!
Yes for → νD and → νD - maximal
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 8 / 12
Back to n = 2 = 1 + 1, x = 1: P(ω) ∝ 2#loops
Coloured loop measure: uniform on pairs (ωr , ωb) 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: νD(σr ) = 1Z
∑σb
1{σr⊥σb} = 1Z 2#free faces. Has FKG!!!
Spatial Markov: Generally NO!
Yes for → νD and → νD - maximal
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 8 / 12
Back to n = 2 = 1 + 1, x = 1: P(ω) ∝ 2#loops
Coloured loop measure: uniform on pairs (ωr , ωb) 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: νD(σr ) = 1Z
∑σb
1{σr⊥σb} = 1Z 2#free faces. Has FKG!!!
Spatial Markov: Generally NO!
Yes for → νD and → νD - maximal
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 8 / 12
Back to n = 2 = 1 + 1, x = 1: P(ω) ∝ 2#loops
Coloured loop measure: uniform on pairs (ωr , ωb) 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: νD(σr ) = 1Z
∑σb
1{σr⊥σb} = 1Z 2#free faces. Has FKG!!!
Spatial Markov: Generally NO!
Yes for → νD and → νD - maximal
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 8 / 12
Back to n = 2 = 1 + 1, x = 1: P(ω) ∝ 2#loops
Coloured loop measure: uniform on pairs (ωr , ωb) 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: νD(σr ) = 1Z
∑σb
1{σr⊥σb} = 1Z 2#free faces. Has FKG!!!
Spatial Markov: Generally NO!
Yes for → νD and → νD - maximal
?
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 8 / 12
Back to n = 2 = 1 + 1, x = 1: P(ω) ∝ 2#loops
Coloured loop measure: uniform on pairs (ωr , ωb) 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: νD(σr ) = 1Z
∑σb
1{σr⊥σb} = 1Z 2#free faces. Has FKG!!!
Spatial Markov: Generally NO!
Yes for → νD and → νD - maximal
?
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 8 / 12
Back to n = 2 = 1 + 1, x = 1: P(ω) ∝ 2#loops
Coloured loop measure: uniform on pairs (ωr , ωb) 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: νD(σr ) = 1Z
∑σb
1{σr⊥σb} = 1Z 2#free faces. Has FKG!!!
Spatial Markov: Generally NO! Yes for → νD
and → νD - maximal
?constant blue spin
constant blue spin
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 8 / 12
Back to n = 2 = 1 + 1, x = 1: P(ω) ∝ 2#loops
Coloured loop measure: uniform on pairs (ωr , ωb) 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: νD(σr ) = 1Z
∑σb
1{σr⊥σb} = 1Z 2#free faces. Has FKG!!!
Spatial Markov: Generally NO! Yes for → νD and → νD - maximal
?any blue spins
any blue spins
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 8 / 12
A taste of the proof. Step 1: infinite vol. measure
Red marginal: νH = limD→H↓ νD
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 19th June 2019 9 / 12
A taste of the proof. Step 1: infinite vol. measure
Red marginal: νH = limD→H↓ νD
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 19th June 2019 9 / 12
A taste of the proof. Step 2: ergodicity (via RSW)
Red marginal: νH = limD→H↓ νD
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 19th June 2019 9 / 12
A taste of the proof. Step 2: ergodicity (via RSW)
Red marginal: νH = limD→H↓ νD
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 19th June 2019 9 / 12
A taste of the proof. Step 2: ergodicity (via RSW)
Red marginal: νH = limD→H↓ νD
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 19th June 2019 9 / 12
A taste of the proof. Step 2: ergodicity (via RSW)
Red marginal: νH = limD→H↓ νD
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 19th June 2019 9 / 12
A taste of the proof. Step 2: ergodicity (via RSW)
Red marginal: νH = limD→H↓ νD
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:
µH ( )≥ 1
µH ( )µH ( )µH ( )
+
+
+
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 9 / 12
A taste of the proof. Step 2: ergodicity (via RSW)
Red marginal: νH = limD→H↓ νD
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:
µH ( )≥ 14
n
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 9 / 12
A taste of the proof. Step 2: ergodicity (via RSW)
Red marginal: νH = limD→H↓ νD
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:
( )≥ cn
2n
µH
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 9 / 12
A taste of the proof. Step 2: ergodicity (via RSW)
Red marginal: νH = limD→H↓ νD
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:
( )≥ c′Λn
Λn/2µH
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 9 / 12
A taste of the proof. Step 3: µH = µH
Red marginal: νH = limD→H↓ νD
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;
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 19th June 2019 9 / 12
A taste of the proof. Step 3: µH = µH
Red marginal: νH = limD→H↓ νD
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;
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 19th June 2019 9 / 12
A taste of the proof. Step 3: µH = µH
Red marginal: νH = limD→H↓ νD
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;
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 19th June 2019 9 / 12
A taste of the proof. Step 3: µH = µH
Red marginal: νH = limD→H↓ νD
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;
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:
Λn
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 9 / 12
A taste of the proof. Step 3: µH = µH
Red marginal: νH = limD→H↓ νD
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;
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:
Λn
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 9 / 12
A taste of the proof. Step 3: µH = µH
Red marginal: νH = limD→H↓ νD
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;
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⇒ µH unique infinite-volume measure: µH = lim
D→HµD = lim
D→HµD
Weak RSW result for percolation:
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 9 / 12
A taste of the proof. Step 4: delocalization
Red marginal: νH = limD→H↓ νD
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;
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⇒ µH unique infinite-volume measure: µH = lim
D→HµD = lim
D→HµD
infinitely many loops around 0 ⇒ delocalisation for µH.
