The Cusp-Airy Process

At a typical cusp point of the disordered region in a random tiling model we expect to see a determinantal process called the Pearcey process in the appropriate scaling limit. However, in certain situations another limiting point process appears that we call the Cusp-Airy process, which is a kind of two sided extension of the Airy kernel point process. We will study this problem in a class of random lozenge tiling models coming from interlacing particle systems. The situation was briefly studied previously by Okounkov and Reshetikhin under the name cuspidal turning point.


The Cusp-Airy kernel
In this paper we will study random discrete interlacing particle systems which can also be interpreted as certain random lozenge tiling models. The particles, or lozenges, form a random point process which is determinantal. We are particularly interested in the limiting point process around the type of cusp point we see in the arctic curve in figure 1, see figure 1 in [16] or [17] for a simulation. Figure 1 illustrates the liquid region L and its boundary E, the arctic curve. Inside the liquid region one expects to see the extended sine-kernel point process in the limit. At the tangency points of the polygon and the boundary E one expects to see the GUE-corner process. At all other points of the curve E, except the cusp, one expects either the Airy kernel or Id-Airy kernel point processes in the appropriate scaling limit. To clarify the situation around the cusp point consider figure 2. All possible tiling configurations of the polygon can be encoded by the red rhombi. This is illustrated in figure 3, where we see that the positions of the red rhombi form two interlacing regions that meet at the dashed line. The rhombi at the common line have been coloured in purple. The purple rhombi form a discrete orthogonal polynomial ensemble, DOPE, see Remark 1.5. The dashed line is also a symmetry line (coloured blue in figure 2). The fact that we have two symmetric interlacing systems meeting at common line, will imply that the frozen boundary has a reflection symmetry in the symmetry line. It will also imply that the particle system consisting of red rhombi will have no horizontal oscillations. Therefore, when considering a scaling limit at the cusp on the symmetry line of this determinantal point process, the correct scaling is discrete in the horizontal direction and continuous of size n 1{3 in the vertical direction, where n is the size of the hexagon. Going back to figure 2, we see that directly above the tip of the cusp the blue and yellow rhombi form a corner. This implies that the height function in [17] will have a jump above the tip of the cusp. This should be contrasted to the situation where one expects to find a Pearcey process in the scaling limit around a cusp point of the arctic curve. Then one has only one type of rhombi in the frozen configuration inside the cusp. This also implies that the height function is flat inside the cusp.   At the cusp, in an appropriate scaling limit, we will see that the correlation kernel for the determinantal point process given by the red rhombi converges to a process with a kernel that we call the Cusp-Airy kernel. We will show this for the model corresponding to figure 1 up to certain natural technical conditions for a particular DOPE. In a simpler model of the type studied by Petrov in [20] we will give the full proof. This type of cusp situation in a random lozenge tiling model was discovered and discussed briefly by Okounkov and Reshetikhin in [18] who called it a Cuspidal turning point. The interpretation of their formula is however not completely clear, see remark 1.1 below.
Let us give the expression for the Cusp-Airy kernel.
Definition 1.1. For r, s P Z and ξ, τ P R we define the Cusp-Airy kernel by K CA ppξ, rq, pτ, sqq "´1 τ ěξ 1 sąr pτ´ξq s´r´1 ps´r´1q!`1 p2πiq 2ˆL L`Cout where the contours are defined in figure 4, and 1 aăb is the indicator function for a ă b. In section 3.2 we give a different formula for the kernel in terms of r-Airy integrals and some polynomials. When r " s " 0 we see that we get the Airy kernel. Hence, we expect the last red particle on the line r " 0 to have Tracy-Widom fluctuations. In the case r " s ‰ 0 we interestingly get the r-Airy kernel, which has appeared in previous work, see [1] and [3].
Remark 1.1. In [18], Okounkov and Reshetikhin give a formula, without proof, for the correlation kernel in the appropriate scaling limit around the type of cusp point studied in the present paper, but in a different model. The definition of the kernel in [18] is somewhat formal due to the fact that the factor 1 w´z w r z s in formula (1.1) above is interpreted via a "time-ordered expansion", see (13) in [18]. However, for the contours used in their formula (18) these expansions are not convergent. Similarly, in our formula (1.1) above we can not expand 1 w´z in a power series when z P L L and w P L R . In this case it seems more natural to rewrite 1 w´z in a different way, see (3.9) in section 3.2 and compare with the formulas derived there, see Proposition 3.2.

Random Lozenge Tiling Model
Consider three different types of lozenges (rhombi with angles π 3 and 2π 3 ) with sides of length 1. We label these as types Y, B, and R as shown in figure 5.     : An example of a tiling and its equivalent interlaced particle configuration when n " 8 and m " 9. The unfilled circles represent the deterministic lozenges/particles. By considering the positions of the yellow tiles, such a tessellation can be encoded as an interlaced particle system. More precisely, let y prq i denote the position of the i:th particle on the r:th row. Then the particles on row r`1 will interlace with the particles on row r according to

Interlacing Model
We begin by briefly recalling the underlying probabilistic model described in [8]. A discrete Gelfand-Tsetlin pattern of depth n is an n-tuple, denoted py p1q , y p2q , . . . , y pnq q P ZˆZ 2ˆ¨¨¨ˆZn , which satisfies the interlacing constraint denoted y pr`1q ą y prq , for all r P t1, . . . , n´1u. For each n ě 1, fix β pnq P Z n with β pnq 1 ą β pnq 2 ą¨¨¨ą β pnq n , and consider the following probability measure on the set of patterns of depth n: q n py p1q , . . . , y pnq q :" 1 Z n¨" 1 ; when β 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 Zˆt1, . . . , nu by placing a particle at position px, rq P Zˆt1, . . . , nu whenever x is an element of y prq . The measure q n is then 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 β pnq . This measure also arises naturally from tiling models as was indicated above. In [8] and [20] it was shown that this process is determinantal. Note that the fixed top row and the interlacing constraint implies that it is sufficient to restrict to those positions, px 1 , y 1 q, px 2 , y 2 q P Zˆt1, . . . , n´1u, with x 1 ě β pnq n`n´y1 and x 2 ě β pnq n`n´y2 . For all such px 1 , y 1 q and px 2 , y 2 q, we give an integral representation of the correlation kernel K n ppx 1 , y 1 q, px 2 , y 2 qq in section 4.1.
In terms of tiling models, K n ppx 1 , y 1 qpx 2 , y 2 qq is equal to a correlation kernel for the yellow particles K pnq Y ppx 1 , y 1 q, px 2 , y 2 qq. However at the cusp one should not consider the correlation kernel of the yellow particles, but the correlation kernel of the red particles instead. The correlation kernels of the different particle species are related according to Lemma 4.1 below, which we will prove in sec. 4.2.
Proposition 1.1. The red tiles (particles) form a determinantal point process with correlation kernel where Z n is a counterclockwise oriented contour containing tβ

