Eigenvalues of GUE Minors

Consider an infinite random matrix $H=(h_{ij})_{0<i,j}$ picked from the Gaussian Unitary Ensemble (GUE). Denote its main minors by $H_i=(h_{rs})_{1\leq r,s\leq i}$ and let the $j$:th largest eigenvalue of $H_i$ be $\mu^i_j$. We show that the configuration of all these eigenvalues $(i,\mu_j^i)$ form a determinantal point process on $\mathbb{N}\times\mathbb{R}$. Furthermore we show that this process can be obtained as the scaling limit in random tilings of the Aztec diamond close to the boundary. We also discuss the corresponding limit for random lozenge tilings of a hexagon.


Introduction
The distribution of eigenvalues induced by some measure on matrices has been the study of random matrix theory for decades. These distributions have been found to be universal in the sense that they turn up in various unrelated problems, some of which do not contain a matrix in any obvious way, or contain a matrix that does not look like a random matrix. In this article, we propose to study the eigenvalues of the minors of a random matrix, and argue that this distribution also is universal in some sense by showing that it is the scaling limit of three apparently unrelated discrete models.
The largest eigenvalues of minors of GUE-matrices have been studied in [Bar01], connecting these to a certain queueing model. It is a special case of the very general class of models analysed in [Joh03]. The large N limit of this model will yield the distribution of all the eigenvalues of a GUE-matrix and its minors.
This process will turn out also to be the scaling limit of a point process related to random tilings of the Aztec diamond, studied in [Joh05a] and of a process related to random lozengetilings of a hexagon, studied in [Joh05b].
1.1. Eigenvalues of the GUE. Consider the following point process on Λ = N × R. There is a point at (n, µ) iff the n:th main minor of H, i.e. H n , has an eigenvalue µ. We will call this process the GUE minor process. A central result in this article is that this process is a determinantal point process with a certain kernel K GUE .
Theorem 1.3. The GUE minor process is determinantal with kernel K GUE .
This will be proved in section 3.
1.4. Aztec Diamond. The Aztec diamond of size N is the largest region of the plane that is the union of squares with corners in lattice points and is contained in the region |x| + |y| ≤ N + 1, see figure 1. It can be covered with 2 × 1 dominoes in 2 N (N +1)/2 ways, [EKLP92a,EKLP92b]. Probability distributions on the set of all these possible tilings have been studied in several references, for example [Joh05a,Pro03]. Typical samples are characterized by having so called frozen regions in the north, south, east and west, regions where the tiles are layed out like brickwork. In the middle there is a disordered region, the so called temperate region. It is for example known that for large N , the shape of the temperate region tends to a circle, see [JPS98] for precise statement.
The key to analyzing this model is to colour all squares black or white in a checkerboard fashion. Let us chose colour white for the left square on the top row. A horizontal tile is of type N, or north, whenever its left square is white. All other horizontal tiles are of type S, or south. Likewise, a vertical domino is of type W, or west, precisely if its top square is white. Other vertical dominoes are of type E.
In figure 1, tiles of type N and E have been shaded. Notice that along the line i = 1, there is precisely one white tile, and its position is a stochastic variable that we denote x 1 1 . Along the line i = 2 there are precisely two white tiles, at positions x 2 1 and x 2 2 respectively, etc. In general, let x i k denote the j-coordinate of the k:th white tile along line i. These white points can be considered a particle configuration, and this particle configuration uniquely determines the tiling. It is shown in [Joh05a] that this process is a determinantal point process on N 2 = {1, 2, . . . , N } 2 , and the kernel is computed. We show that this particle process, properly rescaled, converges weakly to the distribution for eigenvalues of GUE described above. More precisely we have the following theorem that will be proved in section 4.
Theorem 1.5. Let µ i j be the eigenvalues of a GUE matrix and its minors. For each N , let {x i j } be the position of the particles, as defined above, in a random tiling of the Aztec Diamond of size j=0  j=1  j=2  j=3  j=4  j=5  j=6  j=7  j=8  j=9  j=10  j=11  j=12  j=13  j=14  j=15  j=16  j=17  j=18 j=19 j=20  N . Then for each continuous function of compact support φ : 1.6. Rhombus Tilings. Consider an (a, b, c)-hexagon, i.e. a hexagon with side lengths a, b, c, a, b, c. It can be covered by rhombus-shaped tiles with angles π/3 and 2π/3 and side length 1, so called lozenges. The number of possible such tilings is This formula was proved by Percy MacMahon (1854-1928), see [Sta99, page 401] for historical remarks. Thus, we can chose a tiling randomly, each possible tiling assigned equal probability. A typical such tiling is shown in figure 2. Just like in the case of the Aztec diamond, there are frozen regions in the corners of the shape and a disordered region in the middle. It has been shown, that when a = b = c = N → ∞, this so called temperate region, tends to a circle, see [CLP98] for precise statement and other similar results.
