Asymptotics and Universal Edge Fluctuations with Applications to Lozenge Tiling Models

ERIK DUSE

Doctoral Thesis Stockholm, Sweden 2015

ISRN KTH/MAT/A–15/12-SE ISBN 978-91-7595-732-6

100 44 Stockholm SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen i matematik veckodag den 4 december 2015 kl 13.00 i D3, Kungl Tekniska högskolan, Lindstedts- vägen 5, Stockholm.

Erik Duse, 2015c

Tryck: Universitetsservice US AB

**Abstract**

This thesis concerns uniformly random discrete interlacing particle sys- tems and their connections to certain random lozenge tiling models. In par- ticular it contains the first derivation of a relatively unknown universal scaling limit, which we call the Cusp-Airy process, of certain lozenge tiling models at a cusp point. In addition it contains a characterization of the geometry of the macroscopic behavior of uniformly random discrete interlaced parti- cle systems that, although not complete, shows many new and interesting features.

**Sammanfattning**

Denna avhandling behandlar likformigt slumpmässiga system av samman- flätade partiklar (eng: interlacing particles) och deras koppling till vissa rom- biska tesselerings modeller. Den innehåller i synnerhet den första härledning- en av en relativt okänd universell skalningsgräns, som vi kallar Cusp-Airy processen, för vissa rombiska tesselerings modeller vid en spetspunkt. Den innehåller dessutom en karakterisering av geometrin för det makroskopiska beteendet hos likformigt slumpmässiga system av sammanflätade partiklar, som trots att den inte är fullständig, visar på nya och intressanta egenskaper.

**Contents** **v**

**Acknowledgements** **vii**

**Part I: Introduction and Summary**

**1** **Introduction** **2**

1.1 Random Lozenge Tilings . . . . 2 1.2 Cut-Corner Hexagon Dimer Model and Related Models . . . . 4 1.3 Discrete Interlacing Systems . . . . 10 1.4 Discrete Orthogonal Polynomial Ensembles and Equilibrium Measures 38 1.5 Relation To Continuous Interlacing Model . . . . 49

**2** **Summary of Results** **53**

2.1 Paper A . . . . 53 2.2 Paper B . . . . 53 2.3 Paper C . . . . 53

**References** **55**

v

First and foremost I would like to express my gratitude to my scientific advisor Kurt Johansson for all his support and encouragement during these past years. In particular I thank him for his generosity in sharing his mathematical knowledge and his friendship. I would also like to thank Anthony Metcalfe for a fruitful cooperation. In addition I would like to thank Maurice Duits and Sunil Chhita for many useful and interesting discussions regarding my research. I am also grateful to Professor Richard Kenyon at Brown University for letting me use his figure of a simulation of a random tiling of the cut-corner hexagon, figure 1 in the introduction.

My graduate studies were funded by the grant KAW 2010.0063 from the Knut and Alice Wallenberg Foundation. For this I am very grateful.

I would like to thank all the friends I have gained at the department of mathe- matics at KTH during my years as a graduate student. In particular, I would like to thank Martin Strömqvist, Joel Andersson, Andreas Minne, Gaultier Lambert, Mauriusz Hynek, Christopher Svedberg, Oscar Forsman, Katharina Heinrich, Se- bastian Öberg, Samuel Holmin, André Laestedius, Gustav Sændén Ståhl and Antti Haimi for all interesting discussions over the years.

Last but not least, I would like to thank my family for all support during these years. In particular I would like to express my gratitude and love to my fiancée Frida, my parents Alfred and Maria and my grandmother Margareta. This thesis would not have been made possible without you.

Finally I would like to dedicate this thesis to my late grandfather Erik Öhlin, who served as a great inspiration to me during my childhood.

vii

1

**1.1** **Random Lozenge Tilings**

Figure 1.1: A simulation of a random lozenge tiling of the cut-corner hexagon model.

**The Cut-Corner Hexagon Dimer Model**

This thesis was born out of an attempt to understand a special feature of the cut- corner hexagon dimer model. More precisely, consider a regular hexagon with unit side length, and remove from one corner a rhombi of side length 1{2. We call this polygon the cut-corner hexagon. See figure 1.2 below.

Figure 1.2: The cut-corner regular hexagon.

Consider tiling the cut-corner hexagon by the rhombi forming the faces of a
*cube under the projection in the p1, 1, 1q direction with side lengths 1{p2nq. The*
rhombi, or lozenges, are shown in figure 1.3.

Y B R

Figure 1.3: Three types of lozenges with sides of length 1.

