## Seminar information archive

Seminar information archive ～04/12｜Today's seminar 04/13 | Future seminars 04/14～

#### Seminar on Probability and Statistics

16:30-17:40 Room #122 (Graduate School of Math. Sci. Bldg.)

ASYMPTOTICALLY EFFICIENT DISCRETE HEDGING

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/07.html

**深澤 正彰**(大阪大学 金融・保険教育研究センター)ASYMPTOTICALLY EFFICIENT DISCRETE HEDGING

[ Abstract ]

The notion of asymptotic efficiency for discrete hedging is introduced and a discretizing strategy which is asymptotically efficient is given explicitly. A lower bound for asymptotic risk of discrete hedging is given, which is attained by a simple discretization scheme. Numerical results for delta hedging in the Black-Scholes model are also presented.

[ Reference URL ]The notion of asymptotic efficiency for discrete hedging is introduced and a discretizing strategy which is asymptotically efficient is given explicitly. A lower bound for asymptotic risk of discrete hedging is given, which is attained by a simple discretization scheme. Numerical results for delta hedging in the Black-Scholes model are also presented.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/07.html

#### thesis presentations

13:00-14:15 Room #122 (Graduate School of Math. Sci. Bldg.)

Asymptotic Analysis for Stochastic Volatility (確率的ボラティティの漸近解析)

**深澤 正彰**(大阪大学 金融・保険教育研究センター)Asymptotic Analysis for Stochastic Volatility (確率的ボラティティの漸近解析)

### 2009/10/21

#### GCOE lecture series

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Mini course on the gradient models, Ⅲ: Non convex potentials at high temperature

**Jean-Dominique Deuschel**(TU Berlin)Mini course on the gradient models, Ⅲ: Non convex potentials at high temperature

[ Abstract ]

In the non convex case, the situation is much more complicated. In fact Biskup and Kotecky describe a non convex model with several ergodic components. We investigate a model with non convex interaction for which unicity of the ergodic component, scaling limits and large deviations can be proved at sufficiently high temperature. We show how integration can generate strictly convex potential, more precisely that marginal measure of the even sites satisfies the random walk representation. This is a joint work with Codina Cotar and Nicolas Petrelis.

In the non convex case, the situation is much more complicated. In fact Biskup and Kotecky describe a non convex model with several ergodic components. We investigate a model with non convex interaction for which unicity of the ergodic component, scaling limits and large deviations can be proved at sufficiently high temperature. We show how integration can generate strictly convex potential, more precisely that marginal measure of the even sites satisfies the random walk representation. This is a joint work with Codina Cotar and Nicolas Petrelis.

#### Number Theory Seminar

16:30-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

The local Simpson correspondence in positive characteristic

**Bernard Le Stum**(Université de Rennes 1)The local Simpson correspondence in positive characteristic

[ Abstract ]

A Simpson correspondance should relate Higgs bundles to differential modules (or local systems). We stick here to positive characteristic and recall some old and recent results : Cartier isomorphism, Van der Put's classification, Kaneda's theorem and Ogus-Vologodsky local theory. We'll try to explain how the notion of Azumaya algebra is a convenient tool to unify these results. Our main theorem is the equivalence between quasi-nilpotent differential modules of level m and quasi-nilpotent Higgs Bundles (depending on a lifting of Frobenius mod p-squared). This result is a direct generalization of the previous ones. The main point is to understand the Azumaya nature of the ring of differential operators of level m. Following Berthelot, we actually use the dual theory and study the partial divided power neighborhood of the diagonal.

A Simpson correspondance should relate Higgs bundles to differential modules (or local systems). We stick here to positive characteristic and recall some old and recent results : Cartier isomorphism, Van der Put's classification, Kaneda's theorem and Ogus-Vologodsky local theory. We'll try to explain how the notion of Azumaya algebra is a convenient tool to unify these results. Our main theorem is the equivalence between quasi-nilpotent differential modules of level m and quasi-nilpotent Higgs Bundles (depending on a lifting of Frobenius mod p-squared). This result is a direct generalization of the previous ones. The main point is to understand the Azumaya nature of the ring of differential operators of level m. Following Berthelot, we actually use the dual theory and study the partial divided power neighborhood of the diagonal.

