The Arctic Circle Revisited
arXiv:0704.0362v1  [math-ph]  3 Apr 2007
The Arctic Circle Revisited
F. Colomo and A.G. Pronko
Abstract. The problem of limit shapes in the six-vertex model with domain
wall boundary conditions is addressed by considering a specially tailored bulk
correlation function, the emptiness formation probability. A closed expression
of this correlation function is given, both in terms of certain determinant and
multiple integral, which allows for a systematic treatment of the limit shapes
of the model for full range of values of vertex weights. Specifically, we show
that for vertex weights corresponding to the free-fermion line on the phase
diagram, the emptiness formation probability is related to a one-matrix model
with a triple logarithmic singularity, or Triple Penner model. The saddle-point
analysis of this model leads to the Arctic Circle Theorem, and its generalization
to the Arctic Ellipses, known previously from domino tilings.
1. Introduction
The Arctic Circle has first appeared in the study of domino tilings of large
Aztec diamonds [EKLP, JPS]. The name originates from the fact that in most
configurations the dominoes are ‘frozen’ outside the circle inscribed into the dia-
mond, while the interior of the circle is a disordered, or ‘temperate’, zone. Further
investigations of the domino tilings of Aztec diamonds, such as details of statistics
near the circle, can be found in [CEP, J1, J2]. Here we mention that the Arctic
Circle is a particular example of a limit shape in dimer models, in the sense that
it describes the shape of a spatial phase separation of order and disorder. Apart
from domino tilings, many more examples have been discussed recently, see, among
others, papers [CKP, CLP, KO, KOS, OR].
As long as only dimer models are considered, this amounts to restrict to dis-
crete free-fermionic models, although with nontrivial boundary conditions. Indeed,
many of them can be viewed as a six-vertex model at its Free Fermion point (the
correspondence being however usually not bijective), with suitably chosen fixed
boundary conditions. In particular, this is the case of domino tilings of Aztec dia-
monds [EKLP], and the corresponding boundary conditions of the six-vertex model
are the so-called Domain Wall Boundary Conditions (DWBC). Hence the problem
of limit shapes extends to the six-vertex model with generic weights, and with fixed
boundary conditions, among which the case of DWBC is the most interesting.
Historically, the six-vertex model with DWBC was first considered in paper [K]
within the framework of Quantum Inverse Scattering Method [KBI] to prove the
Gaudin hypothesis for norms of Bethe states. The model was subsequently solved
2000 Mathematics Subject Classification. 15A52, 82B05, 82B20, 82B23.
http://arxiv.org/abs/0704.0362v1
2 F. COLOMO AND A.G. PRONKO
in paper [I] where a determinant formula for the partition function was given; see
also [ICK] for a detailed exposition. Quite independently, the model was later
found, under certain restrictions on the vertex weights, to be deeply related with
enumerations of alternating sign matrices (see, e.g., [Br] for a review) and, as
already mentioned, to domino tilings of Aztec diamonds [EKLP].
Concerning the problem of limit shapes for the six-vertex model with DWBC,
as far as the Free Fermion point is considered, the relation with domino tilings
provided apparently an indirect proof of the corresponding Arctic Circle. The non-
bijective nature of the correspondence between the two models asked for more direct
results, purposely for the free-fermion six-vertex model, see [Zi1, FS, KP]. Out
of the Free Fermion point, however, only very few analytical results are available,
such as exact expressions for boundary one-point [BPZ] and two-point [FP, CP1]
correlation functions. The present knowledge on the subject is based mainly on
numerics [E, SZ, AR]; some steps towards finding the limit shapes of the model
have been done recently in [PR].
In the present note we propose a rather direct strategy to address the problem:
after briefly reviewing the six-vertex model with DWBC, we define a bulk corre-
lation function, the Emptiness Formation Probability (EFP), which discriminates
the ordered and disordered phase regions. We give for this correlation function two
equivalent representations, in terms of a determinant and of a multiple integral.
The core derivation of EFP is heavily based on the Quantum Inverse Scattering
Method [KBI], along the lines of papers [BPZ, CP1]; it is out of the scope of the
present paper, corresponding details being given in a separate publication [CP4].
Here our aim is to demonstrate how the limit shapes for the considered model
can be extracted from EFP in a suitable scaling limit, by making use of ideas and
techniques of Random Matrix Models.
To be more specific, and to establish a contact with previous results, we spe-
cialize here our further discussion to the case of free-fermion six-vertex model. We
show that the asymptotic analysis of multiple integral formula for EFP in the scal-
ing limit reduces to a saddle-point problem for a one-matrix model with a triple
logarithmic singularity, or triple Penner model. We argue that the limit shape cor-
responds to condensation of all saddle-point solutions to a single point. This allows
us to recover the known Arctic Circle and Ellipses.