Now, consider all possible such tessellations. Put the uniform probability distri-
bution on the set of all tessellations. One may now ask many interesting questions
about this model. For example, one may ask what a “typical” tessellation look like
*when n is very large, if there is any such thing. Indeed, a simulation of a typical*
tessellation is shown in figure 1 in [KO07] and reprinted in this introduction as
figure 1.1 by permission of Professor Richard Kenyon. From, this figure one may
immediately observe some special features. To start with one observes a brick-like
pattern of only one type of lozenge in each of the corners. These frozen regions
are interrupted by a curve. Inside the region of this curve one observes all types
of lozenges. One also observes that the curve touches the sides of the polygon tan-
*gentially, see figure 1.1. By letting n become large, this curve will look more and*

more like a smooth curve. In particular, this smooth curve will look very similar to
a cardioid curve, again see figure 1.1. Moreover, this curve has a singularity in the
form of a cusp. The starting point of this thesis was to attempt to understand the
local microscopic behavior of the stochastic process of the lozenges in the vicinity
*of the cusp as n Ñ 8. Before discussing how this question was resolved, we will in*
the next sections first describe more general lozenge tiling models and their relation
to interlaced particle systems.

**1.2** **Cut-Corner Hexagon Dimer Model and Related Models**

**Decomposition of Polygons into Interlacing Systems**

Consider first a tessellation of a regular hexagon. An example of such is shown in figure 1.4. In figure 1.5 we also depict the “frozen regions” and the “frozen boundary” separating the frozen region from from the “liquid region”. Also, the asymptotic shape of the frozen boundary is shown.

1

1 1 1

1

1

Y B R

Figure 1.4: Left: A regular hexagon with sides of length 1.

*Middle: The three different types of lozenges with sides of length 1{n.*

*Right: An example tiling when n “ 4.*

B R Y B R

Y

1

1 1 1

1

1

Figure 1.5: Left: The frozen boundary of the example tiling of figure 1.4.

Right: The asymptotic shape of the frozen boundary of a “typical random tiling”

*as n Ñ 8.*

We will now restrict our attention to the configuration of yellow tiles. We see that we may encode the configuration of yellow tiles as interlacing systems of particles. In in figure 1.6 this is done in two different ways. In the figure to the left in figure 1.6, we encode the configuration of yellow tiles as two interlacing systems. One interlacing system between row 1 and row 4, and one interlacing system between row 7 and row 4, being glued together along their common row, row 4. Moreover, since we pick tessellations of the regular hexagon uniformly at random, the configuration of tiles at row 4 is a random configuration. On the other hand, in the figure to the right in figure 1.6 we encode the configuration of yellow tiles as one interlacing system by adding virtual particles on the side of the polygon.

Now, the configuration of yellow tiles at row 8, the “top line”, is deterministic as opposed to the configuration of tiles on row 4.

We may now note that the configuration of yellow tiles entirely fixes the config- urations of the other tiles, the red and the blue tiles. This can be seen as follows.

In between every row of yellow tiles we may we have a row of red and blue tiles.

Firstly, we see that the position of the yellow tile on the first row fixes the positions of the tiles on first row of red and blue tiles. Secondly, we see that the position of yellow tiles on the first and the second row fixes the position of red and blue tiles on the the second row of blue and red tiles. The process may be continued until we reach the “top row” of the first interlacing system, row 4, and the position of the tiles on row 3 and row 4 have determined the position of the red and blue tiles on the fourth row of the rows of red and blue tiles. We have now fixed the position of all the tiles in the lower interlacing system. We may now repeat the process in the upper interlacing system, using that the position of the yellow tile on row 7 fixes position of red and blue tiles on row 8 and so on. This completes the tessellation of the regular hexagon, given the position of the yellow tiles. In fact we notice that the choice of considering yellow tiles was arbitrary, we may in fact equally well have

chosen to consider the configurations of red or the blue tiles instead.

row 1 row 2 row 3 row 4 row 5 row 6 row 7 row 8

Figure 1.6: Left: Equivalent interlaced particle configuration of the example tiling of figure 1.4.

Right: Equivalent interlaced particle configuration with added deterministic lozenges/particles. The unfilled circles represent the deterministic particles.

We will now be more precise on how we encode the positions of the yellow tiles
*as an interlaced particle system. Let y*_{i}^{prq}*denote the position of the i:th particle*
*on the r:th row and let β*_{i}^{pnq}*:“ y*_{i}^{pnq}*denote the position of the particles on the top*
*line, indicated by unfilled circles inside the tiles. Then the particles on row r ` 1*
*will interlace with the particles on row r according to*

*y*^{pr`1q}_{1} *ą y*_{1}^{prq}*ą y*_{2}^{pr`1q}*ą y*_{2}^{prq}*... ą y**r*^{prq}*ą y*_{r`1}^{pr`1q}*,*

*for every r “ 1, ..., n ´ 1.*

We may now try to repeat the idea of encoding lozenge tessellations of the regular cut-corner hexagon by interlacing systems. This can be done in the three different ways depicted in figure (1.7).