#### Lectures

14:40-16:10 Room #128 (Graduate School of Math. Sci. Bldg.)

冨田竹崎理論とその応用 (2)

**竹崎正道**(UCLA)冨田竹崎理論とその応用 (2)

#### Seminar on Probability and Statistics

15:00-16:10 Room #002 (Graduate School of Math. Sci. Bldg.)

AR過程の優調和事前分布と偏自己相関係数による表示

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/06.html

**田中 冬彦**(科学技術振興機構さきがけ)AR過程の優調和事前分布と偏自己相関係数による表示

[ Abstract ]

Tanaka and Komaki(2008)では時系列データが2次の自己回帰過程(AR過程)に従う 時のスペクトル密度の推定を考え、優調和事前分布に基づいたベイズスペクトル 密度の方がジェフリーズ事前分布に基づいたベイズスペクトル密度よりも精度よ く推定できることを示している。高次のAR過程での優調和事前分布はTanaka( 2009)によって初めて与えられたが、特性方程式の根を用いた表示のため、数値 実験を行う上では取り扱いづらかった。本発表では高次のAR過程への応用を念頭 において偏自己相関係数(PAC)によるパラメータ表示を導入し数値実験した結 果を紹介する。 また、PACパラメータによる表示は解析的な取扱いをする上でも利点があり、AR 過程の優調和事前分布に関して新しく得られた結果も幾つか紹介したい。

[ Reference URL ]Tanaka and Komaki(2008)では時系列データが2次の自己回帰過程(AR過程)に従う 時のスペクトル密度の推定を考え、優調和事前分布に基づいたベイズスペクトル 密度の方がジェフリーズ事前分布に基づいたベイズスペクトル密度よりも精度よ く推定できることを示している。高次のAR過程での優調和事前分布はTanaka( 2009)によって初めて与えられたが、特性方程式の根を用いた表示のため、数値 実験を行う上では取り扱いづらかった。本発表では高次のAR過程への応用を念頭 において偏自己相関係数(PAC)によるパラメータ表示を導入し数値実験した結 果を紹介する。 また、PACパラメータによる表示は解析的な取扱いをする上でも利点があり、AR 過程の優調和事前分布に関して新しく得られた結果も幾つか紹介したい。

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/06.html

### 2009/10/20

#### Tuesday Seminar on Topology

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Torus fibrations and localization of index

**吉田 尚彦**(明治大学大学院理工学研究科)Torus fibrations and localization of index

[ Abstract ]

I will describe a localization of index of a Dirac type operator.

We make use of a structure of torus fibration, but the mechanism

of the localization does not rely on any group action. In the case of

Lagrangian fibration, we show that the index is described as a sum of

the contributions from Bohr-Sommerfeld fibers and singular fibers.

To show the localization we introduce a deformation of a Dirac type

operator for a manifold equipped with a fiber bundle structure which

satisfies a kind of acyclic condition. The deformation allows an

interpretation as an adiabatic limit or an infinite dimensional analogue

of Witten deformation.

Joint work with Hajime Fujita and Mikio Furuta.

I will describe a localization of index of a Dirac type operator.

We make use of a structure of torus fibration, but the mechanism

of the localization does not rely on any group action. In the case of

Lagrangian fibration, we show that the index is described as a sum of

the contributions from Bohr-Sommerfeld fibers and singular fibers.

To show the localization we introduce a deformation of a Dirac type

operator for a manifold equipped with a fiber bundle structure which

satisfies a kind of acyclic condition. The deformation allows an

interpretation as an adiabatic limit or an infinite dimensional analogue

of Witten deformation.

Joint work with Hajime Fujita and Mikio Furuta.

#### Lectures

14:40-16:10 Room #128 (Graduate School of Math. Sci. Bldg.)