Equivalently, consider a simple, symmetric, random walks, started at positions (0, 2j), 1 ≤ j ≤ a. At each step in discrete time, each walker moves up or down, with equal probability. They are conditioned never to intersect and to end at positions (c + b, c − b + 2j). Figure 3, the red lines illustrate such a family of walkers, and shows the correspondence between this process and tilings of the hexagon. These red dots in the figure define a point process. [Joh05b] shows that uniform measure on tilings of the hexagon (or equivalently, uniform measure on the possible configurations of simple, symmetric, random walks) induces a measure on this point process that is determinantal, and computes the kernel. We will show, in theorem 5.4, that the complement of this process, the blue dots in the figure, is also a determinantal process and compute its kernel. Let us introduce some notation. Observe that along the line t = 1, there is exactly one blue dot. Let its position be x 1 1 . Along line t = 2 there are two blue dots, at positions x 2 1 and x 2 2 respectively, and so on. All these x i j are stochastic variables, and they are of course not independent of each other.
We expect that the scaling limit of the process {x i j } i,j , as the size of the hexagon tends to infinity, is the GUE minor process with kernel K GUE . More precisely, let µ i j be the eigenvalues of a GUE matrix and its minors. Then for each continuous function of compact support φ : We will outline a proof of this result by going to the limit in the formula for the correlation kernel, which involves the Hahn polynomials. A complete proof requires some further estimates of these polynomials. The GUE minor process has also been obtained as a limit at "turning points" in a 3D partition model by Okounkov and Reshetikhin [OR06]. We expect that the GUE minor process should be the universal limit in random tilings where the disordered region touches the boundary. Acknowledgement: We thank A. Okounkov for helpful comments and for sending the preprint [OR06].

Determinantal point processes
Let Λ be a complete separable metric space with some reference measure λ. For example R with Lebesgue measure or N with counting measure. Let M (Λ) be the space of integer-valued and locally finite measures on Λ. A point process x is a probability measure on M (Λ). For example, let x be a point process. A realisation x(ω) is an element of M (Λ). It will assign positive measure to certain points, {x i (ω)} 1≤i≤N (ω) , sometimes called particles, or just points in the process. In the processes that we will study the number of particles in a compact set will have a uniform upper bound.
Many point processes can be specified by giving their correlation functions, ρ n : Λ n → R, n = 1, . . . , ∞. We will not go into the precise definition of these or when a process is uniquely determined by its correlation functions. For that we refer to any or all of the following references: [DVJ88, Ch. 9.1, A2.1], [Joh05c,Sos00].
Suffice it to say that correlation functions have the following useful property. For any bounded measurable function φ with bounded support B, satisfying the following holds: Correlation functions are thus useful in computing various expectations. For example, if A is some set and χ A is the characteristic function of that set, then 1 − E[ (1 − χ A (x i ))] is the probability of at least one particle in the set A. If the correlation functions of a process exist and are known, this probability can then readily be computed with the above formula.
We will study point processes of a certain type, namely those whose correlation functions exist and are of the form i.e. the n:th correlation function is given as a n × n determinant where K : Λ 2 → R is some, not necessarily smooth, measurable function. Such a process is called a determinantal point process and the function K is called the kernel of the point process.
Let x 1 , x 2 , . . . , x N ,. . . be a sequence of point processes on Λ. Say that x N assigns positive measure to the points {x N i (ω)} 1≤i≤N N (ω) . Then we say that this sequence of point processes converges weakly to a point process x, written x N → x, N → ∞, if for any continuous function φ of compact support, 0 ≤ φ ≤ 1, The next proposition gives sufficient conditions for weak convergence of a sequence of determinantal processes in terms of the kernels.
Then there exists some determinantal point process x with correlation functions K such that Proof. We start by showing that there exists such a determinantal point process x. In [Sos00], the following necessary and sufficient conditions for the existence of a random point process with given correlation functions is given.
