Journal of the American Mathematical Society

The Schur process is a time-dependent analog of the Schur measure on partitions studied by A. Okounkov inInfinite wedge and random partitions, Selecta Math., New Ser.7(2001), 57–81. Our first result is that the correlation functions of the Schur process are determinants with a kernel that has a nice contour integral representation in terms of the parameters of the process. This general result is then applied to a particular specialization of the Schur process, namely to random 3-dimensional Young diagrams. The local geometry of a large random 3-dimensional diagram is described in terms of a determinantal point process on a 2-dimensional lattice with the incomplete beta function kernel (which generalizes the discrete sine kernel). A brief discussion of the universality of this answer concludes the paper.

This is the first of a series of three papers which prove the fact that a K-stable Fano manifold admits a Kähler-Einstein metric. The main result of this paper is that a Kähler-Einstein metric with cone singularities along a divisor can be approximated by a sequence of smooth Kähler metrics with controlled geometry in the Gromov-Hausdorff sense.

We formulate and prove a variational principle (in the sense of thermodynamics) for random domino tilings, or equivalently for the dimer model on a square grid. This principle states that a typical tiling of an arbitrary finite region can be described by a function that maximizes an entropy integral. We associate an entropy to every sort of local behavior domino tilings can exhibit, and prove that almost all tilings lie within ε \varepsilon (for an appropriate metric) of the unique entropy-maximizing solution. This gives a solution to the dimer problem with fully general boundary conditions, thereby resolving an issue first raised by Kasteleyn. Our methods also apply to dimer models on other grids and their associated tiling models, such as tilings of the plane by three orientations of unit lozenges.

The variational techniques (e.g. the total variation based method) are well established and effective for image restoration, as well as many other applications, while the wavelet frame based approach is relatively new and came from a different school. This paper is designed to establish a connection between these two major approaches for image restoration. The main result of this paper shows that when spline wavelet frames of are used, a special model of a wavelet frame method, called the analysis based approach, can be viewed as a discrete approximation at a given resolution to variational methods. A convergence analysis as image resolution increases is given in terms of objective functionals and their approximate minimizers. This analysis goes beyond the establishment of the connections between these two approaches, since it leads to new understandings for both approaches. First, it provides geometric interpretations to the wavelet frame based approach as well as its solutions. On the other hand, for any given variational model, wavelet frame based approaches provide various and flexible discretizations which immediately lead to fast numerical algorithms for both wavelet frame based approaches and the corresponding variational model. Furthermore, the built-in multiresolution structure of wavelet frames can be utilized to adaptively choose proper differential operators in different regions of a given image according to the order of the singularity of the underlying solutions. This is important when multiple orders of differential operators are used in various models that generalize the total variation based method. These observations will enable us to design new methods according to the problems at hand, hence, lead to wider applications of both the variational and wavelet frame based approaches. Links of wavelet frame based approaches to some more general variational methods developed recently will also be discussed.

We derive a new upper bound for the diameter of a k k -regular graph G G as a function of the eigenvalues of the adjacency matrix. Namely, suppose the adjacency matrix of G G has eigenvalues λ 1 , λ 2 , … , λ n {\lambda _1},{\lambda _2}, \ldots ,{\lambda _n} with | λ 1 | ≥ | λ 2 | ≥ ⋯ ≥ | λ n | \left | {{\lambda _1}} \right | \geq \left | {{\lambda _2}} \right | \geq \cdots \geq \left | {{\lambda _n}} \right | where λ 1 = k {\lambda _1} = k , λ = | λ 2 | \lambda = \left | {{\lambda _2}} \right | . Then the diameter D ( G ) D(G) must satisfy \[ D ( G ) ≤ ⌈ log ⁡ ( n − 1 ) / log ( k / λ ) ⌉ D(G) \leq \left \lceil {\log (n - 1)/{\text {log}}(k/\lambda )} \right \rceil \] . We will consider families of graphs whose eigenvalues can be explicitly determined. These graphs are determined by sums or differences of vertex labels. Namely, the pair { i , j } \left \{ {i,j} \right \} being an edge depends only on the value i + j i + j (or i − j i - j for directed graphs). We will show that these graphs are expander graphs with small diameters by using an inequality on character sums, which was recently proved by N. M. Katz.