Figure 1.7: The picture to the left depicts a decomposition of the cut-corner hexagon into two interlacing regions of yellow tiles. The picture in the middle depicts a decomposition of the cut-corner hexagon into two interlacing regions of red tiles.

The picture to the right depicts a decomposition of the cut-corner hexagon into two interlacing regions of blue tiles

In particular, the methods developed in this thesis will apply to those polygons
*that can be decomposed into two interlacing regions for at least one of the three*
*different types of tiles. For a more general decomposition of such a polygon see*
figure 1.10 in the next section.

It will be convenient when considering interlacing system to make a coordinate transformation according to figure 1.8. For more details see section 1.4 in [DM15a]

and section 2.1 in [Pet14] .

Figure 1.8: Coordinate transformation of lozenge tiles.

After the coordinate transformation, figure 1.4 becomes figure 1.9. Furthermore
*the interlacing condition between row r ` 1 and row r has changed into*

*y*^{pr`1q}_{1} *ě y*_{1}^{prq}*ą y*_{2}^{pr`1q}*ě y*_{2}^{prq}*... ě y*_{r}^{prq}*ą y*_{r`1}^{pr`1q}*.*

Figure 1.9: An example tiling and its equivalent interlaced particle configuration after the coordinate transformation. The unfilled circles represent the deterministic lozenges/particles.

**Discrete Orthogonal Polynomial Ensembles**

As was described in the previous section we are interested in those tiling models
*that can be decomposed into two regions, such that after possibly adding virtual*
*particles, these regions become interlacing regions of the type describe in section*
1.2, glued together along a common line as depicted in figure 1.10.

Interlacing direction
*T*_{1}

*T*2

Figure 1.10: *Decomposition of a polygon into two interlacing regions T*_{1} *and T*_{2}
glued together along the thick black line. The blue dots indicate the positions of
*virtual particles/tiles and the black dots indicate the positions of ordinary parti-*
cles/tiles.

Recall that the number of interlacing configurations with a given top line con-

*figuration py*1*, y*_{2}*, ...., y**N*q P Z* ^{N}* is given by Weyl’s dimension formula in [Wey39] for

*the irreducible characters of the unitary group U pnq,*

N^{7}*py*1*, ..., y** _{N}*q “
ś

*1ďiăjďN**|y**i**´ y**j*|
ś

*1ďiăjďN**|i ´ j|* :“ 1
*2C**N*

ź

*1ďi,jďN*
*i‰j*

*|y**i**´ y**j**|.* (1.2.1)

*Let py*1*, y*2*, ...., y**N*q be the positions of the particles/tiles on the intersecting thick
*black line as in figure 1.10. Let V be the index set for the virtual or frozen particles*
*and let F be the index set for the free particles, so that |V| ` |F | “ N and y**i*

*is a virtual particle if i P V and free otherwise. Assume that |F | “ n, and let*
*g : t1, ...., nu Ñ F be a set bijection such that x**i* *:“ y**gpiq**, and x*1*ă x*2*ă ... ă x**n*.
*Furthermore, the virtual particles are densely packed, which implies that they vill*
form wedge shaped frozen regions. However, the fact that the two interlacing regions
*T*_{1} *and T*_{2} need not be symmetrical implies that we need not have frozen regions
on both sides of the intersecting black line. Let V* _{L}* Ď V be the index set of those

*virtual particles such that they form a frozen region to the left, and let V*

*R*Ď V be

*the index set of those virtual particles such that they form a frozen region to the*right. Then by (1.2.1), the number of interlacing configuration with a given fixed

*configuration of free particles at positions px*1

*, ..., x*

*n*q is given by

N^{7}*px*1*, ..., x**n*q “ 1
*2C*_{N}

ź

*1ďiăjďN*

*|y**i**´ y**j*|

“ 1

*2C**N*

ź

*i,jPF*
*i‰j*

*|y**i**´ y**j*|^{2} ź

*jPV**iPF**L*

*|y**i**´ y**j*| ź

*jPV**iPF**R*

*|y**i**´ y**j*| ź

*i,jPV**L*

*i‰j*

*|y**i**´ y**j*| ź

*i,jPV**R*

*i‰j*

*|y**i**´ y**j*|

“ 1

*2C**N*

ź

*1ďk,lďn*
*k‰l*

*|x**k**´ x**l*|^{2}

*n*

ź

*k“1*

ź

*jPV**L*

*|x**k**´ y**j*|

*n*

ź

*k“1*

ź

*jPV**R*

*|x**k**´ y**j*| ź

*i,jPV**L*

*i‰j*

*|y**i**´ y**j*| ź

*i,jPV**R*

*i‰j*

*|y**i**´ y**j**|.*

Now, consider the set of all possible lozenge tessellations of the the original polygon.