As a comment to our approach, it is to be stressed that it is directly tailored on
the six-vertex model, rather than domino tilings. For this reason it is not restricted
to the free-fermion models, even if, of course, further significant efforts might be
necessary, essentially from the point of view of Random Matrix Model reformu-
lation, for application to more general situations. On the basis of our previous
results in [CP2], however, the application of the method to the particular case of
the so-called Ice Point of the model should be straightforward. This would provide
the limit shape of alternating sign matrices.
2. The model
2.1. The six-vertex model. The six-vertex model (for reviews, see [LW,
Ba]) is formulated on a square lattice with arrows lying on edges, and obeying the
so-called ‘ice-rule’, namely, the only admitted configurations are such that there are
always two arrows pointing away from, and two arrows pointing into, each lattice
vertex. An equivalent and graphically simpler description of the configurations of
THE ARCTIC CIRCLE REVISITED 3
w1 w2 w3 w4 w5 w6
Figure 1. The six allowed types of vertices in terms of arrows
and lines, and their Boltzmann weights.
Figure 2. A possible configuration of the six-vertex model with
DWBC at N = 4, in terms of arrows and lines.
the model can be given in terms of lines flowing through the vertices: for each arrow
pointing downward or to the left, draw a thick line on the corresponding edge. This
line picture implements the ‘ice-rule’ in an automated way. The six possible vertex
states and the Boltzmann weights w1, w2, . . . , w6 assigned to each vertex according
to its state are shown in Figure 1.
2.2. Domain Wall Boundary Conditions. The Domain Wall Boundary
Conditions (DWBC) are imposed on the N×N square lattice by fixing the direction
of all arrows on the boundaries in a specific way. Namely, the vertical arrows on the
top and bottom of the lattice point inward, while the horizontal arrows on the left
and right sides point outward. Equivalently, a generic configuration of the model
with DWBC can be depicted by N lines flowing from the upper boundary to the
left one. A possible state of the model both in terms of arrows and of lines is shown
in Figure 2.
2.3. Partition function. The partition function is defined, as usual, as a
sum over all possible arrow configurations, compatible with the imposed DWBC,
each configuration being assigned its Boltzmann weight, given as the product of all
the corresponding vertex weights,
arrow configurations
with DWBC
wn11 w
2 . . . w
Here n1, n2, . . . , n6 denote the numbers of vertices with weights w1, w2, . . . , w6,
respectively, in each arrow configuration (n1 + n2 + · · ·+ n6 = N
2.4. Anisotropy parameter and phases of the model. The six-vertex
model with DWBC can be considered, with no loss of generality, with its weights
invariant under the simultaneous reversal of all arrows,
w1 = w2 =: a , w3 = w4 =: b , w5 = w6 =: c .
4 F. COLOMO AND A.G. PRONKO
Under different choices of Boltzmann weights the six-vertex model exhibits different
behaviours, according to the value of the parameter ∆, defined as
a2 + b2 − c2
It is well known that there are three physical regions or phases for the six-vertex
model: the ferroelectric phase, ∆ > 1; the anti-ferroelectric phase, ∆ < −1; the
disordered phase, −1 < ∆ < 1. Here we restrict ourselves to the disordered phase,
where the Boltzmann weights are conveniently parameterized as
a = sin(λ+ η) , b = sin(λ− η) , c = sin 2η . (2.1)
With this choice one has ∆ = cos 2η. The parameter λ is the so-called spectral
parameter and η is the crossing parameter. The physical requirement of positive
Boltzmann weights, in the disordered regime, restricts the values of the crossing
and spectral parameters to 0 < η < π/2 and η < λ < π − η.
The special case η = π/4 (or ∆ = 0) is related to free fermions on a lattice, and
there is a well-known correspondence with dimers and domino tilings. In particular,
at λ = π/2, the ∆ = 0 six-vertex model with DWBC is related to the domino tilings
of Aztec diamond. For arbitrary λ ∈ [π/4, 3π/4], we shall refer to the ∆ = 0 case
as the Free Fermion line.
The case η = π/6 (i.e. ∆ = 1/2) and λ = π/2, where all weights are equal,
a = b = c, is known as the Ice Point; all configurations are given the same weight.
In this case there is a one to one correspondence between configurations of the
model with DWBC and N ×N alternating sign matrices.
2.5. Phase separation and limit shapes. The six-vertex model exhibits
spatial separation of phases for a wide choice of fixed boundary conditions, and,
in particular, in the case of DWBC. Roughly speaking, the effect is related to the
fact that ordered configurations on the boundary can induce, through the ice-rule,
a macroscopic order inside the lattice.
