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On the Kertèsz line: Thermody- namic versus Geometric phase transitions

Jean RUIZ Centre de Physique Théorique Marseille

works with

Ph. Blanchard, H. Satz (Bielefeld)
D. Gandolfo (Marseille Toulon)
L. Laanait (Rabat)

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Summary

What is the Kertèsz (2D–Ising model)
Some Results for Kertèsz line for Potts model

(analytical and numerical) − Potts model and some known results − Various representations of Potts models − Results − Scheme of proofs

“Kertèsz line” for Potts model on the complete

graph − Erdös–Renyi random graph − Known results for Curie–Weiss Potts model − Various representations − Known results for Fortuin–Kasteleyn representation − What’s we expect

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The Kertèsz Line

(Kertèsz 1989, Stauffer and Aharony, Percolation Theory: the trouble with Kertèsz) 2 − D Ising model σi = ± 1, i ∈ Λ ⊂ Z2

+ − − + − + + + − − + + − − + + + − + + − − + − + − + + + −

Boltzmann weight ωIsing(σ) =

i,j

e

β 2(σiσj−1)

i

e

h 2 (σi−1)

=

i,j

eβ(δσi,σj−1)

i

eh(δσi,1−1) β =

1 k T, inverse temperature

h external magnetic field product is over nearest neighbour pairs.

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h = 0.

Phase transition (eβc − 1)2 = 2, βc = ln (1 + 2 √ ) − β > βc → positive spontaneous magnetization → Two translation invariant Gibbs states → Positive surface tension between the two states − β < βc → no spontaneous magnetization → Unique phase

h > 0

“Nothing happens” − For all β → Analytic free energy → Unique phase

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Edwards–Sokal representation ωES(σ, η) =

i,j

[e−βδηij,0 + (1 − e−β)δηij,1δσi,σj] ×

i

eh(δσi,1−1) where the RV ηij ∈ {0, 1} : write for each edge i, j: eβ(δσi,σj−1) = e−β + (1 − e−β)δσi,σj =

ηij=0,1

e−βδηij,0 + (1 − e−β)δσi,σjδηij,1

+ − − + − + + + − − + + − − + + + − + + − − + − + − + + + −

h = ∞

all σi = + 1 ωES(η) =

i,j

[e−βδηij,0 + (1 − e−β)δηij,1] → Usual 2 − D bond Percolation problem with parameter e−β → Percolation transition at e−βp = 1/2, βp = ln 2

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h

βc = ln(1 + √ 2) βp = ln 2 No percolation Percolation

β

✉ ✉ 6

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Potts model

σi ∈ {1,

, q}

i ∈ Λ ⊂ Zd

3 4 4 1 2 2 3 4 2 6 5 4 2 6 3 3 2 1 4 1 1 4 4 3 1 1 3 6 7 2

Boltzmann weight ωPotts(σ) =

i,j

eβ(δσi,σj−1)

i

eh(δσi,1−1)

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h = 0.

q large First order phase transition d = 2: (eβc − 1)2 = q, βc = ln (1 + q √ ) d 3: βc ≃ 1 d ln q − β > βc → positive spontaneous magnetization → q ordered translation invariant Gibbs states → Positive surface tension between the states → Vanishing mass gap (exponential decrease of correlations) − β < βc → no spontaneous magnetization → Unique phase: disordered state → Positive mass gap (finite correlation length) − β = βc → discontinuity of mean energy and magnetiza- tion → q + 1 phases → Positive surface tension between the phases → Positive mass gap Kotecky Shlosman (1982),...

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h > 0 small

q large : First order transition at some βc(h) − β > βc(h) → 1 ordered translation invariant Gibbs state − β < βc(h) → Unique phase: disordered state − β = βc(h) → discontinuity of mean energy → 2 phases → Positive surface tension between the phases Bakchich, Benyoussef, Laanait (1989),...