Asymptotic Geometry of Discrete Interlaced Patterns
It is natural to consider the asymptotic behaviour of the determinantal process introduced in the previous section as n Ñ 8, under the assumption that the (rescaled) empirical distribution of the fixed particles on the top row converges weakly to a measure with compact support. More exactly: as n Ñ 8, in the sense of weak convergence of measures, where µ is a positive Borel measure on R.
We see that µ ď λ where λ is Lebesgue measure (recall β pnq P Z n ), }µ} " 1, µ has compact support, and b´a ą 1 where ra, bs is the convex hull of supppµq. We write µ P M λ c,1 pRq. Additionally we note that µ admits a density w.r.t. λ, which is uniquely defined up to a set of zero Lebesgue measure. Denoting the density by ϕ, it satisfies ϕ P L 8 pRq, ϕpxq " 0 for all x P Rzra, bs, and 0 ď ϕpxq ď 1 for all x P ra, bs. We write ϕ P ρ λ c,1 pRq. Note that Rzsupppµq is the largest open set on which ϕ " 0 almost everywhere, and Rzsupppλ´µq is the largest open set on which ϕ " 1 almost everywhere. Note that, rescaling the vertical and horizontal positions of the particles of the Gelfand-Tsetlin patterns by 1 n , the above assumption and the interlacing constraint imply that the rescaled particles lie asymptotically in the the following set: Fixing pχ, ηq P P, the local asymptotic behaviour of particles near pχ, ηq can be examined by considering the asymptotic behaviour of K n ppx for w P CzR. In [8], we define the liquid region, L, as the set of all pχ, ηq P P for which f 1 pχ,ηq has a unique root in the upper-half plane H :" tw P C : Impwq ą 0u. Whenever pχ, ηq P L, one expects universal bulk asymptotic behaviour, i.e., that the local asymptotic behaviour of the particles near pχ, ηq are governed by the extended discrete sine kernel as n Ñ`8. Also, one expects that the particles are not asymptotically densely packed. Moreover, when considering the corresponding tiling model and its associated height function, one would expect to see the Gaussian Free Field asymptotically. For a special case see [21].
Let W L : L Ñ H map pχ, ηq P L to the corresponding unique root of f 1 pχ,ηq in H. In [8], we show that W L is a homeomorphism with inverse W´1 L pwq " pχ L pwq, η L pwqq for all w P H, where and C : Czsupppµq Ñ C is the Cauchy transform of µ: Thus L is a non-empty, open (w.r.t to R 2 ), simply connected subset of P.
In [8] a subset of BL called the edge E was determined and its geometry classified. More precisely, the boundary behavior of W´1 L for the open subset R Ă BH " R was studied, where where • R µ :" tt P Rzsupppµq : Cptq ‰ 0u.
• R 1 is the set of all t P BpRzsupppµqq X BpRzsupppλ´µqq for which there exists an ǫ ą 0 such that pt, t`ǫq Ă Rzsupppµq and pt´ǫ, tq Ă Rzsupppλ´µq.
• R 2 is the set of all t P BpRzsupppµqq X BpRzsupppλ´µqq for which there exists an ǫ ą 0 such that pt, t`ǫq Ă Rzsupppλ´µq and pt´ǫ, tq Ă Rzsupppµq.
Note that R 1 X R 2 " H. R 1 Y R 2 :" BpRzsupppµqq X BpRzsupppλ´µqq, the set of all common boundary points of the disjoint open sets Rzsupppµq and Rzsupppλ´µq. Also R 1 Y R 2 is a discrete subset of ra, bs.
In words, R µ Y R 0 is the interior of the set where the density ϕpxq " 0 almost everywhere, and R λ´µ is the interior of the set where ϕpxq " 1 almost everywhere. We see that R 1 are the jumps from 1 to 0 and R 2 are the jumps from 0 to 1.
Definition 1.2. Let t˚P R 2 . We set and Then in particular ϕptq " 0 for all t P pt 2 , t˚q and ϕptq " 1 for all t P pt˚, t 1 q. If t " t˚P R 1 we interchange R λ´µ and R µ Y R 0 in (1.11) and (1.12) It was shown in [8] that by considering a sequence tw n u n P H such that lim nÑ`8 w n " t P R " BH one gets a parametrization of the edge E, for t P R µ and lim nÑ8 χpw n q " t`1´p t´t1 t´t2 qe´C I ptq C 1 I ptq`1 t´t2´1 t´t1 :" χ E ptq (1.15) lim nÑ8 ηpw n q " 1`p t´t2 t´t1 qe CI ptq`p t´t1 t´t2 qe´C I ptq C 1 I ptq`1 t´t2´1 t´t1 :" η E ptq (1. 16) for t P R λ´µ , where C I ptq "´R zI dµpxq t´x , and I is any open interval such that µˇˇI " λ and t P I. Moreover the collection of these smooth parametrized curves has analytic extensions across the set R 1 Y R 2 . In particular, pχ E ptq, η E ptqq " pt, 1´pt´t 2 qe CI ptq q (1.17) for t P R 1 with I " pt 2 , tq, and pχ E ptq, η E ptqq " pt`pt 1´t qe´C I ptq , 1´pt 1´t qe´C I ptq q (1.18) for t P R 2 , with I " pt, t 1 q. Finally, it is proven in [8] that this gives a bijection W E : E Ñ R, with inverse W´1 E ptq " pχ E ptq, η E ptqq, where χ E ptq and η E ptq are real analytic functions.
The importance of the edge E is that one expects universal edge fluctuations at E. In particular one expects that the local asymptotics in a neighborhood of a generic point of E is either given by the Airy kernel or the Id´Airy kernel. For a special case see [20], and more generally [9]. In this paper we will consider certain singular points on the curve E. At these points the curve will have a cusp. Typically one would expect that the local fluctuations at these points in the limit n Ñ 8 is a Pearcey process. However, for the situations considered in this paper this will not be the case. In fact we will show that at these points one gets the Cusp-Airy process given by the kernel (1.1).