The notion of phase separation acquires a precise meaning in the scaling limit,
that is the thermodynamic/continuum limit, performed by sending the number of
sites N to infinity and the lattice spacing to zero, while keeping the total size of the
lattice fixed, e.g., to 1. On a finite lattice, several macroscopic regions may appear,
which in the scaling limit are expected to be sharply separated by some curves, the
so-called Arctic curves.
For the six-vertex model with DWBC the shape of the Artic curve, or limit
shape, has been found rigorously only on the Free Fermion line, and for the closely
related domino tilings of Aztec diamond [JPS, CEP, Zi1, FS, KP]. For generic
values of weights the limit shapes are not known, but the whole picture is strongly
supported both numerically [E, SZ, AR] and analytically [KZ, Zi2, BF, PR].
3. Emptiness Formation Probability
3.1. Definition. We shall use the following coordinates on the lattice: r =
1, . . . , N labels the vertical lines from right to left; s = 1, . . . , N labels the horizontal
lines from top to bottom. We may now introduce the correlation function FN (r, s),
measuring the probability for the first s horizontal edges between the r-th and
r+1-th line to be all ‘full’ (i.e. thick in the line picture, or with a left arrow in the
THE ARCTIC CIRCLE REVISITED 5
Figure 3. Emptiness Formation Probability. The sum in (3.1) is
performed over all configurations compatible with the drawn ar-
rows.
standard picture of the six-vertex model):
FN (r, s) =
‘constrained’
arrow configurations
with DWBC
wn11 w
2 . . . w
6 . (3.1)
Here the sum is performed over all arrow configurations on the N × N lattice,
subjected to the restriction of DWBC, and to the condition that all arrows on the
first s edges between the r-th and r + 1-th line should point left, see Figure 3.
Although this correlation function may appear rather sophisticated, it is com-
putable in some closed form by means of the Quantum Inverse Scattering Method,
on which DWBC are indeed tailored. It is the natural adaptation of the Empti-
ness Formation Probability of quantum spin chains to the present model. For this
reason, and to link to the common practice in the quantum integrable models com-
munity, even if FN (r, s) actually describes ‘fullness’ formation probability, we shall
call it Emptiness Formation Probability (EFP).
3.2. Qualitative discussion of FN (r, s). Let us restrict ourselves to the dis-
ordered regime, −1 < ∆ < 1, for definiteness. From previous analytical and nu-
merical work, in the large N limit the emergence of a limit shape, in the form of
a continuous closed curve touching once each of the four sides of the lattice, is ex-
pected. It follows that five regions emerge in the lattice: a central region, enclosed
by the curve, and four corner regions, lying outside the closed curve and delimited
by the sides of the lattice. The central region is disordered, while the four corners
are frozen, with mainly vertices of type 1, 3, 2, 4 (see Figure 1) appearing in the
top-left, top-right, bottom-right and bottom-left corner, respectively.
By construction, EFP is expected to be almost one in frozen regions of type 1,
or 3, bordering the top side of the lattice, and to be rather small otherwise. DWBC
exclude a region of type 3 to emerge in the upper part of the lattice. Hence FN (r, s)
describes, at a given value of r, as s increases, a transition from a frozen region of
vertices of type 1, where FN (r, s) ∼ 1, to a generic region where FN (r, s) ∼ 0.
It follows that FN (r, s) can describe only the upper left portion of the closed
curve, between its top and left contact points. Nevertheless, it should be mentioned
that the full curve can be built from the knowledge of its top left portion, just
6 F. COLOMO AND A.G. PRONKO
exploiting the crossing symmetry of the six-vertex model. Hence EFP, FN (r, s), is
well suited to describe limit shapes.
3.3. Some notations. For a given choice of parameters λ, η we define
sin 2η
sin(λ+ η) sin(λ− η)
and the integration measure on the real line
µ(x) := ex(λ−π/2)
sinh(ηx)
sinh(πx/2)
related to ϕ as follows:
µ(x) dx .
Let us introduce the complete set of monic orthogonal polynomial {Pn(x)}n=0,1,...
associated to the integration measure µ(x), with the orthogonality relation
Pn(x)Pm(x)µ(x) dx = hnδnm .
The square norms hn are completely determined by the measure µ(x), and may
be expressed, in principle, in terms of its moments. In the following we shall be
interested in the complete set of orthogonal polynomials {Kn(x)}n=0,1,... defined as
Kn(x) = n!ϕ
n+1 1
Pn(x) .
