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J.Propp, The shape of a typical boxed plane partition
 J. of Math
, 1998
"... Abstract. Using a calculus of variations approach, we determine the shape of a typical plane partition in a large box (i.e., a plane partition chosen at random according to the uniform distribution on all plane partitions whose solid Young diagrams fit inside the box). Equivalently, we describe the ..."
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Cited by 51 (5 self)
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Abstract. Using a calculus of variations approach, we determine the shape of a typical plane partition in a large box (i.e., a plane partition chosen at random according to the uniform distribution on all plane partitions whose solid Young diagrams fit inside the box). Equivalently, we describe the distribution of the three different orientations of lozenges in a random lozenge tiling of a large hexagon. We prove a generalization of the classical formula of MacMahon for the number of plane partitions in a box; for each of the possible ways in which the tilings of a region can behave when restricted to certain lines, our formula tells the number of tilings that behave in that way. When we take a suitable limit, this formula gives us a functional which we must maximize to determine the asymptotic behavior of a plane partition in a box. Once the variational problem has been set up, we analyze it using a modification of the methods employed by Logan and Shepp and by Vershik and Kerov in their studies of random Young tableaux. 1.
Conformal Invariance of Domino Tiling
 Ann. Probab
, 1999
"... this paper we deal with the twodimensional lattice dimer model, or domino tiling model (a domino tiling is a tiling with 2 \Theta 1 and 1 \Theta 2 rectangles). We prove that in the limit as the lattice spacing ffl tends to zero, certain macroscopic properties of the tiling are conformally invariant ..."
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Cited by 35 (10 self)
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this paper we deal with the twodimensional lattice dimer model, or domino tiling model (a domino tiling is a tiling with 2 \Theta 1 and 1 \Theta 2 rectangles). We prove that in the limit as the lattice spacing ffl tends to zero, certain macroscopic properties of the tiling are conformally invariant. The height function h on a domino tiling is an integervalued function on the vertices in a tiling. It is defined below in section 2.2; see also [4, 19]. One can think of a domino tiling of U as a map h from U
The Planar Dimer Model With Boundary: A Survey.
 CRM Proceedings and Lecture Notes
, 1998
"... this paper we would like to give a short survey of some of these new results. ..."
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Cited by 15 (1 self)
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this paper we would like to give a short survey of some of these new results.
The asymptotic determinant of the discrete Laplacian
, 1998
"... We compute the asymptotic determinant of the discrete Laplacian on a simplyconnected rectilinear region in R 2 . Specifically, for each ffl ? 0 let H ffl be the subgraph of fflZ 2 whose vertices lie in a fixed rectilinear polygon U . Let N(H ffl ) denote the number of vertices of H ffl and B(H ..."
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We compute the asymptotic determinant of the discrete Laplacian on a simplyconnected rectilinear region in R 2 . Specifically, for each ffl ? 0 let H ffl be the subgraph of fflZ 2 whose vertices lie in a fixed rectilinear polygon U . Let N(H ffl ) denote the number of vertices of H ffl and B(H ffl ) the number of vertices on the boundary (the outer face). Then the log of the determinant of the Laplacian on H ffl has the following asymptotic expansion in ffl: 4G N(H ffl ) + log( p 2 \Gamma 1) 2 B(H ffl ) \Gamma 48 r2 (ffl; U) + o(1) where G is Catalan's constant and r2(ffl; U ), which is O(log 1 ffl ), is the Dirichlet energy of a certain canonical harmonic function h on U . As an application of this result, we prove that the growth exponent of the looperased random walk in Z 2 is 5=4. R'esum'e Nous calculons le d'eveloppement asymptotique du d'eterminant du Laplacien discret sur une r'egion rectilin'eaire de R 2 . Comme application, nous montrons que l"esp`eran...
Project Summary
"... research assistants have in the past been quite helpful both in writing code and in nding patterns in the data generated by the code. On the other hand, some of the more mathematically sophisticated undergraduate research assistants have been able to collaborate on a theoretical level, both in desig ..."
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research assistants have in the past been quite helpful both in writing code and in nding patterns in the data generated by the code. On the other hand, some of the more mathematically sophisticated undergraduate research assistants have been able to collaborate on a theoretical level, both in designing algorithms and in nding rigorous proofs of phenomena after these phenomena were observed empirically. For nearly all of the undergraduates, the joint research eort was their rst deep experience of what mathematical research is like. It is intended that undergraduates will continue to play an important role in the research. 1 Recent Prior Research by the Principal Investigator A major goal of the PI is to understand in detail the way the boundary of a region aects the statistical behavior of random tilings of that region. The articles [C1] and [C3], published by the PI during the past three years, carry out this program in two especially tractable cases, namely, d
Entropy and Boundary Conditions in Random Lozenge Tilings
, 2008
"... The tilings of lozenges in 2 dimensions and of rhomboedra in 3 dimensions are studied when they are constrained by fixed boundary conditions. We establish a link between those conditions and free or periodic boundary ones: the entropy is written as a functional integral which is treated via a saddle ..."
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The tilings of lozenges in 2 dimensions and of rhomboedra in 3 dimensions are studied when they are constrained by fixed boundary conditions. We establish a link between those conditions and free or periodic boundary ones: the entropy is written as a functional integral which is treated via a saddlepoint method. Then we can exhibit the dominant states of the statistical ensemble of tilings and show that they can display a strong structural inhomogeneity caused by the boundary. This inhomogeneity is responsible for a difference of entropy between the studied fixed boundary tilings and free boundary ones. This method uses a representation of tilings by membranes embedded in a higherdimensional hypercubic lattice. It is illustrated in the case of 60 degree lozenge tilings.