One easily sees that each such tessellation is in a bijective correspondence with two
*interlacing configurations on T*1*and T*2*with the same configuration of free particles*
*x*^{pnq}*“ px*1*, x*2*, ..., x**n**q on their common top line. In particular, x**i**P nΣ X Z for each*
*i “ 1, ..., n, where Σ is a finite union of intervals defined by the tiling model. Let*
C*n* *denote the set of all configurations of free particles. Consider the set of all*
lozenge tessellations of the polygon with uniform distribution. Then this induces a
probability distribution on C*n*, given by

*p*^{pnq}*rpx*1*, ...., x*_{n}*qs “ Prparticles at positions x*1*, x*_{2}*, ..., x** _{n}*s “ N

^{7}

*px*1

*, ..., x*

*n*q ÿ

*px*1*,...,x**n*qPC*n*

N^{7}*px*1*, ..., x**n*q
*.*

Let

*w**n**pxq :“* ź

*jPV**L*

*|x ´ y**j*| ź

*jPV**R*

*|x ´ y**j**|.*

Then

*p*^{pnq}*rpx*1*, ...., x**n*qs “ 1
*Z*_{n}

ź

*1ďiăjďn*

*px**i**´ x**j*q^{2}

*n*

ź

*i“1*

*w**n**px**i**q,* (1.2.2)

*where Z**n*“ř

*px*1*,...,x** _{n}*qPC

*n*

ś

*1ďiăjďn**px**i**´ x**j*q^{2}ś*n*

*i“1**w*_{n}*px**i*q. Associated with a par-
*ticular weight function w*_{n}*pxq is a class of discrete orthogonal polynomials tp**n,k**pxqu**k*

defined according to

ÿ

*x**i**PnΣXZ*

*p**n,k**pxqp**n,l**pxqw**n**pxq “ δ**kl**.* (1.2.3)

*Such particle processes are called discrete orthogonal polynomial ensembles, DOPE,*
*and have been studied e.g. in [BKMM07]. In particular if one consider the random*
*empirical measure µ**n* “ ^{1}* _{n}*ř

*n*

*i“1**δ*_{x}_{i}_{{n}*, then µ**n* *á µ*^{λ}* _{V}* in probability, where the

*measure µ*

^{λ}*P M*

_{V}

^{λ}_{1}pΣq is the unique solution of the constrained variational problem

min

*νPM*^{λ}_{1}pΣq

*tI**V**rνsu “* min

*νPM*^{λ}_{1}pΣq

"ˆ

ΣˆΣ

*log |x ´ y|*^{´1}*dνpxqdνpyq `*
ˆ

Σ

*V pxqdνpxq*

*
*,*

(1.2.4)
*where V pxq “ lim*_{nÑ8}*´n*^{´1}*logpw**n**pxqq. Here, M*^{λ}_{1}pΣq is the set of positive Borel
*measures ν, such that supppνq Ă Σ, }ν} “ 1 and ν ď λ, where λ is the Lebesgue*
measure. The constrained variational problem is discussed in more detail in section
1.4.

Again we emphasize that in these models we have a random empirical top line measure. We will not discuss the issues arising from this fact further in introduc- tion, but refer the interested reader to [DJM15]. For more on discrete orthogonal polynomial ensembles see [BKMM07] and [Fer].

**1.3** **Discrete Interlacing Systems**

**Discrete Interlacing Systems and Determinantal Point Processes**
We saw in the previous section that the study of certain random lozenge tiling
models with uniform probability could be reduced to the study of certain discrete
interlacing models. In this section we therefore give a more careful introduction to
*these systems. A discrete Gelfand-Tsetlin pattern of depth n is an n-tuple, denoted*
*py*^{p1q}*, y*^{p2q}*, . . . , y** ^{pnq}*q P Z ˆ Z

^{2}ˆ ¨ ¨ ¨ ˆ Z

*, which satisfies the interlacing constraint*

^{n}*y*_{1}^{pr`1q}*ě y*^{prq}_{1} *ą y*^{pr`1q}_{2} *ě y*^{prq}_{2} *ą ¨ ¨ ¨ ě y*_{r}^{prq}*ą y*_{r`1}^{pr`1q}*,*

*denoted y*^{pr`1q}*ą y*^{prq}*, for all r P t1, . . . , n ´ 1u. For each n ě 1, fix x** ^{pnq}*P Z

*with*

^{n}*x*

^{pnq}_{1}

*ą x*

^{pnq}_{2}

*ą ¨ ¨ ¨ ą x*

^{pnq}*n*

*and a*

^{pnq}*n*

*“ x*

^{pnq}*n*

*and b*

^{pnq}*n*

*“ x*

^{pnq}_{1}, and consider the following

*probability measure on the set of patterns of depth n:*

*ν**n**rpy*^{p1q}*, . . . , y** ^{pnq}*qs :“ 1

*Z*

*¨*

_{n}"