We moreover define
ω(ǫ) :=
sin(ǫ)
sin(ǫ− 2η)
, ω̃(ǫ) :=
sin(ǫ)
sin(ǫ + 2η)
Note that the following relation holds
a2 ω̃ − 2∆ab ω̃ω + b2 ω = 0 , (3.2)
allowing to express ω̃ in terms of ω.
3.4. Determinant representation. For EFP in the six-vertex model with
DWBC, the following representation holds:
FN (r, s) = (−1)
s det
1≤j,k≤s
KN−k(∂ǫj )
[ω(ǫj)]
[ω(ǫj)− 1]
1≤j<k≤s
[ω̃(ǫj)− 1] [ω(ǫk)− 1]
ω̃(ǫj)ω(ǫk)− 1
ǫ1=0,...,ǫs=0
. (3.3)
This representation has been obtained in the framework of the Quantum Inverse
Scattering Method [KBI], along the lines of analogous derivations worked out for
one-point and two-point boundary correlation functions of the model [BPZ, CP1].
The details of the derivation can be found in [CP4].
THE ARCTIC CIRCLE REVISITED 7
3.5. The boundary correlation function. If we consider expression (3.3)
when s = 1, we recover the boundary polarization, introduced and computed in
[BPZ]. It is convenient to consider the closely related boundary correlation function
HN (r) := FN (r, 1)− FN (r − 1, 1) .
As shown in [BPZ, CP1], the following representation holds:
HN (r) = KN−1(∂ǫ)
[ω(ǫ)]N−r
[ω(ǫ)− 1]N−1
We define the corresponding generating function
hN (z) :=
HN (r) z
r−1 . (3.4)
Noticing that ω(ǫ) → 0 as ǫ → 0, it can be shown that, given any arbitrary function
f(z) regular in a neighbourhood of the origin, the following inverse representation
holds
KN−1(∂ǫ)f(ω(ǫ))
(z − 1)N−1
hN(z)f(z) dz . (3.5)
Here C0 is a closed counterclockwise contour in the complex plane, enclosing the
origin, and no other singularity of the integrand.
3.6. Multiple integral representation. Plugging (3.5) into representation
(3.3), we readily obtain the following multiple integral representation for EFP:
FN (r, s) =
· · ·
dsω det
1≤j,k≤s
hN−k+1(ωj)
ωj − 1
ωN−r−1j
(ωj − 1)N
1≤j<k≤s
(ω̃j − 1)(ωk − 1)
ω̃jωk − 1
. (3.6)
Here ω̃j ’s should be expressed in terms of ωj ’s through (3.2). Indeed, due to (3.5),
relation (3.2) for functions ω(ǫ), ω̃(ǫ), translates directly into the same relation
between ωj and ω̃j , j = 1, . . . , s.
Representation (3.6), and all results in this Section hold for any choice of param-
eters λ and η within the disordered regime. Moreover, by analytical continuation
in parameters λ and η, these results can be easily extended to all other regimes.
The determinant in expression (3.6) is a particular representation of the parti-
tion function of the six-vertex model with DWBC, when the homogeneous limit is
performed only on a subset of the spectral parameters [CP3]. The structure of the
previous multiple integral representation therefore closely recalls analogous ones for
the Heisenberg XXZ quantum spin chain correlation functions [JM, KMT].
For generic values of λ and η, the orthogonal polynomialsKn(x), or the generat-
ing function hN (z), are known only in terms of rather implicit representations. For-
tunately, there are three notable exceptions [CP2]: the Free Fermion line (η = π/4,
−π/4 < λ < π/4, ∆ = 0), the Ice Point (η = π/6, λ = π/2, ∆ = 1/2), and the Dual
Ice Point (η = π/3, λ = π/2, ∆ = −1/2). In these three cases, the Kn(x) turn
out to be classical orthogonal polynomials, namely Meixner-Pollaczek, Continuous
Hahn and Continuous Dual Hahn polynomial, respectively. Correspondingly, the
8 F. COLOMO AND A.G. PRONKO
generating function can be represented explicitly in terms of terminating hyperge-
ometric functions that may simplify considerably further evaluation of EFP. In the
next Section we shall focus on the case of Free Fermion line.
4. Multiple integral representation at ∆ = 0
4.1. Specialization to η = π/4. We shall now restrict ourselves to the case
η = π/4. We have ∆ = 0, and the six-vertex model reduces to a model of free
fermions on the lattice. The parameter λ can still assume any value in the interval
(−π/4, π/4). It is convenient to trade λ for the new parameter
τ = tan2(λ− π/4) , 0 < τ < ∞ .
