Tilings of non-convex Polygons, skew-Young Tableaux and determinantal Processes
This paper studies random lozenge tilings of general non-convex polygonal regions. We show that the pairwise interaction of the non-convexities leads asymptotically to new kernels and thus to new statistics for the tiling fluctuations. The precise geometrical figure here consists of a hexagon with cuts along opposite edges. For this model we take limits when the size of the hexagon and the cuts tend to infinity, while keeping certain geometric data fixed in order to guarantee interaction beyond the limit. We show in this paper that the kernel for the finite tiling model can be expressed as a multiple integral, where the number of integrations is related to the fixed geometric data above. The limiting kernel is believed to be a universal master kernel.
- 1 Introduction and main results
- 2 Interlacing and measures on skew-Young Tableaux
- 3 An alternative integral representation for and some determinantal identities
- 4 From Karlin-McGregor to the -kernel
- 5 Transforming into and computing its inverse
- 6 The integral representation of the -kernel
- 7 The -kernel as a limit of the -kernel, for
- 8 The -kernel as a -fold integral, with
1 Introduction and main results
The purpose of this paper is to study random lozenge tilings of non-convex polygonal regions. Non-convex figures are particularly interesting due to the appearance of new statistics for the tiling fluctuations, caused by the non-convexities themselves or by the interaction of these non-convexities. The final goal will be to study the asymptotics of the tiling statistical fluctuations in the neighborhood of these non-convexities, when the polygons tend to an appropriate scaling limit.
The tiling problems of hexagons by lozenges goes back to the celebrated 1911-formula on the enumeration of lozenge tilings of hexagons of sides by the Scottish mathematician MacMahon . This result has been extended in the combinatorics community to many different shapes, including non-convex domains, in particular to shapes with cuts and holes; see e.g., Ciucu, Fischer and Krattenthaler [11, 30].
Tiling problems have been linked to Gelfand-Zetlin cones by Cohn, Larsen, Propp , and to non-intersecting paths, determinantal processes, kernels and random matrices by Johansson [17, 18, 19]. In , Johansson showed that the statistics of the lozenge tilings of hexagons was governed by a kernel consisting of discrete Hahn polynomials; see also Gorin. In  and , it was shown that, in appropriate limits, the tiles near the boundary between the frozen and stochastic region (arctic circle) fluctuate according to the Airy process and near the points of tangency of the arctic circle with the edge as the GUE-minor process.
Tiling of non-convex domains were investigated by Okounkov-Reshetikhin  and Kenyon-Okounkov  from a macroscopic point of view. Further important phenomena for nonconvex domains appear in the work of Borodin, Gorin and Rains , Defosseux , Metcalfe , Petrov [37, 38], Gorin , Novak , Bufetov and Knizel , Duse and Metcalfe [15, 16], and Duse, Johansson and Metcalfe ; see also the recent paper by Betea, Bouttier, Nejjar and Vuletic .
The present study consisting of two papers leads to the so-called Discrete Tacnode Kernel (13), which we believe to be a master kernel, from which many other kernels can be deduced (see Fig.1); namely,
(1) the GUE-tacnode kernel for overlapping Aztec diamonds [2, 1, 4] (also a non-convex geometry), when the size of the overlap remains small compared to the size of the diamonds. See also coupled GUE-matrices .
(2) the Tacnode kernel in the context of colliding Brownian motions and double Aztec diamonds.
(3) The cusp-Airy kernel () should also be a scaling limit of the Discrete Tacnode Kernel (13), etc…
Fig.1. Is the statistics associated with the discrete-tacnode kernel for non-convex hexagons universal ? Does it imply in some appropriate limit all these known statistics? Is it a master-kernel?
This led us to investigate determinantal processes for lozenge tilings of fairly general non-convex polygons, with non-convexities facing each other. This is going much beyond Petrov’s work  on the subject, and yet inspired by some of his techniques. Consider a hexagon with several cuts as in Fig.2, and a tiling with lozenges of the shape as in Fig.3, colored blue, red and green; introducing a cut amounts to covering a region with red tiles. Note there is an affine transformation from our tiles to the usual ones in the literature; see e.g. the simulation of Fig.5. The right-leaning blue tiles turn into our blue ones, the up-right red ones into our red ones and the left-leaning green tiles (30) to our green tiles (45), all as in Fig.2.
Two different determinantal pocesses, a -process and an -process, will be considered, depending on the angle at which one looks at the polygons; south to north for the -process or south-west to north-east for the -process. In this series of two papers, the first one will focus on the -process and its kernel, and the second one  on the -process, its kernel and its asymptotic limit in between the non-convexities. Nevertheless both processes will be introduced in this paper.