1 ; *when x*^{pnq}*“ y*^{pnq}*ą y*^{pn´1q}*ą ¨ ¨ ¨ ą y*^{p1q}*,*
0 ; *otherwise,*

*where Z**n* ą 0 is a normalisation constant. This can equivalently be considered as
*a measure on configurations of interlaced particles in Z ˆ t1, . . . , nu by placing a*
*particle at position pu, rq P Zˆt1, . . . , nu whenever u is an element of y*^{prq}*. Thus, ν**n*

is the uniform probability measure on the set of all such interlaced configurations
*with the particles on the top row in the deterministic positions defined by x** ^{pnq}*.
Also note that due to the interlacing constraint, the interlacing particle system is

*contained inside the polygon nP*

*n*, where

P*n**“ tpχ, ηq P R*^{2}*: 0 ď η ď 1, χ ` η ´ 1 ě a*^{pnq}_{n}*, χ ď b*^{pnq}* _{n}* u (1.3.1)
We now construct a related probability space, the determinantal structure of

*which is more convenient to examine. Consider all tuples, pz*

^{p1q}

*, . . . , z*

*q P Z*

^{pn´1q}*ˆ*

^{n}¨ ¨ ¨ ˆ Z* ^{n}* with

*z*^{pr`1q}_{1} *ě z*^{prq}_{1} *ą z*^{pr`1q}_{2} *ě z*_{2}^{prq}*ą ¨ ¨ ¨ ą z*^{pr`1q}_{n}*ě z*_{n}^{prq}*,*

*for all r, also denoted z*^{pr`1q}*ą z*^{prq}*. Fix z*^{p0q} *:“ px**n**` n ´ 1, . . . , x**n**` 1, x**n*q and
*define the following probability measure on the set of all such pn ´ 1q-tuples:*

*ν*_{n}^{1}*rpz*^{p1q}*, . . . , z** ^{pn´1q}*qs :“ 1

*Z*

_{n}^{1}¨

"

1 ; *when x*^{pnq}*ą z*^{pn´1q}*ą ¨ ¨ ¨ ą z*^{p1q}*ą z*^{p0q}*,*
0 ; *otherwise,*

(1.3.2)
*where Z*_{n}^{1} ą 0 is a normalisation constant.

*Consider the relationship between the spaces. First note that, whenever x** ^{pnq}*“

*y*

^{pnq}*ą y*

^{pn´1q}*ą ¨ ¨ ¨ ą y*

^{p1q}

*for some py*

^{p1q}

*, y*

^{p2q}

*, . . . , y*

*q P Z ˆ Z*

^{pnq}^{2}ˆ ¨ ¨ ¨ ˆ Z

*, then*

^{n}*x*1*ě y*_{1}^{prq}*ą ¨ ¨ ¨ ą y*^{prq}_{r}*ą x**n**` n ´ r ´ 1,*

*for all r ď n. Whenever x ą z*^{pn´1q}*ą ¨ ¨ ¨ ą z*^{p1q}*ą z*^{p0q} *for some pz*^{p1q}*, . . . , z** ^{pn´1q}*q P
Z

*ˆ Z*

^{n}*ˆ ¨ ¨ ¨ ˆ Z*

^{n}*,*

^{n}*x*1*ě z*^{prq}_{1} *ą ¨ ¨ ¨ ą z*_{r}^{prq}*ą z*_{r`1}^{prq}*ą z*_{r`2}^{prq}*ą ¨ ¨ ¨ ą z*^{prq}* _{n}* (1.3.3)

=

*x*_{n}*`n´r´1*

=

*x*_{n}*`n´r´2* *. . .* =

*x*_{n}

*for all r ď n ´ 1.* *We refer to z*^{prq}_{1} *, . . . , z*^{prq}*r* *as the free particles of z** ^{prq}*, and

*z*

^{prq}

_{r`1}*, . . . , z*

*n*

^{prq}*as the deterministic particles. Note the natural bijection between*

*tpy*

^{p1q}

*, . . . , y*

*q P Z ˆ Z*

^{pnq}^{2}ˆ ¨ ¨ ¨ ˆ Z

^{n}*: x “ y*

^{pnq}*ą y*

^{pn´1q}*ą ¨ ¨ ¨ ą y*

^{p1q}u and

*tpz*

^{p1q}

*, . . . , z*

*q P Z*

^{pn´1q}*ˆ Z*

^{n}*ˆ ¨ ¨ ¨ ˆ Z*

^{n}

^{n}*: x ą z*

^{pn´1q}*ą ¨ ¨ ¨ ą z*

^{p1q}

*ą z*

^{p0q}u: re-

*move y*

^{pnq}*from each n-tuple py*

^{p1q}

*, . . . , y*

*q and map the remaining components,*

^{pnq}*y*

^{prq}*“ py*

_{1}

^{prq}*, . . . , y*

*r*

^{prq}*q for each r ď n ´ 1, individually as,*

*y*^{prq}*ÞÑ py*_{1}^{prq}*, . . . , y*_{r}^{prq}*, x*_{n}*` n ´ r ´ 1, x**n**` n ´ r ´ 2, . . . , x**n**q.*