ρ k (x σ(1) , . . . , x σ(k) ) = ρ k (x 1 , . . . , x k ) (2) Positivity. For any finite set of measurable bounded functions φ k : Λ k → R, k = 0, . . . , M , with compact support, such that The first condition is satisfied for all correlation functions coming from determinantal kernels because permuting the rows and the columns of a matrix with the same permutation leaves the determinant unchanged. For the positivity condition consider the kernels K N . They are kernels of determinantal processes so As N → ∞, this converges to the same expression with K instead of K N by Lebesgue's bounded convergence theorem with assumption (2). Positivity of this expression for all N then implies positivity of the limit. So now we know that x exists. We need to show that x N → x Take some test function φ : Λ → R with bounded support B. For this function we check the condition in (2.1). The assumption (3) in this theorem implies that the sum is a finite one. Also, ||φ|| ∞ ≤ 1. Assumption (2) is that the functions K N are uniformly bounded, so in particular they are bounded on B 2 , so ρ k is bounded on B k . The integral of a bounded function over a bounded set is finite, so this is the finite sum of finite real numbers, which is finite. Therefore, for each N , by (2.2), (3) guarantees that the sum is finite. Lebesgue's bounded convergence theorem applies because the support of φ is compact and the correlation functions are bounded on compact sets. Thus the limit exists and is This implies that indeed x N → x, weakly, as N → ∞.

The GUE Minor Kernel
3.1. Performance Table. Consider the following model. Let {w(i, j)} (i,j)∈Z 2 + , be independent geometric random variables with parameter q 2 . I.e. there is one i.i.d. variable sitting at each integer lattice point in the first quadrant of the plane. Let where the maximum is over all up/right paths from (1, 1) to (M, N ). The array Each such up/right path must pass through precisely one of (M − 1, N ) and (M, It is known from [Bar01], that (G(N, 1), G(N, 2), . . . , G(N, M )) for fixed M , properly rescaled, jointly tends to the distribution of (µ 1 1 , µ 2 1 , . . . , µ M 1 ) as N → ∞ in the sense of weak convergence of probability measures. We will show that it is possible to define stochastic variables in terms of the values w that jointly converge weakly to the distribution of all the eigenvalues µ i j of GUE-matrices.
3.2. Notation. We will use the following notation from [Bar01].
(1) W M,N is set of M × N integer matrices.
(2) W M,N,k is set of M × N integer matrices whose entries sum up to k.
(3) V M = R M(M−1)/2 where the components of each element x are indexed in the following way.
x = It follows that only finitely many of the λ i :s are non-zero.
A partition can be represented by a Young diagram, drawn as a configuration of boxes aligned in rows. The i:th row of boxes is λ i boxes long. A semi-standard Young tableau (SSYT) is a filling of the boxes of a Young diagram with natural numbers, increasing from left to right in rows and strictly increasing from the top down in columns. The Robinsson-Schensted-Knuth algorithm (RSK algorithm) is an algorithm that bijectively maps W M,N to pairs of semi-standard Young tableau. For details of this algorithm, see for example [Sag01,Sta99].
Fix a matrix w ∈ W M,N . This matrix is mapped by RSK to a pair of SSYT, (P (w), Q(w)). The P tableau will contain elements of M := {1, 2, . . . , M } only. Construct a triangular array where x i j is the coordinate of the rightmost box filled with a number at most i in the j-th row of the P (w)-tableau. This is a map from W M,N to C GC,N .
3.4. A Measure on Semistandard Young Tableau. Consider the following probability measure on W M,N . The elements in the matrix are i.i.d. geometric random variables with parameter q 2 . Recall that a variable X is geometrically distributed with parameter q 2 , written The square here will save a lot of root signs later. Such a stochastic variable has expectation a = q 2 /(1 − q 2 ) and variance Applying the RSK algorithm to this array induces a measure on SSYT:s, and by the correspondence above, a measure on C GC,N . Call this measure π RSK q 2 ,M,N . The following is shown in [Joh00].
Proposition 3.5. Let W M,N contain i.i.d. Ge(q 2 ) random variables in each position. The probability that the RSK correspondence, when applied to this matrix, will yield Young diagrams In other words, the measure π RSK q 2 ,M,M , integrating out all variables not on the last row, is ρ RSK q 2 ,M,N . This, together with the following result characterizes the measure π RSK q 2 ,M,M completely.
3.6. Uniform lift. Proposition 3.2 in [Bar01] states that the probability measure π RSK q 2 ,M,N , conditioned on the last row of the triangular array being λ, is uniform on the cone q −1 (λ) : In formulas, this can be formulated as follows.