冨田竹崎理論とその応用 (1)

**竹崎正道**(UCLA)冨田竹崎理論とその応用 (1)

### 2009/10/19

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Variation formulas for principal functions (II)

**濱野佐知子**(松江高専)Variation formulas for principal functions (II)

#### Algebraic Geometry Seminar

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

ファノ多様体上の有理曲線の鎖の長さについて

**渡辺 究**(早稲田大学基幹理工学研究科)ファノ多様体上の有理曲線の鎖の長さについて

[ Abstract ]

ピカール数1のファノ多様体に対し、一般の二点を結ぶために必要な

極小有理曲線の本数を「長さ」と呼び、それについて考える。特に、5次元以下の

ファノ多様体や余指数が3以下のファノ多様体などに対し、長さを求める。

ピカール数1のファノ多様体に対し、一般の二点を結ぶために必要な

極小有理曲線の本数を「長さ」と呼び、それについて考える。特に、5次元以下の

ファノ多様体や余指数が3以下のファノ多様体などに対し、長さを求める。

### 2009/10/15

#### Lecture Series on Mathematical Sciences in Soceity

16:20-17:50 Room #056 (Graduate School of Math. Sci. Bldg.)

社会における学位取得者の役割Ⅰ

**藤原 洋**(インターネット総合研究所代表取締役所長)社会における学位取得者の役割Ⅰ

#### Lie Groups and Representation Theory

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Hecke-Clifford superalgebras and crystals of type $D^{(2)}_{l}$

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

**土岡俊介**(RIMS, Kyoto University)Hecke-Clifford superalgebras and crystals of type $D^{(2)}_{l}$

[ Abstract ]

It is known that we can sometimes describe the representation theory of ``Hecke algebra'' by ``Lie theory''. Famous examples that involve the Lie theory of type $A^{(1)}_n$ are Lascoux-Leclerc-Thibon's interpretation of Kleshchev's modular branching rule for the symmetric groups and Ariki's theorem generalizing Lascoux-Leclerc-Thibon's conjecture for the Iwahori-Hecke algebras of type A.

Brundan and Kleshchev showed that some parts of the representation theory of the affine Hecke-Clifford superalgebras and its finite-dimensional ``cyclotomic'' quotients are controlled by the Lie theory of type $A^{(2)}_{2l}$ when the quantum parameter $q$ is a primitive $(2l+1)$-th root of unity.

In this talk, we show that similar theorems hold when $q$ is a primitive $4l$-th root of unity by replacing the Lie theory of type $A^{(2)}_{2l}$ with that of type $D^{(2)}_{l}$.

[ Reference URL ]It is known that we can sometimes describe the representation theory of ``Hecke algebra'' by ``Lie theory''. Famous examples that involve the Lie theory of type $A^{(1)}_n$ are Lascoux-Leclerc-Thibon's interpretation of Kleshchev's modular branching rule for the symmetric groups and Ariki's theorem generalizing Lascoux-Leclerc-Thibon's conjecture for the Iwahori-Hecke algebras of type A.

Brundan and Kleshchev showed that some parts of the representation theory of the affine Hecke-Clifford superalgebras and its finite-dimensional ``cyclotomic'' quotients are controlled by the Lie theory of type $A^{(2)}_{2l}$ when the quantum parameter $q$ is a primitive $(2l+1)$-th root of unity.

In this talk, we show that similar theorems hold when $q$ is a primitive $4l$-th root of unity by replacing the Lie theory of type $A^{(2)}_{2l}$ with that of type $D^{(2)}_{l}$.