*The measure ν*_{n}^{1} *is induced by the measure ν**n* under this bijective map. The prob-
*abilistic structure of particles in the first space (measure ν**n*) is therefore identical
*to the probabilistic structure of the free particles in the second space (measure ν*_{n}^{1}).

From now on we restrict to the second space.

*A more convenient expression for ν*_{n}^{1} can be obtained from the work of Warren,
[War07]:

det

”
1_{z}*pr`1q*

*j* *ěz*^{prq}_{i}

ı*n*
*i,j“1*

“

"

1 *; when z*^{pr`1q}*ą z*^{prq}*,*
0 ; otherwise,

*for all r. Equation (1.3.2) thus gives,*

*ν*_{n}^{1}*rpz*^{p1q}*, . . . , z** ^{pn´1q}*qs “ 1

*Z*

_{n}^{1}

*n´1*ź

*r“0*

det

”

*φ*_{r,r`1}*pz*_{i}^{prq}*, z*_{j}* ^{pr`1q}*q
ı

*n*

*i,j“1**,* (1.3.4)
*where z*^{pnq}*:“ x, and*

*φ*_{r,r`1}*pu, vq :“ 1**věu**,*

*for all r and u, v P Z. We may define coupling matrices A**r,r`1**“ A*_{r,r`1}*pz*^{prq}*, z** ^{pr`1q}*q,

*connecting configurations z*

^{prq}*on row r with configurations z*

^{pr`1q}*on row r ` 1 by*letting

*pA*_{r,r`1}*pz*^{prq}*, z*^{pr`1q}*ij* *:“ φ*_{r,r`1}*pz*_{i}^{prq}*, z*_{j}^{pr`1q}*q.*

Using the coupling matrices we see that (1.3.4) can be written as

*ν*_{n}^{1}*rpz*^{p1q}*, . . . , z** ^{pn´1q}*qs “ 1

*Z*

_{n}^{1}det

„*n´1*

ź

*r“0*

*A*_{r,r`1}*pz*^{prq}*, z** ^{pr`1q}*q

*.* (1.3.5)

*Note, each pz*^{p1q}*, . . . , z** ^{pn´1q}*q P Z

*ˆ Z*

^{n}*ˆ ¨ ¨ ¨ ˆ Z*

^{n}*can be equivalently considered*

^{n}*as a configuration of particles in Z ˆ t1, . . . , n ´ 1u by placing a particle at position*

*pu, rq P Z ˆ t1, . . . , n ´ 1u whenever u is an element of z*

*.*

^{prq}*The measure ν*

_{n}^{1}in equation (1.3.5) therefore defines a random point process on configurations of

*particles in Z ˆ t1, . . . , n ´ 1u. The Eynard-Mehta theorem, see Proposition 2.13*of Johansson, [Joh06], proves that this process is determinantal with correlation kernel,

*K**n**ppu, rq, pv, sqq “ ˜**K**n**ppu, rq, pv, sqq ´ φ**r,s**pu, vq,* (1.3.6)
*for all r, s P t1, . . . , n ´ 1u and u, v P Z, where*

*K*˜_{n}*ppu, rq, pv, sqq :“*

*n*

ÿ

*k,l“1*

*φ*_{r,n}*pu, z*_{k}^{pnq}*qpA*^{´1}q*kl**φ*_{0,s}*pz*_{l}^{p0q}*, vq,*

and

*φ*_{r,s}*pu, vq :“ 0 when s ď r,*

*φ**r,s**pu, vq :“ 1**věu**when s “ r ` 1,*
*φ*_{r,s}*pu, vq :“* ÿ

*z*_{1}*,...,z*_{s´r´1}

*φ*_{r,r`1}*pu, z*1*qφ*_{r`1,r`2}*pz*1*, z*_{2}*q ¨ ¨ ¨ φ*_{s´1,s}*pz*_{s´r´1}*, vq when s ą r ` 1,*

*A P C*^{nˆn}*with A**kl**:“ φ**0,n**pz*_{k}^{p0q}*, z*^{pnq}_{l}*q for all k, l.*

*Note that, for all r, s P t1, . . . , nu and u, v P Z, one may in fact compute φ**r,s**pu, vq*
to get