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Various representations

Sarting from ωES(σ, η)

1. Fortuin-Kasteleyn in external field

ehδσi,1 = 1 + (eh − 1)δσi,1

θi=0,1

δθi,0 + (eh − 1)δσi,1δθi,1 Sum over spin variables ωES ωFK(η, θ) =

i,j

e

−βδηij,0(1 − e−β) δηij,1

×

i

e

−hδθi,0(1 − e−h) δθi,1 qC(η|θ)

2. Sum directly over spins ωES

ω(η) =

i,j

e

−βδηij,0(1 − e−β) δηij,1

×

i

C(η)

(1 + (q − 1)e−h)Si(η)

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Colored–Edwards–Sokal representation

Write δσi,σj = χ(σi = σj = 1) + χ(σi = σj

1)

Replace edge variables ηij ∈ {0, 1} by nij ∈ {0, 1, 2} (white red blue). Then ωCES(σ, n) =

i,j
e−βδnij,0

+ (1 − e−β)δnij,1 χ(σi = σj = 1) + (1 − e−β)δnij,2 χ(σi = σj

1)

×

i

ehδσi,1

3 4 4 1 2 2 3 4 2 6 5 4 2 6 3 3 2 1 4 1 1 4 4 3 1 1 3 6 7 2

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Tri–Clor–Edge representation

Sum over spin variables ωTER(n) =

i,j

e

−βδnij,0 (1 − e−β)(δnij,1+δnij,2)

× ehS1(n) (q − 1)C2(n) × (q − 1 + eh)|Λ|−S1(n)−S2(n) − S1(n) (resp. S2(n)) denotes the number of sites that belong to edges of color 1 (resp. of color 2) − C2(n) denotes the number of connected com- ponents of the set of edges of color 2 − |Λ| is the number of sites of the box under consideration

Geometric order parameter

Let pΛ(i ↔ j) be the probability that the site i is connected to j by a path of edges of color 1. mass–gap (inverse correlation length) m(β, h) = − lim

|i−j|→∞

1 |i − j|ln lim

Λ↑Zd pΛ(i↔ j)

− i and j belong to some line parallel to an axis

f the lattice.

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Diagram of ground states

ba be the value of the Boltzmann weight (in

the TER representation) of the ground state configuration of color a = 0, 1, 2 per unit site b0 = e−βd(q − 1 + eh) b1 = (1 − e−β)deh b2 = (1 − e−β)d

h β β0(h) 1 2

β0(h) = ln [1 + (1 + (q − 1)e−h)1/d] All the ground states coexist at (0, β0(0)). Below β0(h) only the 0–state dominates. Above β0(h) the 1–state dominates: it coexists with the 0–state on β0(h) and with the 2– state on the line h = 0, β ≥ β0(0)

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Analytic results

q is large enough and h not too large Using Pirogov–Sinaï theory

the phase diagram of the TER model mimics

the diagram of ground state configurations

the model undergoes a thermodynamic first
rder phase transition in the sense that the

derivative of its free energy with respect to β (or h) is discontinuous at some βc(h) ∼ β0(h)

the model exhibits a geometric (first order)

transition, in the sense that, on the critical line, the mass gap is discontinuous. Theorem 1. Assume d ≥ 2, q and h such that cd(1 + (q − 1)e−h)−1/2d < 1 holds, where cd is a given number (depending only on the dimension), then there exists a unique βc(h) = β0(h) + O(1 + (q − 1)e−h)−1/2d) such that m(β, h) > 0 for β ≤ βc(h) and m(β, h) = 0 for β > βc(h). Since the free energies of Potts model and of the TER model are the same, the critical lines coincides with that of Potts model.

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Numerical simulations

Generalization

f

the Swendsen–Wang algorithm inherited from colored Edwards–Sokal model.

0.5 0.7 0.9 1.1 1.3 1.5 1 2 3 4 5

h β q = 10

5 4 2

d = 2.