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2009/10/14

#### GCOE lecture series

15:30-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Macroscopic fluctuation theory for nonequilibrium stationary states, Ⅳ

**Claudio Landim**(IMPA, Brazil)Macroscopic fluctuation theory for nonequilibrium stationary states, Ⅳ

#### GCOE lecture series

13:30-15:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Mini course on the gradient models, Ⅱ: Convex interaction potential

**Jean-Dominique Deuschel**(TU Berlin)Mini course on the gradient models, Ⅱ: Convex interaction potential

[ Abstract ]

Much is known for strictly convex interactions, which under rescaling behave much like the harmonic model. In particular the unicity of the ergodic component have been established by Funaki and Spohn, and the scaling limit to the gradient of the continuous gaussian free field by Naddaf and Spencer. The results are based on special analytical and probabilistic tools such as the Brascamp-Lieb inequality and the Hellfer-Sj\\"osstrand random walk representation. These techniques rely on the strict convexity of the interaction potential.

Much is known for strictly convex interactions, which under rescaling behave much like the harmonic model. In particular the unicity of the ergodic component have been established by Funaki and Spohn, and the scaling limit to the gradient of the continuous gaussian free field by Naddaf and Spencer. The results are based on special analytical and probabilistic tools such as the Brascamp-Lieb inequality and the Hellfer-Sj\\"osstrand random walk representation. These techniques rely on the strict convexity of the interaction potential.

#### Geometry Seminar

14:45-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Fixed point theorems for non-positively curved spaces and random groups

Lagrangian mean curvature flow and symplectic area

**近藤剛史 (Kondo Takefumi)**(神戸大学大学院理学研究科) 14:45-16:15Fixed point theorems for non-positively curved spaces and random groups

[ Abstract ]

It is not easy to construct a finitely generated group with a fixed point property for non-positively curved spaces. However, if we randomly choose relators, then we can get examples of such groups. To show this, we need a criterion for deducing a fixed point property from a local property of a group. In this talk, we will introduce one such criterion, and our approach is via a scaling limit argument.

It is not easy to construct a finitely generated group with a fixed point property for non-positively curved spaces. However, if we randomly choose relators, then we can get examples of such groups. To show this, we need a criterion for deducing a fixed point property from a local property of a group. In this talk, we will introduce one such criterion, and our approach is via a scaling limit argument.

**赤穂まなぶ (Akaho Manabu)**(首都大学東京大学院理工学研究科) 16:30-18:00Lagrangian mean curvature flow and symplectic area

[ Abstract ]

In this talk, we consider symplectic area of smooth maps from a Riemann surface with boundary on embedded Lagrangian mean curvature flow in Kahler-Einstein manifolds. As an application, we observe a relation between embedded Lagrangian mean curvature flow and Floer theory of monotone Lagrangian submanifolds in Kahler-Einstein manifolds; in this case non-trivial holomorphic discs turn out to be an obstruction to the existence of long time solution of the flow.

In this talk, we consider symplectic area of smooth maps from a Riemann surface with boundary on embedded Lagrangian mean curvature flow in Kahler-Einstein manifolds. As an application, we observe a relation between embedded Lagrangian mean curvature flow and Floer theory of monotone Lagrangian submanifolds in Kahler-Einstein manifolds; in this case non-trivial holomorphic discs turn out to be an obstruction to the existence of long time solution of the flow.

#### GCOE Seminars

16:30-17:30 Room #370 (Graduate School of Math. Sci. Bldg.)

Partial Cauchy data for general second order elliptic operators in two dimensions

**O. Emanouilov**(Colorado State University)Partial Cauchy data for general second order elliptic operators in two dimensions

[ Abstract ]

We consider the problem of determining the coefficients of a first-order perturbation of the Laplacian in two dimensions by measuring the corresponding Cauchy data on an arbitrary open subset of the boundary. From this information we obtained a coupled PDE system of first order which the coefficients satisfy. As a corollary we show for the magnetic Schr"odinger equation that the magnetic field and the electric potential are uniquely determined by measuring the partial Cauchy data on an arbitrary part of the boundary. We also show that the coefficients of any real vector field perturbation of the Laplacian, the convection terms, are uniquely determined by their partial Cauchy data.

