Double-sided estimates of hitting probabilitiesin discrete planar domains

Dmitry Chelkak (Steklov Institute (PDMI RAS) &

Chebyshev Lab (SPbSU), St.Petersburg)

based on Robust discrete complex analysis: a toolbox,

arXiv:1212.6205

Russian-Chinese Seminar on Asymptotic Methods

in Probability Theory and Mathematical Statistics

St. Petersburg, June 10, 2013

Motivation:

I geometric point of view on scaling limits of 2D lattice models

(e.g., recent progress in mathematical understanding of

a conformally invariant limit of the critical Ising model)

Motivation:

I geometric point of view on scaling limits of 2D lattice models

(e.g., recent progress in mathematical understanding of

a conformally invariant limit of the critical Ising model)

(spin representation, c⃝ C. Hongler)(random cluster representation,

c⃝ S. Smirnov)

Motivation:

I geometric point of view on scaling limits of 2D lattice models

(e.g., recent progress in mathematical understanding of

a conformally invariant limit of the critical Ising model):

a priori estimates for crossing-type events via reductions

to discrete holomorphic and discrete harmonic functions

Motivation:

I geometric point of view on scaling limits of 2D lattice models

(e.g., recent progress in mathematical understanding of

a conformally invariant limit of the critical Ising model):

a priori estimates for crossing-type events via reductions

to discrete holomorphic and discrete harmonic functions;

I by-product: (uniform wrt all discrete domains) analogues of

classical estimates available in geometric complex analysis.

Motivation:

I geometric point of view on scaling limits of 2D lattice models

(e.g., recent progress in mathematical understanding of

a conformally invariant limit of the critical Ising model):

a priori estimates for crossing-type events via reductions

to discrete holomorphic and discrete harmonic functions;

I by-product: (uniform wrt all discrete domains) analogues of

classical estimates available in geometric complex analysis.

Example: (harmonic measure ωΩ(z ; (ab)) of a far boundary arc)

Motivation:

I by-product: (uniform wrt all discrete domains) analogues of

classical estimates available in geometric complex analysis.

Example: (harmonic measure ωΩ(z ; (ab)) of a far boundary arc)

Theorem: (Ahlfors, Beurling, (Carleman))

ωΩ(z ; (ab)) 68

πexp

[−π

∫ x1

x0

dx

ϑ(x)

].

Motivation:

I by-product: (uniform wrt all discrete domains) analogues of

classical estimates available in geometric complex analysis.

Example: (harmonic measure ωΩ(z ; (ab)) of a far boundary arc)

Theorem: (Ahlfors, Beurling, (Carleman))

ωΩ(z ; (ab)) ≍ exp[−πLΩ(z ; (ab))], LΩ(z ; (ab)) >∫ x1x0(ϑ(x))−1dx .

Remark: ⇑ conformal invariance of ωΩ(z ; (ab)) and LΩ(z ; (ab)).

Notation:

Let (Γ;EΓ) be an ininite planar graph embedded into C so that all

its edges (uv) ∈ EΓ are straight segments, wuv = wvu > 0 be some

xed edge weights (conductances), and µu :=∑

v∼u wuv for u ∈ Γ.

Notation:

Let (Γ;EΓ) be an ininite planar graph embedded into C so that all

its edges (uv) ∈ EΓ are straight segments, wuv = wvu > 0 be some

xed edge weights (conductances), and µu :=∑

v∼u wuv for u ∈ Γ.

Discrete domains:

Let (VΩ;EΩint) be a bounded connected subgraph of (Γ;EΓ).

Denote by EΩbd the set of all (oriented) edges (ainta) ∈ EΩ

int

such that aint ∈ VΩ and a ∈ VΩ. We set Ω := IntΩ ∪ ∂Ω,

IntΩ := VΩ, ∂Ω := (a ; (ainta)) : (ainta) ∈ EΩbd.

Formally, the boundary ∂Ω of a discrete domain Ω should be

treated as the set of oriented edges (ainta), but we usually identify

it with the set of corresponding vertices a, and think about IntΩand ∂Ω as subsets of Γ, if no confusion arises.

Notation:

Let (Γ;EΓ) be an ininite planar graph embedded into C so that all

its edges (uv) ∈ EΓ are straight segments, wuv = wvu > 0 be some

xed edge weights (conductances), and µu :=∑

v∼u wuv for u ∈ Γ.