1. For q ≤ 4:

− a whole geometric transition line for which m(β, h) > 0 when β βc(h) m(β, h) = 0 when β > βc(h) − The mass gap is continuous at βc(h) − For β βc(h) the mean cluster sizes remain finite − For β > βc(h) the size of 1–edge clus- ters diverges. − The mean energy as well as the mag- netisation do not show any singular behavior

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0.5 0.7 0.9 1.1 1.3 1.5 1 2 3 4 5

h β q = 10

5 4 2

2. For q ≥ 5: some critical hc appears

− the transition becomes first order when h < hc in accordance with the previous analytic results − both the mass gap and the mean energy exhibit discontinuities at βc(h > hc). − However when h ≥ hc, the scenario is the same as for q ≤ 4. Numerics are in accordance with theory for van- ishing and infinite fields: βc(0) = ln(1 + q √ ) and βc(∞) = ln 2.

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Description of the algorithm

inherited from colored Edwards–Sokal model. ωCES(σ, n) =

i,j

e−βδnij,0

+ (1 − e−β)δnij,1 χ(σi = σj = 1) + (1 − e−β)δnij,2 χ(σi = σj

1)

×

i ehδσi,1

1. Given a spin configuration:

put between any two neighbouring spins of the same color:

an edge colored 0 with probability e−β
w.p. 1 − e−β, an edge colored 1 if these

spins are of colour 1, and coloured 2 other- wise. When two neighbouring spins disagree, the corresponding edge is colored 0.

2. Starting from an edge configuration, a spin

configuration is constructed as follows.

Isolated sites (endpoints of 0–bonds only)

are coloured 1 w.p. eh/(q − 1 + eh) and coloured c ∈ {2, ..., q} w.p. 1/(q − 1 + eh).

Non–isolated sites

are colored 1 (w.p. 1) if they are endpoints of 1–bonds and colored c ∈ {2, ..., q} w.p. 1/(q − 1).

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Scheme of proof

Dilute partition function partition function Za =

n

ωTER(n)χa(n) up to a boundary term Za(Λ) =

n
i∈Λ

ωi(n)qC2(n)−δa,2

i∈∂Λ

j∼i

δnij,a sum is over all configurations n = {nij}ij∩

∅,

∂Λ is the boundary of Λ i∼ j means that i and j are n.n. ωi(n) = (1 − e−β)(δnij,1+δnij,2)/2e−βδnij,0/2ehχ(i∈1′′) × (q − 1 + eh)

j∼iδnij,0

χ(i ∈ 1′′) means that the site i belongs to some edge

f color 1.

Contours (signed) − n a configuration on : E(Λ) = {i, j ∩ Λ

∅}

− i ∈ Λ is called correct if for all j ∼ i, nij take the same value − i ∈ Λ is called incorrect if for all j ∼ i, nij take the same value otherwise − Γ = {SuppΓ, n(Γ)} is called contour if : SuppΓ) is a maximal connected subset of incorrect sites n(Γ) the restriction of n to E(Supp Γ)

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Contour expansions

ver external contours

Za(Λ) =

θ={Γ1,

,Γn}e x t

ba

|ExtΛθ|

×

k=1

n

ρ(Γk)

m=0,1,2

Zm(IntmΓk) where ρ(Γ) =

i∈suppΓ

ωi(nΓ)qC(nΓ)−δa,2

ver compatible contours

Za(Λ) = ba

|Λ|

{Γ1,

,Γn}c o m p

k=1

n

za(Γk) with activities za(Γ) = ρ(Γ)ba

−|suppΓ| m

a

Zm(IntmΓ) Za(IntmΓ)

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Peierls’ estimate ρ(Γ)( max

a=0,1,2 ba)−|SuppΓ| ≤ e−τ |SuppΓ|

(1) where e−τ = (1 + (q − 1)e−h)−1/2d First notice that an incorrect site i is either of color 1

r
f

color 2. In the first case

ne

has

j∼i (δnij,0 + δnij,1) = 2d, so that ωi(nΓ)/b1 = (eβ −

1)−(

j∼iδnij,0)/2 implying

ωi(nΓ)/ max

a=0,1,2 ba ≤ (1 + (q − 1)e−h) −(

j∼iδnij,0)/2d

Since 1 ≤

j∼i δnij,0 ≤ 2d − 1, each incorrect site of

color 1 gives at most a contribution e−τ to the L.H.S.