Fig. 2: A non-convex polygon (hexagon with cuts), and the hexagon . (multicut model)
A good part of the work will consist of reducing the number of integrations in the -kernel to , where , an integer defined in (5), relates to the geometry of the polygon and of the -process. In the second paper, the -kernel will require many more transformations in order to be in the right shape to perform asymptotics. As a preview of the second paper  we merely state here the form of the -kernel and the asymptotics without proof. Incidentally, the -kernel should also lead to interesting open questions related to the Gaussian Free Field (Petrov ) and also to open questions related to Petrov’s  and Gorin’s  work; see comments after Theorem 1.1.
To be precise, and as shown in Figure 2, we consider a general non-convex polygonal region (multicut model) consisting of taking a hexagon where two opposite edges have cuts, cuts cut out of the upper-part and cuts cut out of the lower-part111The and ’s also denote the size of the cuts.; let . Let and be the “cuts” corresponding to the two triangles added to the left and the right of P and let be the size of the lower-oblique side. Then is the distance between the lower and upper edges. The intervals separating the upper-cuts (resp. lower-cuts) are denoted by (resp. ) and we require them to satisfy , which is equivalent to . Define to be the quadrilateral (with two parallel sides) obtained by adding the red triangles to , as in Fig.1.
Introduce the coordinates , where and refer to the lower and upper sides of the polygon, with being the running variable along the lines . The vertices of and all belong to the vertical lines of the grid (in Fig.2). The integer points in are labeled by ; they are the integers in the cuts. We complete that set with the integer points to the left of along ; they are labeled by and we set and . This fixes the origin of the -coordinate. Similarly, the integer points are labeled by . We assume that for all , and that .
Besides the -coordinates, another set of coordinates will also be convenient (see Fig. 3):
Assuming , we define polynomials222 For any integers and we have and .:
The roots of , compared to the roots of , can be subdivided into three sets, the (eft), the (ight), the (enter), and a set (ap) not containing any :
ensuing the decompositions in polynomials
The number , assumed positive, will play an important role:
Referring to contour integration in this paper, the notation will denote a contour encompassing the points in question and no other poles of the integrands; e.g., contours like
The and -processes. Given a covering of this polygonal shape with tiles of three shapes, colored in red, blue and green tiles, as in Fig.2, put a red and blue dot in the middle of the red and blue tiles. The red dots belong to the intersections of the vertical lines integers and the horizontal lines ; they define a point process , which we call the -process. The initial condition at the bottom is given by the fixed red dots at integer locations in the lower-cuts, whereas the final condition at the top is given by the fixed red dots in the upper-cuts, including the red dots to the left and to the right of the figure, all at integer locations. Notice that the process of red dots on form an interlacing set of integers starting from fixed dots (contiguous for the two-cut and non-contiguous for the multi-cut model) and growing linearly to end up with a set of (non-contiguous) fixed dots. This can be viewed as a “truncated” Gel’fand-Zetlin cone!
The blue dots belong to the intersection of the parallel oblique lines with the horizontal lines for ; in terms of the coordinates (1), the blue dots are parametrized by , with as above. It follows that the -coordinates of the blue dots satisfy . This point process defines the -process. The blue dots on the oblique lines also interlace, going from left to right, but their numbers go up, down, up and down again. The number of blue dots per oblique line is given by the difference between the heights computed at the points and along that line ; see Fig.7. A special feature appears in the two-cut model.
One could also consider an -process, by putting a green dot on the green tiles. It would lead to a a determinantal process of green particles on the vertical lines half integers, where the number in (5) would be replaced by ; see Fig.4. In the end, this kernel would be similar to the -kernel.
As it turns out, the two kernels and are highly related, as will be shown in , where also the asymptotics for the -kernel will be carried out. Indeed, both point processes can be described by dimers on the points of the associated bipartite graph dual to ; the two kernels are related by the inverse Kasteleyn matrix of the bipartite graph. The aim of this first paper is to find a kernel for the -process, which involves no more than integrations which is a conditio sine qua non for taking asymptotics for while keeping fixed. As mentioned, the -kernel will require many more contour changes to do so in . This kernel could also be used for showing the Gaussian Free Field result in the bulk, again keeping fixed.
The following rational function,
appears crucially in the following -fold contour integral333Set . A shorthand notation for the Vandermonde is . for , (see (6))
where is a symmetric function of the variables , which depend on the integer points in the lower-cuts, with the number of gaps in that sequence. The precise symmetric function will be given later in (42) and (44). When that sequence is contiguous, we have and the symmetric function equals ; e.g., this is so when has one lower-cut.
For and , the determinantal process of red dots is given by the kernel, involving at most -fold integrals, with as in (5).