Proposition 3.7. For any bounded continuous function φ : where L(λ) is the number of integer points in q −1 (λ).
The number of such integer points is given by 3.8. GUE Eigenvalue measure. It is well know, see for example [Meh91], that the probability measure on the eigenvalues induced by GUE measure on M × M hermitian matrices is for some constant Z M that we need not be concerned with here.
3.9. Uniform lift of GUE measure. [Bar01] shows a result for eigenvalues of minors of GUE matrices that is similar to the above result for partitions. He shows that given the eigenvalues of the whole matrix λ = (λ 1 > · · · > λ M ), the triangular array of eigenvalues of all the minors are uniformly distributed in p −1 (λ) := {x ∈ C GC : p(x) = λ}. Again we can write this more formally.
Proposition 3.10. For any bounded continuous function φ : where Vol(λ) is the volume of the cone p −1 (λ).
This volume is given by This situation is then very similar to the measure π RSK q 2 ,M,N above, in the sense that, conditioned the last row, the rest of the variables is uniformly distributed in a certain cone. 3.11. Scaling limit. We are now in a position to see the connection between the measures π RSK q 2 ,M,N and π GUE Then for any bounded continuous function of compact support φ, Proof. Plug in the expression for the right hand side in proposition 3.7 and for the left hand side in proposition 3.10. Stirling's formula and the convergence of a Riemann sum to an integral proves the theorem.
3.13. Polynuclear growth. The measure π RSK q 2 ,N,M is a version of the Schur process and is a determinantal process on M × N, by [OR03]. We will use the following result from [Joh03].
Proposition 3.14. The process {x i j } with the measure described in 3.4 is determinantal with kernel For r ≤ s, the paths of integration for z and w are anticlockwise along circles centred at zero with radii such that q < |w| < |z| < 1/q. For the case r > s, integrate instead along circles such that q < |z| < |w| < 1/q. This follows immediately from proposition 3.12 and theorem 3.14 in [Joh03].
Having now introduced the PNG-kernel, we can state the following scaling limit result.
(1) For r, s ≤ M , uniformly on compact sets as N → ∞ for a certain function g = 0.
(2) The expression is bounded uniformly for 1 ≤ r, s ≤ M and ξ, η in a compact set.
The proof, given in section 6, is an asymptotic analysis of the integral in (3.3). Now everything is set up so we can prove the main result of this section.
Proof of theorem 1.3. According to proposition 3.12, The point processes on the right hand side of this last expression are determinantal. Their kernels can be written for some function g that cancels out in all determinants, and therfore does not affect the correlation functions. By lemma 3.15, these K N satisfy all the assumptions of proposition 2.1. Thus, the point processes that these define converge weakly to a point process with kernel K GUE . This implies that the measure on the left hand side of equation (3.4), i.e. π GUE M is determinantal with kernel K GUE . The observation that M was arbitrary completes the proof.

Aztec Diamond
The point-process connected to the tilings of this shape, described in the introduction was thoroughly analyzed in [Joh05a]. The following result is shown.
and reference measure µ which is counting measure on N. The paths of integration are as follows: For r ≤ s, integrate w along a contour enclosing its pole at −1 anticlockwise, and z along a contour enclosing w and the pole at 0 but not the pole at 1 anticlockwise. For r > s, switch the contours of z and w.
Based on this integral formula we can prove the following scaling limit analogous to that in lemma 3.15. is uniformly bounded with respect to N for (r, ξ, s, η) contained in any compact set in The proof is based on a saddle point analysis that is presented it section 6. We can now set about proving the main result of this section.
By lemma 4.2, the kernels K N satisfy all the assumptions of proposition 2.1. So they converge to the process with kernel K GUE .

The Hexagon
Consider an (a,b,c)-hexagon, such as the one in figure 3. First we need some coordinate system to describe the position of the dots. Say that the a simple, symmetric, random walks start at t = 0 and y = 0, 2, . . . , 2a − 2. In each unit of time, they move one unit up or down, and are conditioned to end up at y = c − b, c − b + 2, . . . , c − b + 2a − 2 at time t = b + c and never to intersect. One realisation of this process is the red dots in figure 3. At time t, the only possible y-coordinates for the red dots are {α t + 2k} 0≤k≤γt , where the set of all the dots, red and blue. 5.1. A determinantal kernel for the hexagon tiling process. We now need to define the normalised associated Hahn polynomials,q (α,β) n,N (x). These orthogonal polynomials satisfy where the weight function is They can be computed as For convenience, let a r = |c − r| and b r = |b − r|. [Joh05b] shows the following.