Conditions for Cusps
One can rewrite the derivative given by (1.6) as (1. 19) This implies that the function f 1 pw; χ, ηq has an analytic extension to the set Czpsupppµqˇˇr a,χ`η´1,χs Ť supppλ´µqˇˇr χ`η´1,χs Ť supppµqˇˇr χ,bs q. It is shown in Lemma 2.6 in [8] that the edge, E, is the disjoint union E : • E µ is the set of all pχ, ηq P P for which f 1 has a repeated root in Rzrχ`η´1, χs.
• E λ´µ is the set of all pχ, ηq for which f 1 has a repeated root in pχ`η´1, χq.
• E 0 is the set of all pχ, ηq for which η " 1 and f 1 has a root at χ p" χ`η´1q. In particular, E is tangent to the line η " 1.
• E 1 is the set of all pχ, ηq for which η ă 1 and f 1 has a root at χ. In particular, E is tangent to the line χ " t P R 1 .
• E 2 is the set of all pχ, ηq for which η ă 1 and f 1 has a root at χ`η´1. In particular, E is tangent to the line χ`η´1 " t P R 2 .
Moreover it is shown in Lemma 2.6 in [8] that one has following equivalent characterization of the behavior of the roots of f 1 t :" f 1 pw; χ E ptq, η E ptqqˇˇw "t whenever pχ, ηq P E : (a) pχ E ptq, η E ptqq P E µ if and only if t P R µ . Moreover, in this case, t is a root of f 1 t of multiplicity either 2 or 3.
(b) pχ E ptq, η E ptqq P E λ´µ if and only t P R λ´µ . Moreover, in this case, t is a root of f 1 t of multiplicity either 2 or 3.
(c) pχ E ptq, η E ptqq P E 0 if and only if t P R 0 . Moreover, in this case, f 1 t " C and t is a root of f 1 t of multiplicity 1.
(d) pχ E ptq, η E ptqq P E 1 if and only t P R 1 . Moreover, in this case, t is a root of f 1 t of multiplicity either 1 or 2.
(e) pχ E ptq, η E ptqq P E 2 if and only t P R 2 . Moreover, in this case, t is a root of f 1 t of multiplicity either 1 or 2.
If case (a) holds and t is a root of multiplicity 2 of f 1 t one expects to see the extended Airy kernel process. If on the other hand t is a root of multiplicity 3 of f 1 t one expects to see the Pearcey process. If case (b) holds and t is a root of multiplicity 2 of f 1 t one expects to see the Id-extended Airy kernel process. What we mean by this is that we have to make a particle/hole transformation, i.e. change the type of tiles we are considering. If on the other hand t is a root of multiplicity 3 of f 1 t one again expects to see the Id-Pearcey process. If case (c) holds one expects to see the GUE corner process, and similarly if case (d) and (e) holds and t is a root of multiplicity 1 of f 1 t . In the remaining cases (d) and (e) when t is a root of multiplicity 2 of f 1 t one will see the Cusp-Airy process, which will be shown in this paper. In this article we will assume that t " t c P R 1 Y R 2 and that t c is a root of f 1 t of multiplicity 2. If these conditions hold, then it is shown in Lemma 2.9 in [8] that the edge at pχ E ptq, η E ptqq is locally an algebraic cusp of first order, that is the curve locally looks like the algebraic curve y 3 " x 2 in a neighborhood of the origin. Moreover, if t c P R 2 , then by Theorem 3.1 in [8] χ c ą t 1 . Then (1.18) together with the fact that χ c`ηc´1 " t c , where χ c :" χ E pt c q and η c :" η E pt c q, gives f 1 pt c ; χ c , η c q " C I pt c q´ˆχ c t1 dx t c´x " C I pt c q`log |t c´χc |´log |t c´t1 | " C I pt c q`log |pt c´t1 qe´C I ptcq |´log |t c´t1 | " 0.
A similar computation gives f 1 pt c ; χ c , η c q " 0 whenever t c P R 1 . Therefore, f 1 pt c ; χ c , η c q " 0 whenever t c P R 1 Y R 2 . This implies that that pχ c , η c q is a cusp of E for t c P R 1 Y R 2 , if and only if f 2 pt c ; χ c , η c q " 0.
However, a direct computation shows that if t c P R 2 , then and if t c P R 1 , then Therefore, E has a cups at pχ c , η c q, if and only if if t c P R 2 , and if t c P R 1 . From now on we will consider only the case t c P R 2 . The case when t c P R 1 can be treated analogously.
Proof. By Lemma 2.9 in [8], the signed extrinsic curvature k E ptq is negative for all smooth points of the curve E. From this it follows that all cusps point into the liquid region L. Together with Lemma 2.9 case (9) and Lemma 2.8 formula (g) in [8] it follows that f 3 pt c ; χ c , η c q ą 0.
In order to prove convergence to the Cusp-Airy kernel in our discrete model we will have to assume that µ n will jump from empty to full not just asymptotically in terms of ϕ, but already at the discrete level. This is the content of the next assumption.
Assumption 3. Assume that for every ε ą 0 and n large enough we have for t c P R 2 , where txu " maxtm P Z : m ď xu.
Remark 1.2. Assuming only weak convergence of the empirical measures tµ n u n to a limiting measure µ will not be sufficient when considering fluctuation of the edge E. We will need better control of the convergence of sequence of empirical measures. More precisely, it is necessary to assume that their supports converge in an appropriate sense, see [9]. This will not be needed here since the other assumptions that we make are enough.

Rescaled Variables
Introduce the following notation a n :" n´1 mintt P supppµ n qu " n´1β pnq n and b n :" n´1 maxtt P supppµ n qu " n´1β pnq 1 .
Furthermore, we write In words, nt We note that by Assumption 3, it follows that lim nÑ8 a n " a, lim denote the logarithmic potential of σ.
Remark 1.3. Note that sometimes the logarithmic potential is defined with opposite sign. We will however follow the convention in [22].
In particular we note that f is independent of η c . By assumption on µ we have a complete cancelation of the measures µ and λ on rt c , t 1 s, that is λ´µˇˇr tc,t1s " 0. For the exact kernel however, the cancelation of factors is not necessarily complete. Moreover due to the rigidity of the interlacing system, as shown in figure 3, their will be no fluctuations around the frozen boundary at the cusp in the orthogonal direction to the tangential direction of the cusp. It is therefore natural to assume a discrete variation in the orthogonal direction. We therefore assume the following scaling: Definition 1.3. Introduce the fixed limiting variables r, s P Z, and ξ, τ P R. Let (1.31) Define the rescaled variables ξ n , τ n P R, by (1.32) We assume that lim nÑ8 ξ n " ξ, lim nÑ8 τ n " τ. (1.33) When taking limits of the correlation kernel we will always use the scaling (1.32). Note that by (1.26), the rescaled variables satisfy (1.34) It will be convenient to introduce the notation The rescaled coordinate system is depicted in figure 12.

Integrand
In order to write the integrand in (1.2) in a convenient way we introduce some notation. Let We see that q n pw; rq only depends on the parameters through r. Also, since (1.38) q n pnw; rqQ n pnw, ∆x pnq 1 qE n pnwq q n pnz; sqQ n pnz; ∆x u. In addition, γ n is the counterclockwise oriented contour that contains the set n´1tx 1`y1´n , ..., x 1 u and Γ n andγ n is the counterclockwise oriented contour that contains the set n´1tx 1`y1´n , ..., x 1 u andΓ n . See figure 13 and 14. The lemma will be proven in section 2.1. Let ν n and ν n,j , j " 1, 2, be the signed measures Furtheremore, define f n , g n,j and h n,j by In particular, Rerg n,j pwqs "´R log |w´t|dν n,j ptq " U νn,j pwq. Here, we let log be the principal branch of the logaritm for f n pwq. For h n,j pwq, we let the branch cut of log lie along the the positive real axis.
With these choices of branch cuts, it follows that f n is analytic on Czp´8, b n s, and h n,j is analytic in Czrt pnq 1 ,`8q. In particular we note that f n pwq has a jump discontinuity over the real line at t c . However Re[f n ] is real analytic on Czpra n , t pnq 2 s Y rt pnq 1 , b n sq, and f 1 n is homolorphic on Czpra n , t pnq 2 s Y rt pnq 1 , b n sq. Moreover, lim tc˘iεÑtc f pt c˘i εq :" fn pt c q " Rerf n pt c qs˘iπk{n, where k is an integer. Therefore exppnpfǹ pt c q´fń pt c qqq " expp2πikq " 1. Thus the jump discontinuity in the imaginary part of f n does not matter when we perform a steepest descent analysis in a neighborhood of t c . From now on it will be understood that when we Taylor expand f n at w " t c , we look at the branch It follows from our assumptions that f n and g n,j converge uniformly to f on compact subsets of Czpra, t 2 sY rt 1 , bsq, see Lemma 2.1 We need one more assumption which will enable us to replace the non-asymptotic function f n pzq by the asymptotic function f pzq in a neighbourhood of the critical point t c .
uniformly in U .