It is an interesting open problem to investigate the correlation function in the bulk (liquid region) and its limit when the size of the figure tends uniformly to . For a configuration with cuts on one side only, Petrov  has shown that the limit of the correlation function is given by the correlation of the incomplete beta-kernel for a “slope” satisfying the Burger’s equation (translation invariant Gibbs measure); these were introduced in . Gorin  goes beyond by showing that the result remains valid if one allows the location of the cuts on one side to be random, yielding a measure in the limit. For a figure with two-sided cuts (above and below), what is the analogue of the incomplete beta-kernel and the slope? The present project actually deals with a very different limit, as will be explained below.
The -kernel will be given in Theorem 1.2 below, but the proof will appear in another paper . It is not clear how to obtain the -kernel from scratch, due to the intricacy of the interlacing pattern, mentioned earlier. Therefore we must first compute the -kernel and then hope to compute the -kernel by an alternative method. Indeed, we check that the inverse Kasteleyn matrix of the dimers on the associated bipartite graph dual to coincides with the -kernel. This leads us to the following statement to appear in :
The -process of blue dots and the -process of red dots have kernels related as follows:
where and are the same geometric points as and , expressed in the new coordinates (1); see Fig. 6.
We state the asymptotic result (without proof), suggested by the simulation of Fig.5, for the two-cut model (one cut below and one above) and for even . We do not expect the multi-cut case to lead to a fundamentally new universality class in the limit, as the upper- and lower-cuts will only interact pairwise locally. For the two-cut model, we concentrate on the polygonal shape , as in Figs.2&4, with and with equal opposite parallel sides, i.e., and ; it has two cuts of same size , one on top and one at the bottom.
For that model, the oblique strip extending the oblique segments of the upper- and lower-cuts within will play an important role (see Fig.4): that is the region containing the parallel lines (or, what is the same, the lines ) for . Thus the strip has the following width:
and we assume . It will be shown that the parallel oblique lines within the strip each carry the same number of blue dots, with defined in (5). In the simulation of Fig.5, is the “finite” oblique strip separating the two “large” hexagons; see also Fig.4.
The discrete-tacnode kernel in the variables is defined by the following expression, where the integrations are taken along upwards oriented vertical lines to the right of a (counterclock) contour about the origin and with integer :
Fig. 3. Three types of tiles, with the height function and with corresponding level line. The red tiles (blue tiles) have a red dot (blue dot) in the middle.
Fig. 4 : Tiling of a hexagon with two opposite cuts of equal size (Two-cut model), with red, blue and green tiles. Here , , and thus The -coordinates have their origin at the black dot and the -coordinates at the circle given by . Red tiles carry red dots on horizontal lines for (-process) and blue tiles blue dots on oblique lines for (-process). The left and right boundaries of the strip are given by the dotted oblique lines and . Finally, the tilings define non-intersecting paths.
Fig. 5. Computer simulation for and . Courtesy of Antoine Doeraene.
We now state the main Theorem of this project, which will appear in :
Given the polygon P with cuts of equal size , above and below, keeping fixed, as defined in (12),(5), let the polygon and the size of the two cuts go to , according to the following scaling of the geometric variables , in terms of and new parameters , , , ,
The variables with get rescaled into new variables , having their origin at the halfway point along the left boundary of the strip , shifted by (see little circle along the line in Fig.4):
With this scaling and after a conjugation, the kernel (11) of the -process tends to the new kernel , as in (13), depending only on the width of the strip , the number of blue dots on the oblique lines in the strip and the parameter , to be precise,
The kernel satisfies the following involution:
This involution exchanges , with being self-involutive for . Also has support on , has support on and on .
These formulas of Theorems 1.1 and 1.2 can be specialized to known situations: to hexagons with no cuts (Johansson), to non-convex polygons with cuts at the top only (Petrov ), and to the case where the strip reduces to a line (i.e., ), in which the polygon (as in Fig.3) can be viewed as two hexagons glued together along one side (Duse-Metcalfe ).
Outline. Many steps are necessary to prove Theorem 1.1.
Section 2: Instead of putting the uniform distribution on the red dot-configuration on , it will be more convenient to first consider a non-uniform distribution depending on a parameter ; this distribution will tend to the uniform one when . The red dot-configuration will be shown to be equivalent to the set of semi-standard skew-Young Tableaux of a given shape; the latter can be read off the geometry of . The -probability on this set will have a Karlin-McGregor type of formula, which is not surprising due an equivalent formulation in terms of non-intersecting paths. The use of -deformations have been initiated by Okounkov-Reshetikhin  and Kenyon-Okounkov . Also it has been used effectively in the work of Petrov  for lozenge tilings of hexagons with cuts on the upper-side only. Section 2 also contains a brief description of the two-cut model.