f

(1). In the second case,

ne

has

j∼i (δnij,0 + δnij,2) = 2d, so that wi(nΓ)/b2 = (eβ −

1)

−(

j∼iδnij,0)/2 implying

ωi(nΓ)/ max

a=0,1,2 ba ≤ (q − 1 + eh)−(

j∼iδnij,0)/2d

We then use again that 1 ≤

j∼i δnij,0 ≤ 2d − 1 and

that C2(nΓ) ≤

i∈SuppΓ χ(1 ≤ δnij,2)/2

j∼iδnij,2, to
btain that each incorrect site of color 2 gives at

most a contribution (eh + q − 1)−1/2+1/2d ≤ e−τ to the L.H.S. of (1).

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Going on with PS (with Zahradnick formula- tion) Peierls’ estimate with small e−τ (assumption of Thm gives by PS theory good control of the system Truncated activity za

′(Γ) =

za(Γ)

if za(Γ) ≤ e−(τ −τ0)|Supp Γ| e−(τ −τ0)|Supp Γ|

therwise

τ0 some numerical constant

Stable contours: call a contour stable if za(Γ) = za

′(Γ)

Truncated partiton functions: leave out unstable con- tours i.e. take the activities za

′(Γ) in the expansion:

Za(Λ) = ba

|Λ|

{Γ1,

,Γn}c o m p

k=1

n

za

′(Γk)

metastable free energies: fa

met(β, h) = − lim Λ↑Zd (1/|Λ|)ln Za ′(Λ)

Observe fa

met(β, h) = − ln ba + Correction

Correction = O(e−τ), because they are free energies

f contour models which can be controlled by conver-

gent cluster expansions: Stable boundary conditon : bc is called stable if fa

met(β, h) =

min

m=0,1,2 fa met(β, h)

If the bc a is stable then all a–contours are stable

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Thus: as a standard result of Pirogov-Sinai theory,

ne gets that the phase diagram of the system is a

small perturbation of the diagram of ground state configurations. − unique point βc(0) given by the solution of f0

met(β, h) = f1 met(β, h) = f2 met(β, h)

for which all contours are stable and such that Za(Λ) = Za

′(Λ) for a = 0, 1, 2

− line βc(h) given by the solution of f0

met(β, h) = f1 met(β, h)

when h > 0 and such that, Za(Λ) = Za

′(Λ) for

a = 0, 1 − β < βc(h) one has Z0(Λ) = Z0

′(Λ)

− β > βc(h) one has Z1(Λ) = Z1

′(Λ)

− For h = 0 and β ≥ βc(0), one has in addition Z2(Λ) = Z2

′(Λ)

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End of proof: dicontinuity of mass–gap

1. If one impose that the site i is connected to j

by a path made up of edges of color 1, then under the boundary condition 0, there exists necessarily an external contour that encloses both the sites i and j. Probability of external contours Γ decays like (c0e)−τ |SuppΓ| when the 0–contours are stable, i.e. when Z0(Λ) = Z0

′(Λ)

− pΛ(i ↔ j) ≤ (Ctee−τ)|i−j| when β ≤ βc(h)

2. Under the bc. 1, by Peierls type arguments,

the probability that the site i is not connected to j can be bounded from above by a small number O(e−τ) when Z1(Λ) = Z1

′(Λ)

⇒ pΛ(i↔ j) 1 − 0(e−τ) for β ≥ βc(h) ⇒ Under the bc. 0 pΛ(i↔ j) 1 − 0(e−τ) for β > βc(h)

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“Kertèsz line in Curie–Weiss Potts model (Potts model on the complete graph) Erdös–Renyi Random Graph:

Gn,p − Consider n vertices − Put a bond between any two vertices with probability p (and no bond w.p. (1 − p) Take p = c

n

Phase transition: → if c < 1: with high probability only components

f size O(ln n) (or less)