Discrete domains:

(VΩ;EΩint) bounded and

connected,

EΩbd := (ainta) ∈ EΩ

int :aint ∈ VΩ, a ∈ VΩ,

Ω := IntΩ ∪ ∂Ω, IntΩ := VΩ,

∂Ω:=(a ; (ainta)) : (ainta)∈EΩbd.

dashed polygonal representation

Notation:

Let (Γ;EΓ) be an ininite planar graph embedded into C so that all

its edges (uv) ∈ EΓ are straight segments, wuv = wvu > 0 be some

xed edge weights (conductances), and µu :=∑

v∼u wuv for u ∈ Γ.

Partition function of the random walk:

For a bounded discrete domain Ω ⊂ Γ and x , y ∈ Ω,

ZΩ(x ; y) :=∑

γ∈SΩ(x ;y)

w(γ), w(γ) :=

∏n(γ)−1k=0 wukuk+1∏n(γ)

k=0 µuk

,

where SΩ(x ; y) = γ = (x = u0 ∼ u1 ∼ · · · ∼ un(γ) = y) is the

set of all nearest-neighbor paths connecting x and y inside Ω(i.e., u1, . . . , un(γ)−1 ∈ IntΩ while we admit x , y ∈ ∂Ω).

Notation:

Let (Γ;EΓ) be an ininite planar graph embedded into C so that all

its edges (uv) ∈ EΓ are straight segments, wuv = wvu > 0 be some

xed edge weights (conductances), and µu :=∑

v∼u wuv for u ∈ Γ.

Partition function of the random walk:

For a bounded discrete domain Ω ⊂ Γ and x , y ∈ Ω,

ZΩ(x ; y) :=∑

γ∈SΩ(x ;y)

w(γ), w(γ) :=

∏n(γ)−1k=0 wukuk+1∏n(γ)

k=0 µuk

,

where SΩ(x ; y) = γ = (x = u0 ∼ u1 ∼ · · · ∼ un(γ) = y) is the

set of all nearest-neighbor paths connecting x and y inside Ω(i.e., u1, . . . , un(γ)−1 ∈ IntΩ while we admit x , y ∈ ∂Ω).

Further, for A,B ⊂ Ω, let ZΩ(A;B) :=∑

x∈A, y∈B ZΩ(x ; y).

Notation:

Let (Γ;EΓ) be an ininite planar graph embedded into C so that all

its edges (uv) ∈ EΓ are straight segments, wuv = wvu > 0 be some

xed edge weights (conductances), and µu :=∑

v∼u wuv for u ∈ Γ.

Partition function of the random walk:

For a bounded discrete domain Ω ⊂ Γ and x , y ∈ Ω,

ZΩ(x ; y) :=∑

γ∈SΩ(x ;y)

w(γ), w(γ) :=

∏n(γ)−1k=0 wukuk+1∏n(γ)

k=0 µuk

,

where SΩ(x ; y) = γ = (x = u0 ∼ u1 ∼ · · · ∼ un(γ) = y) is the

set of all nearest-neighbor paths connecting x and y inside Ω(i.e., u1, . . . , un(γ)−1 ∈ IntΩ while we admit x , y ∈ ∂Ω).

Examples: x , y ∈ IntΩ: GΩ(x ; y) Green's function in Ω;x ∈ IntΩ,B ⊂ ∂Ω: ωΩ(x ;B) hitting prob. (= harmonic measure).

Assumptions on the graph Γ and the edge weights wuv :

I uniformly bounded degrees: there exists a constant ν0 > 0 such

that, for all u ∈ Γ, µu :=∑

(uv)∈EΓ wuv 6 ν0 and wuv > ν−10 ;

Assumptions on the graph Γ and the edge weights wuv :

I uniformly bounded degrees: there exists a constant ν0 > 0 such

that, for all u ∈ Γ, µu :=∑

(uv)∈EΓ wuv 6 ν0 and wuv > ν−10 ;

I no at angles: there exists a constant η0 > 0 such that all

angles between neighboring edges do not exceed π − η0(NB: ⇒ all degrees of faces of Γ are bounded by 2π/η0);

Assumptions on the graph Γ and the edge weights wuv :

I uniformly bounded degrees: there exists a constant ν0 > 0 such

that, for all u ∈ Γ, µu :=∑

(uv)∈EΓ wuv 6 ν0 and wuv > ν−10 ;