Section 3 deals with some determinantal identities, but also with a useful, but unusual, integral representation of the elementary symmetric function .
Section 4: Adapting Eynard-Mehta techniques, further refined in  and , will lead to the construction a kernel , involving the inverse of a matrix .
Section 5: The matrix can be transformed so that the inverse is readily computable.
Section 6: The transformed kernel has a multiple integral representation, using integral representations of the different ingredients.
Section 7: Taking the limit when yields a kernel , involving contour integrals about , with at the worst -fold integrations, where is the size of the cuts.
Section 8: The kernel will then further be reduced to a sum of contour integrations, mostly about , with at the worst -fold integrals.
2 Interlacing and measures on skew-Young Tableaux
The two-cut model, already mentioned before, deserves some further discussion. Remember for the two-cut model we have and ; see Fig. 2&4. In other terms, this is now a hexagon with edges of size with two cuts, one below and one above, both of same size , satisfying and with (see Fig.4). The quadrilateral associated with is depicted in Fig.7. Note that the most-right point in the lower-cut plays an important role!
In the two-cut model, two strips within will play a special role: (Fig.4) (i) the oblique strip , already introduced in (12).
(ii) a vertical strip extending the vertical segments of the upper- and lower-cuts; that is the region between the lines and The strip has width (again same notation for the name and the width of the strip!)
Fig. 7. The polygon , with upper-cuts and lower-cuts and the parallelogram . The heights along the boundary of are given by the numbers next to the figure.
That amounts to the inequalities:
It is natural to assume that the strips (respectively ) have no point in common with the vertical parts (respectively oblique parts) of the boundary . This condition for implies and , implying . The condition for the strip implies and , implying . These inequalities combined with condition above imply
The four regions mentioned in (3) can now be written as
and so the polynomials and , as in (4), spelled out, are given by:
where are monic polynomials whose roots are given by the sets and , respectively, where and .
The inequalities (19) imply that each of the sets and form a contiguous set of integers, such that each of the three sets are completely separated: and .
Nonintersecting paths, level lines and the - and -processes. The height function on the tiles, given in Fig.3, imply that the heights along the boundary of the polygon are independent of the tiling; for the heights along , see Fig. 7. The level lines of heights pass obliquely through green tiles, vertically through blue tiles, and avoid the red tiles; the level lines are the nonintersecting paths going from top to bottom in Fig.4. It follows that the intersection (within ) of the level lines with the oblique lines determine the blue dots and with the lines the red dot at the integers level lines. In other terms drawing a horizontal line from left to right through the middle of the blue and green tiles (say, in Fig. 4) increases the height by and remains flat along the red tiles. Thus the tilings of the hexagon are equivalent to non-intersecting level-lines. It follows that
When an oblique line traverses a tile, as in Fig.2, the height increases by for a blue tile and stays flat for a red and green tile. Therefore, we have
leading to the following pattern for the blue dots for the two-cut model:
This implies that the number of blue dots are (local minimum) along each of the oblique lines within the strip , including its boundary. Outside, that number starts growing linearly in steps of up to , stays fixed for a while and then goes down to linearly in steps of .
The red dot-configurations and skew-Young tableaux. The red dots on the horizontal lines for are parametrized by
subjected to the interlacing pattern below, in short ,
The interlacing pattern above, with , is equivalent to a red dot configuration of a polygon .
As an example, for the two-cut model, the top consists of the three regions , and of contiguous integers; and the bottom of one contiguous region, given by the lower-cut.
Moreover, setting for , 444 Define for ; in particular for and , we have and . That means, the sum is always taken within the parallelogram . Also define
leads to a sequence of partitions
with prescribed initial and final condition,
and such that each skew diagram is a horizontal strip555An horizontal strip is a Young diagram where each row has at most one box in each column. . Note the condition for all on the cuts, mentioned just before (2), guarantees that . In fact the consecutive diagrams are horizontal strips, if and only if the precise inequalities hold.
Putting the integer ’s in each skew-diagram , for , is equivalent to a skew-Young tableau filled with numbers and exactly . This leads to the following semi-standard skew-Young tableau in the two-cut case, as in Fig.8. An argument, using the height function, which is similar to the one in , shows that all configurations are equally likely. So we have uniform distribution on the set of configurations. To conclude, we have:
with equal probability for each configuration.
Fig. 8: Example for the two-cut model. Semi-standard skew-Young Tableau filled with numbers in one-to one correspondence with the red dot-process. On the right is the precise semi-standard skew-Young Tableau associated with Fig. 4.
Uniform and -probability on the set of red dot-configurations. The last statement above implies that