*φ**r,s**pu, vq “ 1*_{sąr}*h** _{v´u}*pp1q

^{s´r}*q,*

*for all r, s P t1, . . . , nu and u, v P Z, and where h**k* denotes the homogeneous sym-
*metric polynomials and where by convention h*_{k}*:“ 0 whenever k ă 0. Using the*
*special form of φ**r,s**pu, vq, we where in [DM15a] able to compute ˜**K**n**ppu, rq, pv, sqq,*
*and to derive an alternative form for φ**r,s**pu, vq. We got*

*K*˜*n**ppu, rq, pv, sqq “* *pn ´ sq!*

*pn ´ r ´ 1q!*

*n*

ÿ

*k“1*
*v*

ÿ

*l“v´n`s*

1*x*_{k}*ěu*

ś*u´1*

*j“u`r´n`1**px**k**´ jq*
ś*v*

*j“v´n`s,j‰l**pl ´ jq*
ź

*i‰k*

ˆ *l ´ x**i*

*x**k**´ x**i*

˙
*,*

(1.3.7) and

*φ**r,s**pu, vq “ 1*_{věu}*pn ´ sq!*

*pn ´ r ´ 1q!*

*n*

ÿ

*k“1*
*v*

ÿ

*l“v´n`s*

ś*u´1*

*j“u`r´n`1**px**k**´ jq*
ś*v*

*j“v´n`s,j‰l**pl ´ jq*
ź

*i‰k*

ˆ *l ´ x**i*

*x**k**´ x**i*

˙
*.*

It is interesting and surprising to note that both ˜*K*_{n}*ppu, rq, pv, sqq and φ**r,s**pu, vq*
have the same form. Also note that the fixed top row and the interlacing constraint
*implies that it is sufficient to restrict to those positions, pu, rq, pv, sq P Zˆt1, . . . , n´*

*1u, with u ě x*^{pnq}*n* *` n ´ v and v ě x*^{pnq}*n* *` n ´ s. We finally note that in terms of*
*lozenge tilings as depicted in figure 1.9, the correlation kernel K** _{n}* is the correlation

*kernel for the positions of the yellow tiles. When we want to emphasize this fact*

*we will sometimes write K*

_{n}^{Y}. In [DJM15] we showed how to get correlation kernels of the other tiles through the particle transformations

*K*_{n}^{R}*ppu, rq, pv, sqq “ ´K*_{n}^{Y}*ppu, rq, pv, s ´ 1qq* (1.3.8)
*K*_{n}^{B}*ppu, rq, pv, sqq “ K*_{n}^{Y}*ppu, rq, pv ` 1, s ´ 1qq.* (1.3.9)
*Here, K*_{n}^{R}*is a correlation kernel for the red tiles and K*_{n}^{B}is a correlation kernel for
the blue tiles.

At this point it is advantageous to pause for a moment and reflect upon the fact
*why we where able to compute the correlation kernel K**n**ppu, rq, pv, sqq. We see that*
it depended on two critical steps:

*• We were able to compute the n-fold matrix product A and get ‘nice’ formulas*
*for the matrix elements pAq**ij*.

*• We were able to compute the inverse matrix A*^{´1}, which by Cramer’s rule
reduces to computing determinants. The fact that we could compute the
determinants in turn depended critically on the form of the matrix elements
*A**ij*.

For more complicated determinantal point processes, neither the first nor the
second step is usually feasible. One could ask why the uniformly random discrete
*interlacing model has this integrable structure. We will not give a precise answer*
to this question in the thesis but we note that interlaced particle systems fit within
the broader framework of Schur process. See for example [OR03] and [Bor11]. We
also note that there is a close connection between random interlaced particles and
Markov chains, see [BF15].

**Asymptotic Limit Shapes**

What do we mean by asymptotic limit shapes of discrete interlaced particle sys-
tems? In analogy with statistical physics we may view the asymptotic limit shapes
as the boundaries of different types of “phases”, i.e., microscopic behavior of the
*underlying systems as n Ñ 8. The phases that can appear are called liquid or solid*
*state. Of course the limit n Ñ 8 is an idealization of a true system where n is very*
large. Therefore, the asymptotic limit shape only exists in an idealized sense.

For many models from statistical physics it turns out that when one consider
various scaling limits of the microscopic system, i.e. one zoom in on the stochastic
*system at a certain scale, one gets universal scaling limits. By universal scaling*
limits we mean that the limit does not depend on the particular details of the
original statistical model and that the same limits occur in many different models.