I no at angles: there exists a constant η0 > 0 such that all

angles between neighboring edges do not exceed π − η0(NB: ⇒ all degrees of faces of Γ are bounded by 2π/η0);

I edge lengths are locally comparable: there exists a constant

ρ0 > 1 such that, for any vertex u ∈ Γ, one has

max(uv)∈EΓ

|v − u| 6 ρ0ru, where ru := min(uv)∈EΓ

|v − u|;

Assumptions on the graph Γ and the edge weights wuv :

I uniformly bounded degrees: there exists a constant ν0 > 0 such

that, for all u ∈ Γ, µu :=∑

(uv)∈EΓ wuv 6 ν0 and wuv > ν−10 ;

I no at angles: there exists a constant η0 > 0 such that all

angles between neighboring edges do not exceed π − η0(NB: ⇒ all degrees of faces of Γ are bounded by 2π/η0);

I edge lengths are locally comparable: there exists a constant

ρ0 > 1 such that, for any vertex u ∈ Γ, one has

max(uv)∈EΓ

|v − u| 6 ρ0ru, where ru := min(uv)∈EΓ

|v − u|;

I Γ is quantitatively locally nite : for any ρ > 1 there exists

some constant ν(ρ) > 0 such that, uniformly over all u ∈ Γ,

#v ∈ Γ : |v−u| 6 ρru 6 ν(ρ).

Assumptions on the graph Γ and the edge weights wuv :

I Assumption S (space): There exist two positive constants

η0, c0 > 0 such that, uniformly over all discrete discs BΓr (u),

u ∈ Γ, r > ru, and θ ∈ [0, 2π], one has

ωBΓr (u)

( u ; a ∈ ∂BΓr (v) : arg(a−u) ∈ [θ, θ+(π−η0)]) > c0.

Assumptions on the graph Γ and the edge weights wuv :

I Assumption S (space): There exist two positive constants

η0, c0 > 0 such that, uniformly over all discrete discs BΓr (u),

u ∈ Γ, r > ru, and θ ∈ [0, 2π], one has

ωBΓr (u)

( u ; a ∈ ∂BΓr (v) : arg(a−u) ∈ [θ, θ+(π−η0)]) > c0.

In other words, there are no exceptional directions: the random

walk started at the center of any discrete disc BΓr (u) can exit

this disc through any given boundary arc of the angle π−η0with probability uniformly bounded away from 0.

Assumptions on the graph Γ and the edge weights wuv :

I Assumption S (space): There exist two positive constants

η0, c0 > 0 such that, uniformly over all discrete discs BΓr (u),

u ∈ Γ, r > ru, and θ ∈ [0, 2π], one has

ωBΓr (u)

( u ; a ∈ ∂BΓr (v) : arg(a−u) ∈ [θ, θ+(π−η0)]) > c0.

I Assumption T (time): There exist two positive constants

c0,C0 > 0 such that, uniformly over all u ∈ Γ and r > ru,

c0r2 6

∑v∈IntBΓ

r (u)r2v GBΓ

r (u)(v ; u) 6 C0r

2.

Assumptions on the graph Γ and the edge weights wuv :

I Assumption S (space): There exist two positive constants

η0, c0 > 0 such that, uniformly over all discrete discs BΓr (u),

u ∈ Γ, r > ru, and θ ∈ [0, 2π], one has

ωBΓr (u)

( u ; a ∈ ∂BΓr (v) : arg(a−u) ∈ [θ, θ+(π−η0)]) > c0.

I Assumption T (time): There exist two positive constants

c0,C0 > 0 such that, uniformly over all u ∈ Γ and r > ru,

c0r2 6

∑v∈IntBΓ

r (u)r2v GBΓ

r (u)(v ; u) 6 C0r

2.

In other words, if one considers some time parametrization

such that the (expected) time spent by the walk at a vertex vbefore it jumps is of order r2v , then the expected time spent in

a discrete disc BΓr (u) by the random walk started at u before it

hits ∂BΓr (u) should be of order r2, uniformly over all discs.

Assumptions on the graph Γ and the edge weights wuv :

I Assumption S (space): There exist two positive constants

η0, c0 > 0 such that, uniformly over all discrete discs BΓr (u),

u ∈ Γ, r > ru, and θ ∈ [0, 2π], one has

ωBΓr (u)

( u ; a ∈ ∂BΓr (v) : arg(a−u) ∈ [θ, θ+(π−η0)]) > c0.