The symmetric point (related to the domino tiling of Aztec Diamond) corresponds
now to τ = 1. For generic values of τ we have:
ω̃ = −τω .
The generating function (3.4) is known explicitely (see [CP2] for details):
hN (z) =
1 + τz
1 + τ
Plugging this expression into (3.6), we get
FN (r, s) =
· · ·
dsω det
1≤j,k≤s
(1 + τωj)(ωj − 1)
(1 + τ)ωj
ωN−r−1j
(ωj − 1)N
1≤j<k≤s
(1 + τωj)(ωk − 1)
1 + τωjωk
. (4.1)
4.2. Symmetrization. After extracting a common factor
(1 + τωj)(ωj − 1)
(1 + τ)ωj
from the determinant in (4.1), we recognize it to be of Vandermonde type. We can
therefore collect from the integrand of (4.1) the double product
1≤j<k≤s
(1 + τωj)(ωj − 1)
(1 + τ)ωj
(1 + τωk)(ωk − 1)
(1 + τ)ωk
(1 + τωj)(ωk − 1)
1 + τωjωk
Noticing that the integration and the remaining of integrand are fully symmetric
under permutation of variables ω1, . . . , ωj , we can perform total symmetrization of
the previous double product over all its variables, with the result
(−1)s(s−1)/2
ωs−1j
1≤j<k≤s
(ωj − ωk)
THE ARCTIC CIRCLE REVISITED 9
Hence, we finally obtain the following representation for EFP on the Free Fermion
line:
FN (r, s) =
(−1)s(s+1)/2
s!(1 + τ)s(N−s)(2πi)s
· · ·
1≤j<k≤s
(ωj − ωk)
(1 + τωj)
(ωj − 1)s ω
. (4.2)
The appearance of a squared Vandermonde determinant in this expression naturally
recalls the partition functions of s× s Random Matrix Models.
5. Triple Penner model and Arctic Ellipses
5.1. Scaling limit. We shall now address the asymptotic behaviour of expres-
sion (4.2) for EFP in the ∆ = 0 case. We are interested in the limit N, r, s → ∞,
while keeping the ratios
r/N = x , s/N = y ,
fixed. In this limit, x, y ∈ [0, 1] will parameterize the unit square to which the
lattice is rescaled. Correspondingly EFP is expected to approach a limit function
F (x, y) := lim
FN (xN, yN) , x, y ∈ [0, 1] .
We shall exploit the standard approach developed for instance in the investigation
of asymptotic behaviour for Random Matrix Models. Before this let us however
point out some facts which holds already for any finite value of s.
5.2. A useful identity. Consider the quantity
IN (r, s) :=
(−1)s(s+1)/2
s!(1 + τ)s(N−s)(2πi)s
· · ·
1≤j<k≤s
(ωj − ωk)
(1 + τωj)
(ωj − 1)s ω
which differs from (4.2) only in the integration contours. Here C1 is a closed,
clockwise oriented contour (note the change in orientation) in the complex plane
enclosing point ω = 1, and no other singularity of the integrand. We have the
identity
IN (r, s) = 1 (5.1)
for any integer r, s = 1, . . . , N . The simplest way to prove the previous identity is
by shifting ωj → ωj + 1, and rewriting IN (r, s) as an Hankel determinant; indeed
we have
IN (r, s) =
(−1)s(s−1)/2
(1 + τ)s(N−s)
1≤j,k≤s
ωj+k−2−s(1 + τ + τω)N−s
(1 + ω)r
The entries of the Hankel matrix vanish for j+k > s+1, and hence the determinant
is simply given by the product of the antidiagonal entries, j + k = s + 1 (modulo
a sign (−1)s(s−1)/2 emerging from the permutation of all columns). Identity (5.1)
follows immediately.
10 F. COLOMO AND A.G. PRONKO
5.3. Saddle-point evaluation for large N and finite s. When using the
saddle-point method in variables ω1, . . . , ωs to evaluate the behaviour of FN (r, s) for
large N and r, and finite s, it is rather easy to see that the saddle-point equations
decouple at leading order, and that each saddle-point will be on the real axis,
contributing with a factor e−NSj with Sj positive.