We consider the problem of determining the coefficients of a first-order perturbation of the Laplacian in two dimensions by measuring the corresponding Cauchy data on an arbitrary open subset of the boundary. From this information we obtained a coupled PDE system of first order which the coefficients satisfy. As a corollary we show for the magnetic Schr"odinger equation that the magnetic field and the electric potential are uniquely determined by measuring the partial Cauchy data on an arbitrary part of the boundary. We also show that the coefficients of any real vector field perturbation of the Laplacian, the convection terms, are uniquely determined by their partial Cauchy data.

### 2009/10/13

#### Tuesday Seminar on Topology

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Instanton Floer homology for lens spaces

**笹平 裕史**(東京大学大学院数理科学研究科)Instanton Floer homology for lens spaces

[ Abstract ]

Let Y be an oriented closed 3-manifold and P be an SU(2)-bundle on Y. Under a certain condition, instanton Floer homology for Y can be defined as the Morse homology of the Chern-Simons functional. The condition is that all flat connections on P are irreducible. When there is a reducible flat connection on P, instanton Floer homology is not defined in general.

Since the fundamental group of a lens sapce is commutative, all flat connections on the lens space are reducible. In this talk I will introduce instanton Floer homology for lens spaces. I also show calculations for some lens spaces.

Let Y be an oriented closed 3-manifold and P be an SU(2)-bundle on Y. Under a certain condition, instanton Floer homology for Y can be defined as the Morse homology of the Chern-Simons functional. The condition is that all flat connections on P are irreducible. When there is a reducible flat connection on P, instanton Floer homology is not defined in general.

Since the fundamental group of a lens sapce is commutative, all flat connections on the lens space are reducible. In this talk I will introduce instanton Floer homology for lens spaces. I also show calculations for some lens spaces.

#### Lie Groups and Representation Theory

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Extensions between finite-dimensional simple modules over a generalized current Lie algebra

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

**小寺諒介**(東京大学)Extensions between finite-dimensional simple modules over a generalized current Lie algebra

[ Abstract ]

$\\mathfrak{g}$を$\\mathbb{C}$上の有限次元半単純Lie代数,$A$を有限生成可換$\\mathbb {C}$代数とする.

テンソル積$A \\otimes \\mathfrak{g}$に自然にLie代数の構造を与えたものを一般化されたカレントLie代数と呼ぶ.

一般化されたカレントLie代数の任意の2つの有限次元既約表現に対して,その1次のExt群を完全に決定することができたので,その結果について発表する.

[ Reference URL ]$\\mathfrak{g}$を$\\mathbb{C}$上の有限次元半単純Lie代数,$A$を有限生成可換$\\mathbb {C}$代数とする.

テンソル積$A \\otimes \\mathfrak{g}$に自然にLie代数の構造を与えたものを一般化されたカレントLie代数と呼ぶ.

一般化されたカレントLie代数の任意の2つの有限次元既約表現に対して,その1次のExt群を完全に決定することができたので,その結果について発表する.

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2009/10/09

#### Lecture Series on Mathematical Sciences in Soceity

16:20-17:50 Room #117 (Graduate School of Math. Sci. Bldg.)

暗号の実践編

**岡本龍明**(NTT 情報流通プラットフォーム研究所 岡本特別研究室長)暗号の実践編

#### GCOE lecture series

16:30-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Associated varieties for Representations of classical Lie

super-algebras

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

**Michel Duflo**(Paris 7)Associated varieties for Representations of classical Lie

super-algebras

[ Abstract ]

In this lecture, I'll discuss the notion of "Associated

varieties for Representations of classical Lie super-algebras (joint work with Vera Serganova)" and the relation with the degree of atypicality. This is related to a conjecture of Kac and Wakimoto.

[ Reference URL ]In this lecture, I'll discuss the notion of "Associated

varieties for Representations of classical Lie super-algebras (joint work with Vera Serganova)" and the relation with the degree of atypicality. This is related to a conjecture of Kac and Wakimoto.