I Assumption T (time): There exist two positive constants

c0,C0 > 0 such that, uniformly over all u ∈ Γ and r > ru,

c0r2 6

∑v∈IntBΓ

r (u)r2v GBΓ

r (u)(v ; u) 6 C0r

2.

(Open) question: Do assumptions (a)(d) on the graph Γ and the

edge weights wuv listed on the previous page imply (S) and (T)?

Assumptions on the graph Γ and the edge weights wuv :

I Assumption S (space): There exist two positive constants

η0, c0 > 0 such that, uniformly over all discrete discs BΓr (u),

u ∈ Γ, r > ru, and θ ∈ [0, 2π], one has

ωBΓr (u)

( u ; a ∈ ∂BΓr (v) : arg(a−u) ∈ [θ, θ+(π−η0)]) > c0.

I Assumption T (time): There exist two positive constants

c0,C0 > 0 such that, uniformly over all u ∈ Γ and r > ru,

c0r2 6

∑v∈IntBΓ

r (u)r2v GBΓ

r (u)(v ; u) 6 C0r

2.

(Open) question: Do assumptions (a)(d) on the graph Γ and the

edge weights wuv listed on the previous page imply (S) and (T)?

(Closed) answer: (A. Nachmias, private communication): YES.

Uniform estimates of ωΩ(z ; (ab)) in simply connected Ω's:

Let Ω be a simply connected discrete domain, u ∈ IntΩ,and (ab) ⊂ ∂Ω be a (far from z) boundary arc. Let

CΩ(z) be the boundary of a discrete disc BΓr (z), r =

14 dist(z ; ∂Ω).

Uniform estimates of ωΩ(z ; (ab)) in simply connected Ω's:

Let Ω be a simply connected discrete domain, u ∈ IntΩ,and (ab) ⊂ ∂Ω be a (far from z) boundary arc. Let

CΩ(z) be the boundary of a discrete disc BΓr (z), r =

14 dist(z ; ∂Ω).

Theorem: For some constants β1,2,C1,2 > 0, the following

estimates are fullled uniformly for all congurations (Ω, z , a, b):

C1 exp[−β1L(Ω,z,a,b)] 6 ωΩ(z ; (ab)) 6 C2 exp[−β2L(Ω,z,a,b)],

where L(Ω,z,a,b) = LΩ(CΩ(z); (ab)) denotes the extremal length

(aka eective resistance) between CΩ(z) and (ab) in Ω.

Uniform estimates of ωΩ(z ; (ab)) in simply connected Ω's:

Let Ω be a simply connected discrete domain, u ∈ IntΩ,and (ab) ⊂ ∂Ω be a (far from z) boundary arc. Let

CΩ(z) be the boundary of a discrete disc BΓr (z), r =

14 dist(z ; ∂Ω).

Theorem: For some constants β1,2,C1,2 > 0, the following

estimates are fullled uniformly for all congurations (Ω, z , a, b):

C1 exp[−β1L(Ω,z,a,b)] 6 ωΩ(z ; (ab)) 6 C2 exp[−β2L(Ω,z,a,b)],

where L(Ω,z,a,b) = LΩ(CΩ(z); (ab)) denotes the extremal length

(aka eective resistance) between CΩ(z) and (ab) in Ω.

Corollary: Uniformly for all discrete domains (Ω, z , a, b), one has

log(1 + ω−1disc) ≍ Ldisc ≍ Lcont ≍ log(1 + ω−1

cont)

(hardly available by any coupling arguments, if ω's are exp. small)

Extremal Length LΩ(CΩ(z); (ab)):

I Can be dened via the unique solution of some boundary value

problem for discrete harmonic functions: Dirichlet (= 0) on(ab), Dirichlet (= 1) on CΩ(z), Neumann on ∂Ω \ (ab)

Extremal Length LΩ(CΩ(z); (ab)):

I Can be dened via the unique solution of some boundary value

problem for discrete harmonic functions: Dirichlet (= 0) on(ab), Dirichlet (= 1) on CΩ(z), Neumann on ∂Ω \ (ab);

I Equivalently, can be dened via some optimization problem

for discrete metrics (or electric currents) g : EΩ → R+

Extremal Length LΩ(CΩ(z); (ab)):