If a given saddle-point is smaller than 1, the contour C0 can be deformed
through the saddle-point without encountering any pole, and its contribution will
vanish as e−NSj in the large N limit. If however the saddle-point, still on the real
axis, happens to be larger than 1, the deformation of the contour C0 through the
saddle-point will pick up the contribution of the pole at ω = 1 (with a reversed
orientation of the contour), and the j-th integral will behave as 1 + e−NSj . Hence,
in the large N limit (at fixed s) the quantity FN (r, s) will vanish unless all the
saddle-points are greater than 1, in which case FN (r, s) ∼ IN (r, s) = 1. Note that
in the present situation the s saddle-points coincide. A detailed analysis shows that
in this case the position of the s saddle-points depends on the value x = r/N as
τ(1−x)
. In correspondence to the value x0 =
, for which these saddle-
points are exactly 1, the function F (x, 0) has a step discontinuity. More precisely,
it is easy to show that for x ∈ [0, 1], F (x, 0) = θ(x − x0), where θ(x) is Heaviside
step function. From a physical point of view x0 is the contact point between the
limit shape and the boundary. What have been discussed here can easily be verified
in the case s = 1. The extension to finite s > 1 is rather direct as well.
5.4. Saddle-point equation. Having in mind the analogy with s × s Ran-
dom Matrix Models, and the scaling limit specified in Section 5.1, we rewrite our
expression for FN (r, s) at ∆ = 0 as follows:
FN (r, s) =
(−1)s(s+1)/2
s!(1 + τ)s
2(1/y−1)(2πi)s
· · ·
dsω exp
j,k=1
j 6=k
ln |ωj − ωk|
ln(τωj + 1)− ln(ωj − 1)−
ln(ωj)
. (5.2)
Both sums in the exponent are O(s2). The corresponding (coupled) saddle-point
equations read
ωj − 1
(1/y − 1)τ
τωj + 1
k 6=j
ωj − ωk
. (5.3)
A standard physical picture reinterprets the saddle-point equations as the equi-
librium condition for the positions of s charged particle confined to the real axis,
with logarithmic electrostatic repulsion, in an external potential. In the present
case the latter can be seen as generated by three external charges, 1, x/y, and
−(1/y − 1) at positions 1, 0, and −1/τ , respectively. It is natural to refer to this
model as the triple Penner model. Although the simple Penner [P] matrix model
has been widely investigated, not so much is known about the much more compli-
cate double Penner model [M, PW]. We have not been able to trace any previous
study concerning the triple Penner model.
THE ARCTIC CIRCLE REVISITED 11
5.5. The exact Green function at finite s. To investigate the structure of
solutions of the saddle-point equations (5.3) for large s we first introduce the Green
function
Gs(z) =
z − ωj
which, if the ωj ’s solves (5.3), has to satisfy the differential equation:
z(z − 1)(τz + 1)
sG′s(z) + s
2G2s(z)
− s(αz2 + βz + γ)sGs(z)
τs(s− 1)− αs2
z + (1 − τ)s(s− 1)− βs2 +Ω
2τs(s− 1)− αs2
. (5.4)
The coefficients α, β and γ are readily obtained as the coefficients of the second
order polynomial appearing in the numerator, when setting to common denominator
the left hand side of (5.3). We give them explicitly for later convenience:
α = τ
, β =
+ (1− τ)
, γ = −
The derivation of the differential equation is very standard (see, e.g., [SD]). The
left hand side is built by suitably combining the explicit definition of the Green
function and its derivative. The result has to be a polynomial of the first degree in
z, whose coefficients are constructed by matching the leading and first subleading
behaviour of the left hand side as |z| → ∞.
5.6. The first moment Ω. The quantity Ω appearing in (5.4) is defined as
the first moment of the solutions of the saddle-point equations:
It is related in a obvious way to the first subleading coefficient of Gs(z); indeed,
from the definition of the Green function, it is evident that
Gs(z) =
+O(z−3) , |z| → ∞ .
It is worth to emphasize that Ω is still unknown, and that in principle its value
should be determined self consistently by first working out the explicit solution
of Gs(z) (which will depend implicitly on Ω), from (5.4) and then demanding that
j=1 ωj evaluated from this solution coincides with Ω. The appearance of the un-
determined parameter Ω is a manifestation of the ‘two-cuts’ nature of the Random
Matrix Model related to (5.2), see, e.g., par. 6.7 of [D1].
5.7. The asymptotic Green function. We are now in condition to perform
the large s (and large N , r) limit at fixed x, y. In the limit, we can neglect terms of
order O(s) in the differential equation (5.4), which therefore reduces to an algebraic
equation for the limiting Green function G(z):
z(z−1)(τz+1)[G(z)]2−(αz2+βz+γ)G(z) = (τ−α)z+(1−τ−β)+Ω(2τ−α) . (5.5)
The previous algebraic equation has to be supplemented by the normalization con-
dition
G(z) ∼
, |z| → ∞ . (5.6)
12 F. COLOMO AND A.G. PRONKO
Hence the Green function describing the large s asymptotic distribution of solutions
for the saddle equation (5.3) reads:
G(z) =
2z(z − 1)(τz + 1)
(αz2 + βz + γ)
(αz2 + βz + γ)2 + 4z(z − 1)(τz + 1)[(τ − α)z +Ω(2τ − α) + 1− τ − β]
(5.7)
We have selected the positive branch of the square root, to satisfy the normalization
condition (note that the coefficient of z4 under the square root is (α − 2τ)2, and
α − 2τ is negative for any x, y ∈ [0, 1]). However, the expression for G(z) is not
completely specified yet, because Ω is still undetermined.