Main Theorem
We can now give the main theorem about convergence to the Cusp-Airy kernel for our system of red particles in the interlacing model. p n px 2 , y 2 q p n px 1 , y 1 q uniformly for ξ and τ in some fixed compact subset of R, where p n px 1 , y 1 q "ˆd 0 n 2 3˙x Let pξ r j , rq be the rescaled coordinates for particles on line r. Fix r 1 , . . . , r M P Z and let φ : Rt r 1 , . . . , r M u Ñ r0, 1s be a bounded measurable function with compact support. Let E β pnq denote the expectation with respect to the determinantal point process with kernel K pnq R . Then, Example 1.1. Let the limiting measure dµptq " χ r´1´a,´1s ptqdt`χ r0,as ptqdt`χ r2a,1s ptqdt where a " 3`? 17 4 « 0.28. In particular }µ} " 1 and t c " 0 P R 2 . Let t 1 " a. Then, with I " p0, aq, From equation (1.18) we then get pχ E p0q, η E p0qq "ˆ1 2p1`aq , 1´1 2p1`aq˙. Using that a ă 1 2p1`aq ă 2a and (1. 19), we obtain since a is a root of the polynomial 1´x´8x 2´4 x 3 . Thus Assumption 2 is satisfied. We will now construct a sequence of empirical measures tµ n u n that satisfies Assumptions 1,3 and 4. Let where d n is chosen so that´n`d n´t´n p1´aqu`tnau`n´t2nau`3 " n. Clealy, µ n á µ, so Assumption 1 is satisfied. Now, so by construction tµ n u n satisfies Assumption 3. Now, and thus (1.49) Choose r ă a. Then for n large enough, 1 w´t and´1 pw´tq 2 are continuously differentiable in Bp0, rq. Hence (1.48) and (1.49) are Riemann sums of smooth functions with equidistant partitions. Thus, there exists a constant C ą 0, such that holds uniformly in Bp0, rq for n large enough. In particular Assumption 4 is satisfied. This example shows that we indeed can get the Cusp-Airy limit in a natural model of the type considered in [20].
Remark 1.4. The regularity assumption on the sequence of empirical measures made in Assumption 3 are necessary for Theorem 1 to hold, and cannot be substantially relaxed. If one in particular try to perform the cancellation of factors as in Lemma 1.2 without Assumption 3, one immediately sees that pw´kq, in general, and that q n pw; rq would also depend on the sequence tβ pnq i u n . In particular the limit in Theorem 1.1 need not exist.