I Can be dened via the unique solution of some boundary value

problem for discrete harmonic functions: Dirichlet (= 0) on(ab), Dirichlet (= 1) on CΩ(z), Neumann on ∂Ω \ (ab);

I Equivalently, can be dened via some optimization problem

for discrete metrics (or electric currents) g : EΩ → R+:

LΩ(CΩ(z); (ab)) := supg :EΩ→R+

[Lg (CΩ(z); (ab))]2

Ag (Ω)

where Lg (CΩ(z); (ab)) := infγ:CΩ(z)↔(ab)

∑e∈γ ge

and Ag (Ω) :=∑

e∈EΩ weg2e .

Extremal Length LΩ(CΩ(z); (ab)):

I Can be dened via the unique solution of some boundary value

problem for discrete harmonic functions: Dirichlet (= 0) on(ab), Dirichlet (= 1) on CΩ(z), Neumann on ∂Ω \ (ab);

I Equivalently, can be dened via some optimization problem

for discrete metrics (or electric currents) g : EΩ → R+:

LΩ(CΩ(z); (ab)) := supg :EΩ→R+

[Lg (CΩ(z); (ab))]2

Ag (Ω)

where Lg (CΩ(z); (ab)) := infγ:CΩ(z)↔(ab)

∑e∈γ ge

and Ag (Ω) :=∑

e∈EΩ weg2e .

In particular, any function g : EΩ → R+

gives a lower bound for LΩ(CΩ(z); (ab)).

Extremal Length LΩ(CΩ(z); (ab)):

LΩ(CΩ(z); (ab)) := supg :EΩ→R+

[Lg (CΩ(z); (ab))]2

Ag (Ω)

Corollary: For any Ω ⊂ Z2 and some absolute constants β,C > 0,

ωΩ(z ; (ab)) 6 C exp[−β∑k1

k=k0ϑ−1k ].

Proof: take g := ϑ−1k on horizontal edges.

Uniform estimates of ωΩ(z ; (ab)) in simply connected Ω's:

Let Ω be a simply connected discrete domain, u ∈ IntΩ,and (ab) ⊂ ∂Ω be a (far from z) boundary arc. Let

CΩ(z) be the boundary of a discrete disc BΓr (z), r =

14 dist(z ; ∂Ω).

Theorem: For some constants β1,2,C1,2 > 0, the following

estimates are fullled uniformly for all congurations (Ω, z , a, b):

C1 exp[−β1L(Ω,z,a,b)] 6 ωΩ(z ; (ab)) 6 C2 exp[−β2L(Ω,z,a,b)],

where L(Ω,z,a,b) = LΩ(CΩ(z); (ab)) denotes the extremal length

(aka eective resistance) between CΩ(z) and (ab) in Ω.

Corollary: Uniformly for all discrete domains (Ω, z , a, b), one has

log(1 + ω−1disc) ≍ Ldisc ≍ Lcont ≍ log(1 + ω−1

cont)

Uniform estimates of ωΩ(z ; (ab)) in simply connected Ω's:

Let Ω be a simply connected discrete domain, u ∈ IntΩ,and (ab) ⊂ ∂Ω be a (far from z) boundary arc. Let

CΩ(z) be the boundary of a discrete disc BΓr (z), r =

14 dist(z ; ∂Ω).

Theorem: For some constants β1,2,C1,2 > 0, the following

estimates are fullled uniformly for all congurations (Ω, z , a, b):

C1 exp[−β1L(Ω,z,a,b)] 6 ωΩ(z ; (ab)) 6 C2 exp[−β2L(Ω,z,a,b)],

where L(Ω,z,a,b) = LΩ(CΩ(z); (ab)) denotes the extremal length

(aka eective resistance) between CΩ(z) and (ab) in Ω.

Corollary: Uniformly for all discrete domains (Ω, z , a, b), one has

log(1 + ω−1disc) ≍ Ldisc ≍ Lcont ≍ log(1 + ω−1

cont)

THANK YOU!

Uniform estimates of ωΩ(z ; (ab)) in simply connected Ω's:

Let Ω be a simply connected discrete domain, u ∈ IntΩ,and (ab) ⊂ ∂Ω be a (far from z) boundary arc. Let

CΩ(z) be the boundary of a discrete disc BΓr (z), r =

14 dist(z ; ∂Ω).