→ If c 1: A component of size O(n) appears (Giant component) and the other ones have size O(ln n) (w.h.p.) Only transition in the topology: nothing happens for the mean occupation number of bonds

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Curie–Weiss Potts model n vertices σi ∈ {1,

, q}

i = 1,

, n

Boltzmann weight ωCWPotts(σ) =

i<j

e

β n(δσi,σj−1)

For q = 2: Second order phase transition at

βc = 2

For q 3: First order transition at βc =

2q − 1

q − 2 ln (q − 1)

The transition is characterized by the order param- eter s solution of the mean–field equation s = eβs − 1 eβs + q − 1

For q = 2

− β βc s = 0 − β > βc s > 0

For q 3

− β < βc s = 0 − β = βc sc = q − 2

q − 1

− β > βc s > sc Wu (1982),...

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fcan

CWPotts: canonical free energy of the model with

fixed density of colors ρ1,

, ρq (can be explicitely

computed Minima of fcan

CWPotts parametrized s ∈ [0, 1], and com-

ponents of the vector (ρ1,

, ρq) permutations of the

q values:

1 q(1 + (q − 1)s), 1 q(1 − s),

, 1

q(1 − s)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.002 –0.002 0.004 0.006 0.008 0.01 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 –0.002 –0.003 –0.001 0.001 0.002 0.003

β<βc

(2)

β=βc

(2)

β>βc

(2)

β<βc

(q)

β=βc

(q)

β>βc

(q)

q = 2

→ β βc unique minimizer (s = 0) → β > 2 two minima {1

2 + s0, 1 2 − s0} and

{1

2 − s0, 1 2 + s0} with s0 sol. of MFE

q 3

→ β < βc unique minimizer (s = 0) → β = βc q+1 minimizers: (s = 0) and the permutations of the above vector, with s = sc → β > βc q minimizers: permutations of the above vector with s > sc

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Edwads-Sokal and Fortuin–Kasteleyn repre- sentation of CW Potts model ωES

CW(σ, η) =

i<j

e−β/n δηij,0 + (1 − e−β/n)δηij,1δσi,σj ωFK

CW(η) =

i<j

[e

−(β/n)δηij,0 (1 − e−β/n) δηij,1]qC(η)

For q = 1, Erdös–Renyi random graph Gn,p with p = 1 − e−β/n ≃ β n

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Results for FK representation (Bollobas, Grimmet Janson (1996)

q = 1 classical ER transition from No Giant to

Giant component at βc = 1

q = 2

→ Transition from No Giant to Giant component at βc = 2 → Second order transition at βc = 2 in the mean

ccupation

number

f

bonds (corresponding to the second

rder

transition of CWPotts)

q = 3

→ Transition from No Giant to Giant component at βc = 2 → First order transition at βc = 2q − 1

q − 2 ln (q − 1)

in the mean

ccupation

number of bonds (corresponding to the first order transition of CWPotts) Able to study all real (positive) values of q

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Add a magnetic field ωCWES(σ, η) =

i<j

[e−β/nδηij,0 + (1 − e−β/n)δηij,1δσi,σj] ×

i

ehδσi,1

h = ∞

all σi = + 1 ωCWES(η) =

i,j

[e−β/nδηij,0 + (1 − e−β/n)δηij,1] → ER random graph Gn,p with p = 1 − e−β/n → ER transition at βc = ln 1

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Consider CES representation and TER representa- tion on the complete graph

h

βc = 2 or 2q−2

q−1 ln(q − 1)

β0 = 1 No Giant Component Giant component

β

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What we expect (for h > 0)

q = 2 no transition in mean occupation number
f 1–bonds
q 3

− appearance of a critical hc → h < hc First order transition in the mean occupation number of 1–bonds → h hc No transition in MON Why: because with h > 0 the Mean field equation reads (Biskup Chayes Crawford 2006) s = eβs+h − 1 eβs+h + q − 1 − q = 2