Random Top Line Measure
Up to now we have assumed that the top line configuration of yellow particles is fixed. However, for many models it is natural not to assume that the top line configuration is fixed, but instead is a random particle process. Let Σ Ă R be finite union of closed, bounded intervals and write Σ n " Z X nΣ. Let X pnq denote the set X pnq " tβ pnq P Σ n n ; β pnq 1 ă¨¨¨ă β pnq n u.
We will call X pnq the set of all admissible top line configurations. Note that X pnq is a finite set. We will now assume that we have a probability distribution p pnq pβ pnq 1 , ..., β pnq n q on X pnq . Extended to Σ n n , we assume that p pnq pβ pnq 1 , ..., β pnq n q is a symmetric function which vanishes if β pnq i " β pnq j for some i ‰ j. Let P pnq denote the probability and E pnq the corresponding expectation given by p pnq . Also, let E β pnq denote the expectation with respect to the red particles in the uniform interlacing with fixed top line β pnq that we studied above.
Let µ with supp µ Ď Σ be given and let f be defined by (1.5) as previously. We assume that Assumption 2 holds with this f . In order to transfer the Main theorem to the case with a random top line we will use the following assumption. Proof. We can write (1.51) By Assumption 5 the second term in the right hand side of (1.51) goes to 0 as n Ñ 8. There is aβ pnq such that  is assumed at β pnq "β pnq , since the maximum is over a finite set. By Assumption 5 we can apply the Main theorem to the sequenceβ pnq and thus lim sup We can do the analogous argument for the lower limes, and in this way we obtain the desired result.
Remark 1.5. Interlacing particle systems with a random top line occur in certain types of lozenge tiling models. More precisely, we are interested in those tiling models that can be decomposed into two regions, such that after possibly adding virtual particles, these the regions become interlacing regions of the type describe in section 1.2, glued together along a common line as depicted in figure 15. Recall that the number of interlacing configurations with a given top line configuration py 1 , y 2 , ...., y N q P Z N is given by Weyl's dimension formula in [24] for the irreducible characters of the unitary group U pnq, Let py 1 , y 2 , ...., y N q be the positions of the particles/tiles on the intersecting thick black line as in figure 15.
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 (1.52), the number of interlacing configuration with a given fixed configuration of free particles at positions px 1 , ..., x n q is given by where Z n is a normalization constant, and ∆ n pxq is the Vandermonde determinnat. Associated with a particular weight function w n pxq is a class of discrete orthogonal polynomials tp n,k pxqu k satisfying ÿ xPΣn p n,k pxqp n,l pxqw n pxq " δ kl . (1.54) Such particle processes are called discrete orthogonal polynomial ensembles, DOPE, and have been studied e.g. in [4]. In particular if one considers the random empirical measure µ n " 1 n ř n i"1 δ xi , then µ n á µ λ V , where the measure µ λ V P M λ 1 pΣq is the unique solution of the constrained variational problem where V pxq " lim nÑ8´n´1 logpw n pxqq. In the case of those problems originating from a random tiling model as above, the potential will be of the form V pxq " U χ I r pxq`U χ I l pxq, (1.56) where I r and I l are finite unions of closed intervals of R, such that pI r Y I l q˝X Σ˝" ∅. In particular lim nÑ8´n´1 logpw n pxqq " V pxq uniformly on compact subsets of Σ not containing the subset BΣ X BpI r Y I l q. It follows from Theorem 2.1 in [7] that the minimizer µ λ V of the variational problem (1.55) has a unique the characterization in terms of the following variational inequalities. There exists a constant F λ V such that (1.59) It follows from general large deviation estimates for DOPE that we can define X pnq reg so that Assumption 3 is satisfied for large classes of potentials, see [11]. In particular, in the class of potentials of the form (1.56) and associated weights w n coming from certain tiling models, this is proved in the upcoming review article [10].
It is shown in [10] that if R 1 Y R 2 ‰ ∅, with µ " µ λ V , then for t c P R 1 Y R 2 , we must have t c P BΣ. Thus, if in particular t c P R 2 , then there exists an interval rt 2 , t c s, such that rt 2 , t c s X supppµ n q " ∅. This implies that the sequence of empirical measures tµ n u n automatically satisfies the left-sided part of regularity assumption in Assumption 3. Let for some positive constants C V and C S . In particular this implies that we can define X pnq reg so that Assumption 3 holds. Unfortunately, the class of potentials given by (1.56) need not be analytic in a neighborhood of Σ due to the fact that we may have BΣ X BpI r Y I l q ‰ 0. However, the we believe that by additional local arguments around such points, one may generalize the methods used in the book [4] to prove that Theorem 3.3 and Theorem 3.5 in [4] also hold in this case. In particular, the effect of non-analyticity should only matter on a very small neighborhood around the points of BΣ X BpI r Y I l q. Since t c R BΣ X BpI r Y I l q, and therefore is a macroscopic distance away from BΣ X BpI r Y I l q, the effects of non-analyticity should not matter.
Finally, let A Ă R be an interval and let K be a compact subset of tz P C : d H pz, A X supppµ n qq ą 0u. Then, for a "generic" potential V , and a sequence of weight functions tw n pxqu n , such that lim nÑ8´n´1 logpw n pxqq " V pxq uniformly on compact subsets of Σ, we should have for every ε ą 0 and every z P K that In particular, this implies that we can define X pnq reg so that Assumption 4 holds. This result should also follow from Riemann-Hilbert methods adapted to the class of potentials derived from tiling models, with the caveat that some of the local arguments need to be modified due to the possible non-analyticity of V pxq at some of the points of BΣ. This questions has also been studied in [5] for very similar models by means of discrete loop equations, though the assumptions in [5] are not satisfied in our models, and can therefore not be applied directly. However, in the special case when Σ is an interval, the results in [5] do apply. In particular they apply to the model in Example 1.2.
Remark 1.6. Let z prq " py prq 1 , ..., y prq r , x n`n´r´1 , x n`n´r´2 , ..., x n q. Then, using (1.53) the total probability distribution is given by ν tot rpy 1 , ..., y pn´1q , x pnq qs " 1 Z n,tot ∆ n pxq 2 Hence, by the Eynard-Mehta theorem, the total process is also a determinantal process. One could therefore try to derive the correlation kernel for the total process instead of the conditional process. However, the Eynard-Mehta formula for the kernel of this process seems very difficult to analyze. In particular, we can not expect to get such a simple formula for the kernel as (1.2), as it would necessarily need to contain all the information about the DOPE, which is highly non-trivial. 3´2 κ, where κ P p0, 1{ ? 3q. In the figure we have added densely packed virtual particles in the intervals r0, κs and r2κ`2δ, 3κ`2δs. Moreover, the particles contained in the interval rκ`δ, 2κ`2δs are distributed according to a discrete orthogonal polynomial ensemble, DOPE, as is shown above. The fraction of particles contained in this interval as n Ñ 8 equals 1´?3κ. We will now make the symmetric parameter choice κ " δ " 2 3 ?
3 . Associated to the DOPE, there is an equilibrium measure with respect to an external field as above. See also Proposition 2.2 in [5]. Solving the minimization problem gives the density of the measure µ, ρptq " χ r0,   3 ,as " 1. A direct verification that the Cauchy transform of ρ satisfies Assumption 2 for t c " 4 3 ?
3 is very difficult using (1.65)-(1.67) even in the case when the original polygon has an apparent symmetry. This is due to the artificial decomposition of the polygon into two interlacing regions which are glued together along a common boundary. After this decomposition has been done, the original symmetry is no longer apparent in the parametrization of the edge E. We will therefore prove that the asymptotic cusp condition holds by an indirect symmetry argument. Instead of considering figure 16, we consider figure 17. We see the yellow dashed line corresponds to the decomposition in figure 16 into two interlacing regions glued together along the yellow dashed line. However, we may equally well decompose our hexagon into two interlacing regions in blue particles glued together along the dashed blue line instead. Moreover there is bijective correspondence between each configuration of blue and yellow particles given by a reflection in the dashed black symmetry line. This implies in particular that the edge E must posses a reflection symmetry in the dashed black line. Now, the parametrization of the edge E, given by the density (1.65) and (1.13)-(1.16) is smooth by Remark 2.1 in [8], in fact real analytic. Since the density ρ of the limit measure µ satisfies ρˇˇr 4 3 ?
3 . However, by Theorem 2.3 in [8], the parametrization is injective. This together with the fact that the only singularities of the edge E are cusps implies the edge E has a cusp at the tangent point with the line χ`η´1 " 4 3 ?
3 qq " 0. Hence, modulo the technical issues discussed above about how to construct X pnq reg , we get a cusp-point where the scaling limit is given by the Cusp-Airy process.

Outline of the Paper
We now give a brief outline of the paper. In section 2.1 we prove Lemma 1.2. Due to the fact that f 3 pt c , χ c , η c q ą 0, it will be necessary to change the integration contours of the correlation kernel given in Proposition 1.2 to be able to deform the contours to the local steepest ascent/descent contours. This is done section 2.1. In section 2.3 we prove the existence of global ascent/descent contours and finally in sections 2.4 and 2.5 we perform the local asymptotic analysis to arrive at the final result In section 3.1 we prove a certain reflection symmetry of the Cusp-Airy kernel in the axis r " 0. In section 3.2 we proceed to prove an alternative representation of the the Cusp-Airy kernel in terms of r-Airy integrals and certain polynomials. In section 4 we derive the integral representation of the kernel for the interlacing particle system, or the yellow particles and prove Proposition 1.1.

Discrete Cancellation in the Correlation Kernel
In this section we will perform the discrete cancellation of factors in the correlation kernel (1.2). In the continuum limit, this corresponds to the cancellation between the measures µ and λ on the interval rt c , t 1 s, where µˇˇr tc,t1s " λ. The difference is that on the discrete level, the cancellation need no longer be exact. We now prove Lemma 1. where we have used the definitions of q n , Q n and E n . Similarly, we get k"x2`y2´n pz´kq " pz´x 2 qq n pz; rqQ n pz; ∆x pnq 1 qE n pzq.
Finally we rescale the integration variables according to w Ñ 1 n w and z Ñ 1 n z. Set Γ n " 1 n Z n , Γ n " 1 nZ n , and γ n "γ n " 1 n W n . Since we have cancelled out poles we may deform Γ n to be a contour that contains the set tn´1β