Var(ΓD(0))→∞ as D increases to H ⇒ Delocalisation in finite volume.
Weak RSW result for percolation:
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 9 / 12
Dichotomy theorem: idea of proof
Lemma (Pushing lemma)
ρn
nnρ
µ( )≥ c(ρ) > 0
and
5n
nµ( )≥ c
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 10 / 12
Dichotomy theorem: idea of proof
Lemma (Pushing lemma)
ρn
nnρ
µ( )≥ c(ρ) > 0
and
5n
nµ( )≥ c
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 10 / 12
Proof of dichotomy using pushing lemma
Either: αn ≥ c > 0 or αn ≤ exp(−cnδ) , where
αn:=µ ( )Λρn
ΛnΛn/2
With probability
c
αρn:
⇒ Two isolated circuits,
so αρn ≤ c7α2n
Λn Λn
ρn
Λρ2n
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 11 / 12
Proof of dichotomy using pushing lemma
Either: αn ≥ c > 0 or αn ≤ exp(−cnδ) , where
αn:=µ ( )Λρn
ΛnΛn/2
With probability
c
αρn:
⇒ Two isolated circuits,
so αρn ≤ c7α2n
Λn Λn
ρn
Λρ2n
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 11 / 12
Proof of dichotomy using pushing lemma
Either: αn ≥ c > 0 or αn ≤ exp(−cnδ) , where
αn:=µ ( )Λρn
ΛnΛn/2
With probability
c
αρn:
⇒ Two isolated circuits,
so αρn ≤ c7α2n
Λn Λn
ρn
Λρ2n
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 11 / 12
Proof of dichotomy using pushing lemma
Either: αn ≥ c > 0 or αn ≤ exp(−cnδ) , where
αn:=µ ( )Λρn
ΛnΛn/2
With probability c αρn:
⇒ Two isolated circuits,
so αρn ≤ c7α2n
Λn Λn
ρn
Λρ2n
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 11 / 12
Proof of dichotomy using pushing lemma
Either: αn ≥ c > 0 or αn ≤ exp(−cnδ) , where
αn:=µ ( )Λρn
ΛnΛn/2
With probability
c
αρn:
⇒ Two isolated circuits,
so αρn ≤ c7α2n
Λn Λn
ρn
Λρn
Λ2ρn Λρ2n
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 11 / 12
Proof of dichotomy using pushing lemma
Either: αn ≥ c > 0 or αn ≤ exp(−cnδ) , where
αn:=µ ( )Λρn
ΛnΛn/2
With probability c αρn:
⇒ Two isolated circuits,
so αρn ≤ c7α2n
Λn Λn
ρn
Λρn
Λ2ρn Λρ2n
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 11 / 12
Proof of dichotomy using pushing lemma
Either: αn ≥ c > 0 or αn ≤ exp(−cnδ) , where
αn:=µ ( )Λρn
ΛnΛn/2
With probability c αρn:
⇒ Two isolated circuits,
so αρn ≤ c7α2n
Λn Λn
ρn
Λρ2n
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 11 / 12
Proof of dichotomy using pushing lemma
Either: αn ≥ c > 0 or αn ≤ exp(−cnδ) , where
αn:=µ ( )Λρn
ΛnΛn/2
With probability c3αρn:
⇒ Two isolated circuits,
so αρn ≤ c7α2n
Λn Λn
ρn
Λρ2n
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 11 / 12
Proof of dichotomy using pushing lemma
Either: αn ≥ c > 0 or αn ≤ exp(−cnδ) , where
αn:=µ ( )Λρn
ΛnΛn/2
With probability c3αρn:
⇒ Two isolated circuits,
so αρn ≤ c7α2n
Λn Λn
ρn
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 11 / 12
Proof of dichotomy using pushing lemma
Either: αn ≥ c > 0 or αn ≤ exp(−cnδ) , where
αn:=µ ( )Λρn
ΛnΛn/2
With probability c7αρn:
⇒ Two isolated circuits,
so αρn ≤ c7α2n
Λn Λn
ρn
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 11 / 12
Thank you!
Ioan Manolescu (University of Fribourg) Uniform Lipschitz functions 19th June 2019 12 / 12
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