Invoking equation (5.1) again to simplify the left hand side and explicitly calculating the right hand side gives It is easy to check that which proves the lemma.
We now need to introduce the normalized Hahn polynomials q (α,β) n,N (x). These satisfy Theorem 5.4. The blue dots form a determinantal point process on the space Λ a,b,c with kernel  Proof. It is well known that the complement of a determinantal point processes on a finite set with kernel K is also determinantal with kernelK = I − K, i.e.K(x, y) = δ x,y − K(x, y).
Applying this result to our problem, we consider δ x,y δ r,s −K L a,b,c (r, x; s, y). We now separate two cases. When s ≥ r see that n,γr (x)q (bs,as) n,γs (y)ω r (x)ω s (y) is a candidate for the kernel for the blue particles. By lemma 5.3 this simplifies to n,γr (x)q (bs,as) n,γs (y)ω r (x)ω s (y).
We now exploit a useful duality result from [Bor02]. It states that Insert this into the formulas above and define the new kernel γr−n,γr (x)q (bs,as) γs−n,γs (y) otherwise.
The change of variables j := a − 1 − n puts these expressions on a simpler form, thereby proving the theorem. 5.5. Asymptotics. Let 0 < p < 1 be some real number. Let α = γpN , β = γ(1 − p)N , uniformly on compact sets in x as N → ∞. For completeness we give the proof of this result in the appendix. We would like to apply this with p = 1 2 and γ = 2 to our kernel K L and letting a = b = c → ∞, i.e. we would like to take the limit The factor (−3N ) r−s cancels out in all determinants and is thus of no import. For s ≥ r we get and formally, if we ignore the fact that this turns into an infinite sum, for s < r we get This expression can be simplified with the following lemma Lemma 5.6. Let H be the Heaviside function defined by equation (1.1) above. Then, The proof is given in section 6. In view of this result, the infinite series in 5.7 converges and the kernel K is exactly the GUE minor kernel K GUE . The interpretation of this is the following. The distribution of the blue particles, properly rescaled, tends weakly to the distribution of the eigenvalues of GUE minors as the size of the diamond tends to infinity, equation (1.2). The only thing needed to make this a theorem is appropriate estimates of the Hahn polynomials to control the convergence to the infinite sum.

Proof of lemmas
Proof of lemma 5.6. As the Hermite polynomials are orthogonal, there is an expansion of the function in the left hand side of (5.8) of the form where H n is the n:th Hermite polynomial, as defined in for example [KS98], and where the coefficients are given by It is known that e −x 2 H n (x) = − d dx (e −x 2 H n−1 (x)) for n ≥ 1. Integration by parts and limiting the integration interval according to the Heaviside function gives For n ≥ k, repeat this process k − 1 times to get For 0 ≤ n < k, stop doing partial integrations when H 0 is reached, giving Inserting (6.3) and (6.4) in (6.1) and changing to normalized Hermite polynomials proves the lemma.
(Here we have ignored the difference between aN + ξ √ 2bN and its integer part.) Note that f ′ (z) = 0 gives z = 1/q. This leads us to choose a N/2 , and to deform γ r2 to a circle Γ oriented clockwise around 1/q with radius 1/a N/2. The specific choice of radii are convenient for the computations below. Choose g(r, ξ, N ) = 2 −r/2 e −ξ 2 /2 q −ξ √ 2bN q a N/2 r . Then, Parameterize γ r1 by w(t) = r 1 e itEn , −π/E N ≤ t ≤ π/E N , E N = q/a N/2. We have for N large enough. Since cos x ≤ 1 + x 2 /8 when |x| ≤ π, the last expression is for |t| ≤ π/E N , where C > 0 is a constant depending only on q. Hence, for |t| ≤ π/E n with C > 0. In the right hand side of (6.6) we make the change of variables (6.8) z = z(u) = 1/q − u/a N/2 with u on the unit circle oriented anticlockwise. We obtain the integral Note that q √ b/a = 1. Also, (6.10) for |h| small. Hence, for N sufficiently large, where h N (u) is bounded for |u| = 1. We have .
It follows from (6.7), (6.11), (6.12), (6.13) and (6.14) that the part of the integral in (6.9) where the t-integration is restricted to N 1/3 ≤ |t| ