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research
Fluctuations for the Ginzburg-Landau
β
Ο
\nabla \phi
β
Ο
Interface Model on a Bounded Domain
Authors
A. Naddaf
B. Helffer
+20Β more
C. Kipnis
D. Stroock
E. Bolthausen
G. Giacomin
G.B. Arous
H. Brascamp
I. Karatzas
J.-D. Deuschel
J.-D. Deuschel
J.-D. Deuschel
J.-D. Deuschel
Jason Miller
O. Daviaud
O. Schramm
R. Kenyon
S. Sheffield
S. Sheffield
T. Funaki
T. Funaki
T. Funaki
Publication date
8 June 2010
Publisher
'Springer Science and Business Media LLC'
Doi
Cite
View
on
arXiv
Abstract
We study the massless field on
D
n
=
D
β©
1
n
Z
2
D_n = D \cap \tfrac{1}{n} \Z^2
D
n
β
=
D
β©
n
1
β
Z
2
, where
D
β
R
2
D \subseteq \R^2
D
β
R
2
is a bounded domain with smooth boundary, with Hamiltonian
\CH(h) = \sum_{x \sim y} \CV(h(x) - h(y))
. The interaction
\CV
is assumed to be symmetric and uniformly convex. This is a general model for a
(
2
+
1
)
(2+1)
(
2
+
1
)
-dimensional effective interface where
h
h
h
represents the height. We take our boundary conditions to be a continuous perturbation of a macroscopic tilt:
h
(
x
)
=
n
x
β
u
+
f
(
x
)
h(x) = n x \cdot u + f(x)
h
(
x
)
=
n
x
β
u
+
f
(
x
)
for
x
β
β
D
n
x \in \partial D_n
x
β
β
D
n
β
,
u
β
R
2
u \in \R^2
u
β
R
2
, and
f
ββ£
:
R
2
β
R
f \colon \R^2 \to \R
f
:
R
2
β
R
continuous. We prove that the fluctuations of linear functionals of
h
(
x
)
h(x)
h
(
x
)
about the tilt converge in the limit to a Gaussian free field on
D
D
D
, the standard Gaussian with respect to the weighted Dirichlet inner product
(
f
,
g
)
β
Ξ²
=
β«
D
β
i
Ξ²
i
β
i
f
i
β
i
g
i
(f,g)_\nabla^\beta = \int_D \sum_i \beta_i \partial_i f_i \partial_i g_i
(
f
,
g
)
β
Ξ²
β
=
β«
D
β
β
i
β
Ξ²
i
β
β
i
β
f
i
β
β
i
β
g
i
β
for some explicit
Ξ²
=
Ξ²
(
u
)
\beta = \beta(u)
Ξ²
=
Ξ²
(
u
)
. In a subsequent article, we will employ the tools developed here to resolve a conjecture of Sheffield that the zero contour lines of
h
h
h
are asymptotically described by
S
L
E
(
4
)
SLE(4)
S
L
E
(
4
)
, a conformally invariant random curve.Comment: 58 page
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info:doi/10.1007%2Fs00220-011-...
Last time updated on 11/12/2019