5.8. Limit shape and condensation of roots. The polynomial under the
square root is of fourth order, hence G(z) will have in general two cuts in the
complex plane. The emergence of a two-cut problem was already expected from
the appearance of the undetermined first moment Ω in (5.4). The discontinuity of
G(z) across these cuts defines, when positive, the density of solutions of the saddle-
point equations (5.3) when s → ∞. The problem of explicitly finding this density,
for arbitrary α, β, γ (or x, y), is a formidable one, not to mention the evaluation of
the corresponding ‘free energy’, and of the saddle-point contribution to the integral
in (5.2). But our aim is much more modest, since we are presently interested only
in the expression of the limit shape, i.e. in the curve in the square x, y ∈ [0, 1],
delimiting regions where F (x, y) = 0 from regions where F (x, y) = 1. Of course we
are here somehow assuming that the transition of F (x, y) from 0 to 1 is stepwise in
the scaling limit, but this is supported both by the physical interpretation of EFP
(in the disordered region, by definition, the number of ‘thin’ lines is macroscopic,
and the probability of finding no ‘thin’ horizontal edges immediately vanishes in
the scaling limit) and by the discussion of Section 5.3.
As explained in the discussion of the double Penner model in paper [PW], the
logarithmic wells in the potential can behave as condensation germs for the saddle-
point solutions. In our case, this can role can be played only by the ‘charge’ at
ω = 1 in the Penner potential since the charge at ω = −1/τ is always repulsive,
while the one at ω = 0 is larger than 1, at least in the region of interest. [PW]
have shown that condensation can occur only for charges less than or equal to 1,
since this will be the fraction of condensed solutions. This consideration, together
with the expected stepwise behaviour and the discussion in Section 5.3, suggest
the following picture for the evolution of saddle-point solution density from the
disordered region, F (x, y) ∼ 0, to the upper left frozen region, F (x, y) ∼ 1: in the
disordered region there is a macroscopic fraction of solutions which are real and
smaller than 1, while in the upper left frozen region this fraction vanish. On the
basis of the discussion here and in Sections 3.2 and 5.3, we shall assume that at
the transition between the two regions all saddle-point solutions have condensed at
ω = 1.
5.9. Main assumption. We claim that the Arctic curve in the square x, y ∈
[0, 1] separating the disordered phase from the upper left frozen phase is defined by
the condition that all solutions of the saddle-point equation lies at ω = 1.
In the derivation of the limit shape, this is indeed the only assumption to which
we are unable to provide a proof. There is in fact no guarantee, at this level, for
THE ARCTIC CIRCLE REVISITED 13
this possibility to occur, and limit shapes could in principle emerge from a different
condition. But if for some values of x, y ∈ [0, 1] we have all solutions of the saddle-
point equation condensing at ω = 1, then this provides a transition mechanism
from 0 to 1 for F (x, y), and this might correspondingly define some limit shape.
If all saddle-point solutions condensate at ω = 1, then we obviously have:
Ω = 1 .
Moreover, the complicate expression (5.7) for G(z) should simply reduce to
G(z) =
z − 1
, (5.8)
since we expect to have no cuts, and only one pole at z = 1 with unit residue.
5.10. Arctic Ellipses. Consider the quartic polynomial under the square root
in (5.7). It is convenient to rewrite it in terms of
α̃ := 2τ − α = τ
β̃ := 2− β = τ
x+ y − 1
y − x
γ̃ := −γ =
(5.9)
Note that α̃ and γ̃ are always positive for x, y ∈ [0, 1]. When Ω = 1, our quartic
polynomial reads
α̃2z4 + 2α̃β̃z3 + (β̃2 + 2α̃γ̃)z2 + 2β̃γ̃z + γ̃2 ,
which may be equivalently rewritten as
(α̃z2 + β̃z + γ̃)2 .
We see that the quartic polynomial reduces to a perfect square, and hence, when
Ω = 1, the two cuts of G(z) disappear, as expected.