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2009/10/07

#### PDE Real Analysis Seminar

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Wiener measure and Feynman-Kac formula on the Heisenberg group

**劉和平(Liu Heping)**(Beijing University)Wiener measure and Feynman-Kac formula on the Heisenberg group

[ Abstract ]

It is well known that the Feynman-Kac formula on the Euclidean space gives the solution of Schrodinger equation by the Wiener integral. We will discuss the Wiener measure and Feynman-Kac formula on the Heisenberg group. The results hold on the H-type groups.

It is well known that the Feynman-Kac formula on the Euclidean space gives the solution of Schrodinger equation by the Wiener integral. We will discuss the Wiener measure and Feynman-Kac formula on the Heisenberg group. The results hold on the H-type groups.

#### GCOE lecture series

16:30-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Representations of classical Lie super-algebras

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

**Michel Duflo**(Paris 7)Representations of classical Lie super-algebras

[ Abstract ]

In this lecture, I'll survey classical topics on finite dimensional representations of classical Lie super-algebras, in particular the notion of the degree of atypicality.

[ Reference URL ]In this lecture, I'll survey classical topics on finite dimensional representations of classical Lie super-algebras, in particular the notion of the degree of atypicality.

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

#### Number Theory Seminar

16:30-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

On GAGA theorems for the rigide-étale topology

**Ahmed Abbes**(Université de Rennes 1)On GAGA theorems for the rigide-étale topology

[ Abstract ]

Last year, I finished my course in Todai on "Rigide Geometry following M. Raynaud" by stating a GAGA theorem for the rigide-étale topology, due to Gabber and Fujiwara. I will give a new proof of this theorem, inspired by another theorem of Gabber, namely the Affine analog of the proper base change theorem.

Last year, I finished my course in Todai on "Rigide Geometry following M. Raynaud" by stating a GAGA theorem for the rigide-étale topology, due to Gabber and Fujiwara. I will give a new proof of this theorem, inspired by another theorem of Gabber, namely the Affine analog of the proper base change theorem.

### 2009/10/05

#### GCOE lecture series

15:30-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Macroscopic fluctuation theory for nonequilibrium stationary states, Ⅲ

**Claudio Landim**(IMPA, Brazil)Macroscopic fluctuation theory for nonequilibrium stationary states, Ⅲ

#### GCOE lecture series

13:30-15:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Mini course on the gradient models, I: Effective gradient models, definitions and examples

**Jean-Dominique Deuschel**(TU Berlin)Mini course on the gradient models, I: Effective gradient models, definitions and examples

[ Abstract ]

We describe a phase separation in $R^{d+1}$ by an effective interface model with basis in $Z^d$ and height in $R$. We assume that the interaction potential depends only on the discrete gradient and that the a priori measure is the product Lebesgue measure. Note that this is an unbounded massless model with continuous symmetry and this implies that the interface is delocalized for the infinite model in lower lattice dimensions $d=1,2$. Instead of looking at the distribution of the height of the interface itself, we consider the measure on the height differences the so called gradient Gibbs measure, which exists in any dimensions. The gradient field must satisfy the loop condition, that is the sum of the gradient along any closed loop is zero, this implies a long range interaction with a slow decay of the correlations. We are interested in characterizing the ergodic components of this gradient field, in the decay of correlations, large deviations and continuous scaling limits. As an example we consider the harmonic or discrete gaussian free field with quadratic interactions.

We describe a phase separation in $R^{d+1}$ by an effective interface model with basis in $Z^d$ and height in $R$. We assume that the interaction potential depends only on the discrete gradient and that the a priori measure is the product Lebesgue measure. Note that this is an unbounded massless model with continuous symmetry and this implies that the interface is delocalized for the infinite model in lower lattice dimensions $d=1,2$. Instead of looking at the distribution of the height of the interface itself, we consider the measure on the height differences the so called gradient Gibbs measure, which exists in any dimensions. The gradient field must satisfy the loop condition, that is the sum of the gradient along any closed loop is zero, this implies a long range interaction with a slow decay of the correlations. We are interested in characterizing the ergodic components of this gradient field, in the decay of correlations, large deviations and continuous scaling limits. As an example we consider the harmonic or discrete gaussian free field with quadratic interactions.

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