Change of Integration Contours
In order to perform a steepest descent analysis in a later section, it will be necessary to change the integration contours so that they may be suitably deformed around the critical point.
Proposition 2.1. The correlation kernel can be rewritten as K pnq R ppξ n , rqpµ n , sqq "´1 x1ěx2 B n ppx 1 , y 1 q, px 2 , y 2 qq`K pnq R ppx 1 , y 1 q, px 2 , y 2 qq, Here the contours are as in figure 18; more precisely: • Γ 1 n is a counter-clockwise oriented contour that contains the interval rt pnq c´m axt|r|, |s|u, b n s and nothing else of the support of ν n . Furthermore, Γ 1 n contains the contours Γ 2 n , γ 2 n and γ 3 n • Γ 2 n is a clockwise oriented contour that contains γ 2 n and the interval rt pnq c´m axt|r|, |s|u, t pnq cm axt|r|, |s|us and nothing else of the support of ν n .
• γ 1 n is a clockwise oriented contour that contains the interval rt pnq 1 , b n s and nothing else of the support of ν n .
• γ 2 n is a clockwise oriented contour that contains the interval rt  Next, we deform the contour Γ n into the contours Γ 1 n and Γ 2 n according to Figure 21. This gives us We now instead assume that x 1 ě x 2 . We then deform deform the contourΓ n into the contoursΓ 1 n and Γ 2 n according to figure 22. Finally, we deform the contourγ n into the contoursγ 0 n ,γ 2 n andγ 1 n , according to figure 23. However, using that the only residue contained inγ 0 n is the pole pw´zq´1, we get where the contours are shown in figure 24. We now deform the contourΓ 1 n intoΓ 3 n`C R according to figure 25. Clearly the contribution along ℓ vanish and to prove that the contribution from C R vanishes as R Ñ 8 we observe that that g n,2 pzq " ν n,2 pRq log |z|`Op|z|´1q and lim nÑ8 ν n,2 pRq " νpRq ą 0. From this it is not difficult to see that lim RÑ8 |J pnq CRγ i n | " 0.
Since f pw; χ c q " f pw; χ c q it is sufficient to prove the existence of the contours in the upper-half plane H.  Proof. Since f pw; χ c q " f pw; χ c q it is sufficient to prove the existence of the contours in the upper-half plane H. We note that U ν is real analytic in Czsupppνq and that´R argpw´tqdνptq is real analytic in Czp´8, bs. Moreover, the boundary values on the real axis are given by lim wÑxPR wPH U ν pwq " U ν pxq and lim wÑxPR wPHˆR argpx´tqdνptq " πνprx,`8qq.
In particular, U ν pwq has a continuous extension to all of C.
Since f 3 pt c ; χ c , η c q ą 0 the local steepest ascent/descent structure around t c is as in figure 28. Recall that the contours of steepest ascent/descent are those for which Im f pwq " Im f pt c q " πµprt c , bsq. Note that U ν pwq " νpRq log |w|`Op|w|´1q as |w| Ñ 8. Therefore limwÑ8 wPH U ν pwq "`8. This implies that the descent contour have to be contained in some ball Bp0, Rq, R sufficiently large. On the other hand since the function Im f pwq is real analytic, the curve Im f pwq " Im f pt c q has to be either a closed curve in Bp0, Rqzsupppνq or end somewhere in supppνq. Assume the first case. Then, clearly there has to be a point t p ‰ t c on the curve such that f 1 pt p q " 0. By Theorem 3.1 in [8], we must for such a t p have t p P Rzsupppνq and f 2 pt p q ‰ 0. At such a point we have descent contour exiting at angle˘π{2 and ascent contours exiting at 0 and π. This gives a contradiction. We may therefore assume the second case holds It follows from Lemma 2.2 that if πνprt c , 8qq ă 0 then the equation πνprx, 8qq " πµprt c ,`8qq has no solution and the steepest descent contour from t c has to go to infinity, which is impossible. If πνprt c , 8qq " 0, then we have to be in the first case above which is impossible. Thus, πνprt c , 8qq ą 0 and Lemma 2.2 implies that the equation πνprx, 8qq " πµprt c ,`8qq has at least one solution t e for x P pχ c , bq and no solution for x P p´8, t 2 q Y pt 1 , χ c q Y rb, 8q. In figure 29 we give a plot of what the function πνprx,`8qq may look like. If νprx, 8qq is strictly monotonically decreasing at t e , then t e is the unique solution to the equation above. Otherwise, by the monotonicity of νprx, 8qq, there exists an interval rté , tè s such that πνprx, 8qq " πµprt c ,`8qq for all x P rté , tè s. In particular, pté , tè q X supppµq " ∅. By Lemma 2.2 and the discussion above, the only possible end points are t 2 , t 1 and t e or t 2 , t 1 , té and tè . Assume that it ends at t 1 . Then we get a closed contour in H containing the interval rt c , t 1 s, and such that the boundary value of Imrf pwqs equals Imrf pt c qs " πµprt c , bsq everywhere on the curve. However, since Imrf pwqs is harmonic inside the domain bounded by the curve, this implies that Imrf pwqs is constant, a contradiction. Similarly, the descent contour cannot end in t 2 . Thus it has to end either at t e or one of té and tè .
This proves the existence of the global steepest descent path of f pw, χ c q. We now consider the ascent path. Recall that the ascent and descent paths cannot intersect. By considering the local ascent and descent contours we know it cannot end at t 1 . Suppose that it ends a t 2 . However, by a similar argument as before, this is not possible. Moreover, by the continuity of U ν pwq it cannot end at t e . Thus, the ascent contour will become an asymptote of the line tte iθ : t P r0,`8q, θ " πµprt c , bsqu since´R argpw´tqdνptq " νpRq argpwq`Op|w|´1q as |w| Ñ 8. Now assume that the decent path ends at té and that the ascent path ends at tè . Again by forming a closed contour containing the interval rté , tè s and exploiting the harmonicity of Imrf pwqs, we get a contradiction. Thus as before, contour will become an asymptote of the line tte iθ : t P r0,`8q, θ " πµprt c , bsqu.