Theorem: For some constants β1,2,C1,2 > 0, the following

estimates are fullled uniformly for all congurations (Ω, z , a, b):

C1 exp[−β1L(Ω,z,a,b)] 6 ωΩ(z ; (ab)) 6 C2 exp[−β2L(Ω,z,a,b)],

where L(Ω,z,a,b) = LΩ(CΩ(z); (ab)) denotes the extremal length

(aka eective resistance) between CΩ(z) and (ab) in Ω.

Corollary: Uniformly for all discrete domains (Ω, z , a, b), one has

log(1 + ω−1disc) ≍ Ldisc ≍ Lcont ≍ log(1 + ω−1

cont)

If time permits ... some ideas of the proof on the next slides→

Some ideas of the proof:

I Work with discrete quadrilaterals (Ω; a, b, c , d): simply

connected domains with four marked boundary points

(then use some additional reduction to handle

ωΩ(z ; (ab)), RW partition functions in annuli,

and corresponding extremal lengths L(Ω,z,a,b)).

Some ideas of the proof:

I Work with discrete quadrilaterals (Ω; a, b, c , d): simply

connected domains with four marked boundary points;

I Discrete cross-ratios YΩ:

YΩ(a, b; c , d) :=

[ZΩ(a; d)ZΩ(b; c)

ZΩ(a; b)ZΩ(c ; d)

]1/2;

Some ideas of the proof:

I Work with discrete quadrilaterals (Ω; a, b, c , d): simply

connected domains with four marked boundary points;

I Discrete cross-ratios YΩ:

YΩ(a, b; c , d) :=

[ZΩ(a; d)ZΩ(b; c)

ZΩ(a; b)ZΩ(c ; d)

]1/2;

I RW partition function ZΩ = ZΩ((ab); (cd))

Some ideas of the proof:

I Work with discrete quadrilaterals (Ω; a, b, c , d): simply

connected domains with four marked boundary points;

I Discrete cross-ratios YΩ:

YΩ(a, b; c , d) :=

[ZΩ(a; d)ZΩ(b; c)

ZΩ(a; b)ZΩ(c ; d)

]1/2;

I RW partition function ZΩ = ZΩ((ab); (cd));

I Extremal length LΩ = LΩ((ab); (cd)).

Some ideas of the proof:

I Work with discrete quadrilaterals (Ω; a, b, c , d): simply

connected domains with four marked boundary points;

I Discrete cross-ratios YΩ:

YΩ(a, b; c , d) :=

[ZΩ(a; d)ZΩ(b; c)

ZΩ(a; b)ZΩ(c ; d)

]1/2;

I RW partition function ZΩ = ZΩ((ab); (cd));

I Extremal length LΩ = LΩ((ab); (cd)).

Theorem: Uniformly for all discrete quadrilaterals (Ω; a, b, c , d),

log(1 +YΩ)[!]≍ ZΩ 6 L−1

Ω

Some ideas of the proof:

I Work with discrete quadrilaterals (Ω; a, b, c , d): simply

connected domains with four marked boundary points;

I Discrete cross-ratios YΩ:

YΩ(a, b; c , d) :=

[ZΩ(a; d)ZΩ(b; c)

ZΩ(a; b)ZΩ(c ; d)

]1/2;

I RW partition function ZΩ = ZΩ((ab); (cd));

I Extremal length LΩ = LΩ((ab); (cd)).

Theorem: Uniformly for all discrete quadrilaterals (Ω; a, b, c , d),

log(1 +YΩ)[!]≍ ZΩ 6 L−1

Ω

log(1 + YΩ) ≍ ZΩ 6 L−1Ω ,

where YΩ, ZΩ and LΩ denote the same objects for (Ω; b, c , d , a).

Some ideas of the proof:

I Work with discrete quadrilaterals (Ω; a, b, c , d): simply

connected domains with four marked boundary points;

I Discrete cross-ratios YΩ;

I RW partition function ZΩ = ZΩ((ab); (cd));

I Extremal length LΩ = LΩ((ab); (cd)).

Theorem: Uniformly for all discrete quadrilaterals (Ω; a, b, c , d),

log(1 +YΩ)[!]≍ ZΩ 6 L−1

Ω

log(1 + YΩ) ≍ ZΩ 6 L−1Ω ,

where YΩ, ZΩ and LΩ denote the same objects for (Ω; b, c , d , a).