Now, when Ω = 1, in our new notations, the Green function reads:
G(z) =
[(2τ − α̃)z2 + (2− β̃)z − γ̃] +
(α̃z2 + β̃z + γ̃)2
2z(τz + 1)(z − 1)
. (5.10)
We now require the coefficients α̃, β̃, γ̃ to be such that the polynomial under
the square root combines with the first part of the numerator in (5.10) to give
2z(τz+1) and simplify the Green function according to (5.8). Once we have chosen
a given branch of the square root (the positive one, in order to satisfy normalization
condition (5.6)), it is obvious that the required simplification can occur for any z
in the complex plane only if the second order polynomial α̃z2 + β̃z + γ̃ does not
change its sign, i.e. only if its two roots coincide, implying:
β̃2 − 4α̃γ̃ = 0 .
Rewriting the last relation in terms of x, y, through (5.9), we readily get
(1 + τ)2x2 + (1 + τ)2y2 − 2(1− τ2)xy − 2τ(1 + τ)x − 2τ(1 + τ)y + τ2 = 0 .
We have therefore recovered the limit shape, which in this Free Fermion case is the
well-known Arctic Ellipse (Arctic Circle for τ = 1) [JPS, CEP]. We recall that,
as discussed in Section 3.2, F (x, y) is non-vanishing only in the upper left region
14 F. COLOMO AND A.G. PRONKO
of the unit square. Therefore, concerning EFP, only the upper left portion of the
Arctic curve, between the two contact points at ( τ
, 0) and (1, 1
), is relevant.
6. Concluding remarks
Our starting point has been the definition of a relatively simple but relevant
correlation function for the six-vertex model with DWBC, the Emptiness Formation
Probability. We have provided both a determinant representation and a multiple
integral representation for the proposed correlation function. This is the first ex-
ample in literature of a bulk (as opposed to boundary) correlation function for the
considered model, for generic weights.
The multiple integral representation, specialized to the Free Fermion case, has
been studied in the scaling limit. In the standard picture of Random Matrix Mod-
els, we recognize the emergence of a triple Penner model. Assuming condensation of
the roots of saddle point equations in correspondence to a limit shape, we recover
the well-known Arctic Circle and Ellipse. It would be interesting to investigate
whether universality considerations of Random Matrix Models (see, e.g., [D2]) can
be extended to the Penner model in the neighbourhood of its logarithmic singular-
ities. This would imply directly the results of [CEP, J1, J2] on the Tracy-Widom
distribution and the Airy process, emerging in a suitably rescaled neighbourhood
of the Arctic Ellipse.
It is worth to stress that the multiple integral representation for EFP presented
in Section 3 can be studied beyond the usual Free Fermion situation. We expect
that condensation of roots of the saddle point equation in correspondence of the
limit shape is a general phenomenon. We believe that this assumption could be of
importance in addressing the problem of limit shapes in the six-vertex model with
DWBC.
Our derivation of the limit shape in the Free Fermion case uses the explicit
knowledge of function hN (z), standing in the multiple integral representation (3.6).
It is worth mentioning that function hN(z) is also known explicitly at Ice Point,
(∆ = 1/2), and Dual Ice Point, (∆ = −1/2), being expressible in terms of (poly-
nomial) Gauss hypergeometric function [Ze, CP2]. For instance, at Ice Point the
triple Penner model discussed above generalizes to a two-matrix Penner model.
This model can be studied along the lines presented here, thus providing a solution
to the longstanding problem of limit shape for Alternating Sign Matrices.
Acknowledgements
We thank Nicolai Reshetikhin for useful discussion, and for giving us a draft
of [PR] before completion. FC is grateful to Percy Deift, and Courant Institute
of Mathematical Science, for warm hospitality. AGP thanks INFN, Sezione di
Firenze, where part of this work was done. We acknowledge financial support from
MIUR PRIN program (SINTESI 2004). One of us (AGP) is also supported in
part by Civilian Research and Development Foundation (grant RUM1-2622-ST-
04), by Russian Foundation for Basic Research (grant 04-01-00825), and by the
program Mathematical Methods in Nonlinear Dynamics of Russian Academy of
Sciences. This work is partially done within the European Community network
EUCLID (HPRN-CT-2002-00325), and the European Science Foundation program
INSTANS.
THE ARCTIC CIRCLE REVISITED 15
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I.N.F.N., Sezione di Firenze and Dipartimento di Fisica, Università di Firenze, Via
G. Sansone 1, 50019 Sesto Fiorentino (FI), Italy
E-mail address: colomo@fi.infn.it
Saint Petersburg Department of Steklov Mathematical Institute of Russian Acad-
emy of Sciences, Fontanka 27, 191023 Saint Petersburg, Russia
E-mail address: agp@pdmi.ras.ru