Estimates and localization
We start with some preliminary results that we will need. By lemma 2.1 and Assumption 4, if we take δ 0 small enough, then |g n,i pwq| ď C for |w´t c | ď δ 0 , where C is a constant. We have the Taylor expansion g n,i pzq " g n,i pt c q`g 1 n,i pt c qpz´t c q`1 2 g 2 n,i pt c q 2 pz´t c q 2`1 6 g 3 n,i pt c q 3 pz´t c q 3`r n,i pzqpz´t c q 4 , (2.6) where r n,i pzq " 1 2πiˆ| w´tc|"δ3 g n,i pwq pw´t c q 5 p1´z´t c w´tc q dw. (2.7) From (2.7) we see that, if we take δ 1 ď δ 0 {2, then there is a constant Cpδ 0 q so that |r n,i pzq| ď Cpδ 0 q, (2.8) for all z P Bpt c , δ 1 q. Consider a curve zptq " t c`ζ ptq, (2.9) t P I, such that |ζptq| ď Ct ď δ 1 for all t P I, I an interval. From (2.6) we obtain Rerg n,i pzptqq´g n,i pt c qs " g 1 n,i pt c qRerζptqs`1 2 g 2 n,i pt c qRerζptq 2 s`1 6 g 3 n,i pt c qRerζptq 3 s (2.10) Rerr n,i pzptqqζptq 4 s.
We will now discuss the localization of the asymptotic analysis of the kernel to a neighbourhood of t c . Let δ 1 ą 0 and let B 2 " Bpt c , δ 1 q be a (small) ball around t c . The descent contour D from Lemma 2. We now formulate a lemma that will allow us to neglect the contribution from D outside B 2 .
Lemma 2.4. If we choose δ 1 sufficiently small, there is a constant b 0 pδ 1 q ą 0 such that for n large enough Re rg n,i pD 2 q´g n,i pt c qs ď´b 0 pδ 1 q. (2.16) Proof. We take ζptq " te iθpδ1q , 0 ď t ď δ 1 , where θpδ 1 q is chosen so that ζpδ 1 q " D 2 . It follows from Lemma 2.1, f 1 pt c q " f 2 pt c q " 0, and f 3 pt c q ą 0 that, given ε 1 ą 0, we have |g 1 n,i pt c q| ď ε 1 , |g 2 n,i pt c q| ď ε 1 (2.17) for large n, and there is a c 1 ą 0 so that for all sufficiently large n. Also, since f 1 pt c q " f 2 pt c q " 0 and f 3 pt c q ą 0, a local Taylor expansion of f shows that θpδ 1 q Ñ π{3 as δ 1 Ñ 0. Consequently, for 0 ď t ď δ 1 if δ 1 is sufficiently small. From (2.10), (2.11), (2.17), and (2.18) we see that Re rg n,i pD 2 q´g n,i pt c qs ě´1 12 We can now choose δ 1 so that C 1 δ 1 ď 1{48 and then ǫ 1 so that C 1 ε 1 pδ´2 1`δ´1 1 q ď 1{48. We then get (2.16) with b 0 pδ 1 q " c 1 δ 3 1 {24 and the lemma is proved. Let B 3 " Bpt e , δ 2 q be a small ball around t e . The descent contour D intersects BB 3 X H at the point D 3 . Let C be as in figure 31 and D 1 be the part of D outside B 2 and B 3 , so that D 1 lies strictly in H. Hence, from Lemma 2.1, it follows that g n,i pzq Ñ f pzq uniformly on D 1 as n Ñ 8. for all z P C if n is sufficiently large.
We have chosen δ 2 so small that x 0 ą χ c . From (1.42) we see that Note that u pnq 1 ptq ď 0 and u pnq 2 ptq "´d dt 1 2n Re rf n pzq´f n pD 3 qs ď Cδ 2 for all z P C if n is sufficiently large. It remains to estimate 1 n Rerh n,i pzptqqs. Consider h n,1 and assume ∆x pnq 1 ą 0, the other cases are similar. From (1.43) and (1.37) we obtain 1 nˇˇˇˇR e rh n,1 pzptqqsˇˇˇˇď log n n ∆x if n is sufficiently large. Here we used again the fact that x pnq c {n Ñ χ c ă x 0 . This proves the lemma.
By uniform convergence we have that |g n,i pzq´f pzq| ď 1 4 b 0 pδ 1 q for all z P D 1 if n is large enough. Thus, if z P D 1 we have Rerg n,i pzq´g n,i pD 2 qs ď Rerg n,i pzq´f pzqs`Rerf pD 2 q´g n,i pD 2 qs`Rerf pzq´f pD 2 qs loooooooooomoooooooooon where we have used that D 1 is a descent curve. Combining this with Lemma 2.4, we obtain Rerg n,i pzq´g n,i pt c qs ď´1 2 b 0 pδ 1 q.
(2.24) for all z P D 1 . Given δ 1 we choose δ 2 so small that b 1 pδ 2 q ď b 0 pδ 1 q{4. Since D 3 P D 1 we find, using Lemma 2.5 and (2.24) that Rerg n,i pzq´g n,i pt c qs ď´1 for all z P C if n is sufficiently large. Note that Re g n,2 pzq " ν n,2 pRq log |z|`Op|z|´1q (2.26) as |z| Ñ 8, and ν n,2 pRq Ñ νpRq ą 0 as n Ñ 8. In (2.22) we can let Γ 2 n and γ 2 n be small circles around t c inside B 2 and deform γ 1 n to the descent contour D 1`C (and its reflection image in the lower half plane). Using (2.26) we see that we can deform Γ 1 n to the ascent contour A. We can now use the estimates (2.24), (2.25) and (2.26) to see that in the integral (2.22) we can ignore D 1`C and the part of A outside B 2 in the limit. More precisely, we also have to combine this estimate with the estimates and computations inside B 2 that we will do in the next section. We leave out the details.

Local Analysis
Let tM n u ně1 be a sequence satisfying M n Ñ 8, M n n 1{12 Ñ 0, n 2{3 M n |f 1 n pt c q| Ñ 0 (2.27) as n Ñ 8, which exists by Assumption 4. Consider the ball B 1 " Bpt c , M n n´1 {3 q. Again we will only discuss the descent contour; the ascent case is analogous. As above, we take ζptq " te iθpδ1q in (2.9) with M n {n 1{3 ď t ď δ 1 , so that zpδ 1 q " t c`ζ pδ 1 q " D 2 , see figure 32. It follows from (2.10),(2.11) and (2.18) that there is a constant C so that Rerg n,i pzptqq´g n,i pt c qs ď´t 3 (2.28) It follows from Assumption 4 that n 2{3 f 1 n pt c q Ñ 0 and n 2{3 f 2 n pt c q Ñ 0 as n Ñ 8. Combining this with (2.13) we see that, for t P rM n {n 1{3 , δ 1 s,  This estimate can be used together with the corresponding estimate for the ascent contour to control, in the limit, the contribution from the parts of the descent and ascent contours that lie in B 2 zB 1 . We find that we can neglect the contributions from B 2 zB 1 .
Consider the contribution from γ 1 n . Write npg n,1 pwq´g n,1 pt c qq " npf n pwq´f n pt c qq`h n,1 pwq´h n,1 pt c q.