I YΩYΩ = 1, LΩLΩ ≍ 1. Moreover, ZΩ ≍ L−1Ω , if > const.

Some ideas of the proof:

Theorem: Uniformly for all discrete quadrilaterals (Ω; a, b, c , d),

ZΩ((ab); (cd)) ≍ log(1 +YΩ), YΩ =

[ZΩ(a; d)ZΩ(b; c)

ZΩ(a; b)ZΩ(c ; d)

]1/2.

Some ideas of the proof:

Theorem: Uniformly for all discrete quadrilaterals (Ω; a, b, c , d),

ZΩ((ab); (cd)) ≍ log(1 +YΩ), YΩ =

[ZΩ(a; d)ZΩ(b; c)

ZΩ(a; b)ZΩ(c ; d)

]1/2.

I (Factorization) Theorem: Uniformly for all (Ω; a, c , d),

ZΩ(a; (cd)) ≍ [ZΩ(a; c)ZΩ(a; d) /ZΩ(c ; d) ]1/2

Some ideas of the proof:

Theorem: Uniformly for all discrete quadrilaterals (Ω; a, b, c , d),

ZΩ((ab); (cd)) ≍ log(1 +YΩ), YΩ =

[ZΩ(a; d)ZΩ(b; c)

ZΩ(a; b)ZΩ(c ; d)

]1/2.

I (Factorization) Theorem: Uniformly for all (Ω; a, c , d),

ZΩ(a; (cd)) ≍ [ZΩ(a; c)ZΩ(a; d) /ZΩ(c ; d) ]1/2

I ⇒ if YΩ 6 const, then ZΩ ≍ YΩ (... sum along (ab) ...)

Some ideas of the proof:

Theorem: Uniformly for all discrete quadrilaterals (Ω; a, b, c , d),

ZΩ((ab); (cd)) ≍ log(1 +YΩ), YΩ =

[ZΩ(a; d)ZΩ(b; c)

ZΩ(a; b)ZΩ(c ; d)

]1/2.

I (Factorization) Theorem: Uniformly for all (Ω; a, c , d),

ZΩ(a; (cd)) ≍ [ZΩ(a; c)ZΩ(a; d) /ZΩ(c ; d) ]1/2

I ⇒ if YΩ 6 const, then ZΩ ≍ YΩ (... sum along (ab) ...);

I In particular, if YΩ is of order 1, then ZΩ is of order 1 too.

Some ideas of the proof:

Theorem: Uniformly for all discrete quadrilaterals (Ω; a, b, c , d),

ZΩ((ab); (cd)) ≍ log(1 +YΩ), YΩ =

[ZΩ(a; d)ZΩ(b; c)

ZΩ(a; b)ZΩ(c ; d)

]1/2.

I (Factorization) Theorem: Uniformly for all (Ω; a, c , d),

ZΩ(a; (cd)) ≍ [ZΩ(a; c)ZΩ(a; d) /ZΩ(c ; d) ]1/2

I ⇒ if YΩ 6 const, then ZΩ ≍ YΩ (... sum along (ab) ...);

I ⇒ if YΩ > const, then ZΩ ≍ logYΩ: partition functions ZΩ

are additive while cross-ratios YΩ are multiplicative as one

splits the arc (ab) into smaller arcs (a0a1) ∪ · · · ∪ (an−1an).

Some ideas of the proof:

Theorem: Uniformly for all discrete quadrilaterals (Ω; a, b, c , d),

ZΩ((ab); (cd)) ≍ log(1 +YΩ), YΩ =

[ZΩ(a; d)ZΩ(b; c)

ZΩ(a; b)ZΩ(c ; d)

]1/2.

I (Factorization) Theorem: Uniformly for all (Ω; a, c , d),

ZΩ(a; (cd)) ≍ [ZΩ(a; c)ZΩ(a; d) /ZΩ(c ; d) ]1/2

I ⇒ if YΩ 6 const, then ZΩ ≍ YΩ (... sum along (ab) ...);

I ⇒ if YΩ > const, then ZΩ ≍ logYΩ: partition functions ZΩ

are additive while cross-ratios YΩ are multiplicative as one

splits the arc (ab) into smaller arcs (a0a1) ∪ · · · ∪ (an−1an).

THANK YOU ONCE MORE!