*For a physically realistic system the limit n Ñ 8 of the original microscopic system*
can then be approximated by the universal scaling limit in much the same way as
*the the sum of n identically distributed independent random variables, rescaled by*
*one over the square root of n, can be approximated by the normal distribution.*

Which universal scaling limit one gets will depend on where one is looking in the
system. Since these limits can change in a discontinuous way depending on where
one is looking one expects to see different phases. In particular one expects that
*for n large enough the microscopic behavior of system changes in an abrupt way as*
one crosses the asymptotic boundary between different types of phases.

Now how is the asymptotic limit shape of uniformly discrete interlaced patterns
*determined? In what follows we will change notation somewhat. Change x** ^{pnq}* “

*px*1

*, ...., x*

*n*

*q Ñ β*

^{pnq}*“ pβ*1

*, ..., β*

*n*

*q and ppu, rq, pv, sqq Ñ ppx*1

*, y*1

*q, px*2

*, y*2qq. Since this system is determinantal, the correlation functions are given by

*ρppx*1*, y*1*q, px*2*, y*2*q, ..., px**m**, y**m**qq :“ Prparticles at positions px*^{1}*, y*1*q, px*2*, y*2*q, ..., px**m**, y**m*qs

*“ detrK**n**ppx**i**, y*_{i}*q, px**j**, y** _{j}*qqs

^{m}

_{i,j“1}*.*(1.3.10)

*Therefore, various scaling limits of (1.3.10) (with m fixed) are entirely determined*

*by the correlation kernel K*

_{n}*ppx*

*i*

*, y*

_{i}*q, px*

*j*

*, y*

*qq. To analyze the scaling limits, in*

_{j}[DJM15], we gave an integral representation (1.3.11) below of the correlation kernel
*K**n* based on the formula (1.3.7). This reduces the problem of studying limits of
the kernel (1.3.7) to the problem of determining the asymptotics of double contour
integrals. By the method of steepest descent in complex analysis, this problem in
*turn reduces to determining the roots of certain functions f*_{n,1}^{1} *pwq, f*_{n,2}^{1} *pzq. Under*
*certain mild assumptions the function f**n,i* *for i “ 1, 2 converges to an asymptotic*
*function f .* Therefore the problem of determining the limit shapes of discrete
interlacing systems can be put in a bijective correspondence to a certain asymptotic
*function f . This function will of course depend on the coordinates of the position*
of where one is taking this limit and the weak limit of the sequence of point masses

1
*n*

*n*

ÿ

*i“1*

*δ*_{β}*pnq*
*i*

*to be defined below. The double contour integral representation of K**n*

given in [DJM15] is
*K**n**ppx*1*, y*1*q, px*2*, y*2qq “
1*x*_{1}*ăx*2

*pn ´ y*1q!

*pn ´ y*2´ 1q!

1
*p2πiq*^{2}

˛

Γ*n*

*dz*

˛

*γ**n*

*dw*
ś*x*_{2}´1

*k“x*2*`y*2*´n`1**pz ´ kq*
ś*x*_{1}

*k“x*1*`y*1*´n**pw ´ kq*
1
*w ´ z*

*n*

ź

*i“1*

*ˆ w ´ β*_{i}^{pnq}*z ´ β*^{pnq}_{i}

˙

´ 1*x*_{1}*ěx*2

*pn ´ y*1q!

*pn ´ y*2´ 1q!

1
*p2πiq*^{2}

˛

Γ^{1}_{n}

*dz*

˛

*γ**n*

*dw*
ś*x*_{2}´1

*k“x*2*`y*2*´n`1**pz ´ kq*
ś*x*_{1}

*k“x*1*`y*1*´n**pw ´ kq*
1
*w ´ z*

*n*

ź

*i“1*

*ˆ w ´ β*^{pnq}_{i}*z ´ β*_{i}^{pnq}

˙
*,*

(1.3.11)
where Γ*n* *is a counterclockwise oriented contour containing tβ*_{j}^{pnq}*: β*_{j}^{pnq}*ě x*2u but
*not the set tβ*_{j}^{pnq}*ď x*2´1u, and Γ^{1}* _{n}*is a counterclockwise oriented contour containing

*tβ*

^{pnq}

_{j}*: β*

_{j}

^{pnq}*ă x*2

*u but not the set tβ*

^{pnq}

_{j}*ě x*2

*` 1u. In addition, γ*

*n*contains the set

*tx*1

*` y*1

*´ n, ..., x*1u and Γ

*n*or Γ

^{1}

*. The integration contours are shown in figure 1.11.*

_{n}*a**n* *b**n*

Γ^{1}_{n}

*x*2

Γ*n*

*γ**n*

*Figure 1.11: Integration contours. Here, a**n**“ β*^{pnq}*n* *and β*_{1}^{pnq}*“ b**n*.

One should note that the correlation kernel of a determinantal point process is
*not uniquely defined. Indeed, if K*_{n}*ppx*1*, y*_{1}*q, px*2*, y*_{2}qq is a correlation kernel of a de-