Representation of The Cusp-Airy Kernel
In this section we will derive an alternative representation of the Cusp-Airy kernel involving the so called r-Airy integrals and certain polynomials. Define the r-Airy integrals, where r ě 0 and ℓ is a contour from 8e 5πi{6 to 8e πi{6 such that 0 lies above the contour, see [1]; compare also with the functions s pmq and t pmq in [3]. Note that A0 puq " Ai puq, the standard Airy function. Let LL be the contour L L shifted to the right so that 0 lies to the left of it, see figure 35; define LR analogously by shifting L R to the left so that 0 is to the right of it. It is straightforward to check that and p´1q r 2πiˆLR e 1 3 w 3´u w 1 w r dw " for r ě 0.
Define the polynomials P n pw, ξq and p n pξq through P n pw, ξq :" e´1 3 w 3`u w d n dw n e By Cauchy's integral formula, we have for r ě 0, Note that L L`Cout " LL , see figure 34. Thus, p´1q r 2πiˆL R e 1 3 w 3´u w 1 w r dw " 1 2πiˆL L e´1 3 z 3`u z 1 z r dz " Aŕ puq`p´1q r p r´1 puq. We can now give a different formula for the Cusp-Airy kernel in terms of the r-Airy integrals.  Consider the case (i). If r, s ě 0, then C in does not contribute, and using L L`Cout " LL , see figure  34, and the formulaˆ8 0 e´λ pw´zq dλ " 1 w´z (3.9) valid if Re pw´zq ą 0, we find K CA ppξ, rq, pτ, sqq "ˆ8 0˜1 2πiˆLL e´1 3 z 3`p τ`λqz 1 z s dz¸˜1 2πiˆLR e 1 3 w 3´p ξ`λqw w r dw¸dλ "ˆ8 0 Aś pτ`λqAr pξ`λq dλ, by (3.3) and (3.4). Here, we also used the fact that since r ě 0, we can move L R to LR . In the case (ii) we have r ě 0 and s ă 0, and thus neither C in nor C out contribute. Using (3.9) and then moving L L to LL and L R to LR , we obtaiñ K CA ppξ, rq, pτ, sqq "ˆ8 0˜1 2πiˆLL e´1 3 z 3`p τ`λqz z´s dz¸˜1 2πiˆLR e 1 3 w 3´p ξ`λqw w r dw¸dλ " p´1q sˆ8 0 A`spτ`λqAr pξ`λq dλ.
Next, consider the case (iii). If r ă 0, s ě 0, we can writê Since |w| ą |z| if z P C out and w P L R , we can use the identity 1 w´z " 8 ÿ k"0 z k w k`1 to see that 2πiˆC out e´1 3 z 3`τ z 1 z s´k dz˙ˆ1 2πiˆL R e 1 3 w 3´ξ w 1 w´r`k`1 dw" s´1 ÿ k"0 p s´k´1 pτ qˆp´1q´r`k`1A´r`k`1pξq`p´r`kpξq" p´1q s´r s´1 ÿ k"0 p k pτ qA´r`s´kpξq`s´1 ÿ k"0 p´1q k p k pτ qp´r`s´1´kpξq. If z P C out and w P C in , then 1 w´z "´8 ÿ k"0 w k z k`1 and we see that, by (3.7), and the fact that C in is negatively oriented, we have 2πiˆC out e´1 3 z 3`τ z 1 z s`k`1 dz˙ˆ1 2πiˆC in e 1 3 w 3´ξ w 1 w´r´k dw"´r´1 ÿ k"0 p s`k pτ qp´r´k´1pξq " s´r´1 ÿ k"s p k pτ qp´r`s´1´kpξq. (3.12) Adding up (3.10) -(3.12) we have proved (iii). The case (iv) follows from (i) by using Proposition 3.1.
Note that if we take r " s ě 0, then by (i), K CA ppξ, rq, pτ, sqq "ˆ8 0 Aŕ pτ`λqAr pξ`λq dλ :" K prq pτ, ξq, which is called the r-Airy kernel. Note that when r " 0 we get the standard Airy kernel. The r-Airy kernel has appeared previously in the work [3] on largest eigenvalues of sample covariance matrices and in [1] on Dyson's Brownian motion with outliers. See also [2]. Though we do not show it in this paper using the results above and some further estimates it should be possible to prove that the scaling limit of the position of the last (red) particle on line r, close to the cusp point, has the distribution function F r pxq " detpI´K prq q L 2 px,8q . In particular, when r " 0 we should get the Tracy-Widom distribution.
Remark 3.1. It is instructive to compare the Cusp-Airy kernel to the GUE-corner kernel. Recall that the GUE-corner kernel is given by Kpn, x; n 1 , x 1 q "´1 nąn 1 1 xąx 1 2 n´n 1 px´x 1 q n´n 1´1 pn´n 1´1 q!`2 p2πiq 2ˆΓ 0 dzˆL dw w´z w n 1 z n e w 2´z2`2 zx´2wx 1 , where x, x 1 P R and n, n 1 P Z and n, n 1 ě 0, and where the contours are shown in figure 36. In a sense, one can regard the Cusp-Airy kernel as a double sided version of the GUE-corner kernel. If one changes the assumption that f 1 tc has a simple root at t c P R 1 Ť R 2 , then a similar computation as in the Cusp-Airy case will yield the GUE-corner kernel for an appropriate scaling limit of K n ppx

Integral Representation of the Correlation Kernel for the Yellow Particles
In section A.1 in [8] it was shown that the correlation kernel for the interlacing particle system is given by Since the w-contour is outside Z n the sum over the β pnq k ě x 2 can be rewritten using Corlollary 4.1 and we find 1 x1ăx2 1 p2πiq 2˛W n dw˛Z n dz ś x2´1 j"x2`y2´n`1 pz´jq ś x1 j"x1`y1´n pw´jq 1 w´z ś n i"1 pw´β pnq i q ś n i"1 pz´β pnq i q .
The second expression in the right hand side of (4.5) can be rewritten in exactly the same way and we have proved the proposition.

Particle Transformation
From knowledge of a correlation kernel for the yellow particles we now want to derive an expression for a correlation kernel for the red (and blue) particles.
Lemma 4.1. Correlation kernels for the red and blue tiles (particles) are given by K pnq R ppx 1 , y 1 q, px 2 , y 2 qq "´K pnq Y ppx 1 , y 1 q, px 2 , y 2´1 qq K pnq B ppx 1 , y 1 q, px 2 , y 2 qq " K pnq Y ppx 1 , y 1 q, px 2`1 , y 2´1 qq. Proof. Let K P be the Kasteleyn matrix of the adjacency matrix of the honeycomb graph G P of the polygon P. It is defined according to K P ppx, nq; py, mqq " Recall that if py, mq " px, nq we have a yellow particle (rhombi of shape) at position px, nq in our lattice. Similarly, if py, mq " px, n´1q we have a red particle at position px, nq and if py, mq " px`1, n´1q we have a blue particle at position px, nq.
It was shown in [20] Theorem 6.1 that inverse Kasteleyn matrix K´1 P is related to the correlation kernel of the yellow particles K Y according to K´1 P ppy, mq; px, nqq " p´1q y´x`m´n K pnq Y px, n; y, mq. (4.7) From Corollary 3 in [15] one has that the probability of finding a set of edges tb 1 w 1 , ...b k w k u is given by Predges at tw 1 b 1 , ..., w k b k us "ˆk ź i K P pw i , b i q˙detpK´1 P pb i , w j qq k i,j Now finding k red particles at positions tpx i , n i qu k i"1 is equivalent to finding the edges tppx i , n i q, px i , n i1 qqu k i"1 Hence, the probability of finding k red particles at positions tpx i , n i qu k i"1 equals Prred particles at positions tpx i , n i qu k i"1 s " Predges at positions tppx i , n i q, px i , n i´1 qqu k i"1 s " detpK´1 P ppx i , n i´1 q, px j , n j qq k i,j " detpp´1q xi´xj`ni´nj`1 K pnq Y px j , n j ; x i , n i´1 qq k i,j " detp´K pnq Y px i , n i ; x j , n j´1 qq k i,j However, by definition ρ R ppx 1 , n 1 q, px 2 , n 2 q, ..., px k , n k qq " Prred particles at positions tpx i , n i qu k i"1 s " detpK pnq R px i , n i ; y j , n j qq 1ďi,jďk whenever the right hand side is well-defined.
This quantity is referred to as the Fredholm determinant, denoted detrI´φKs L 2 pBq .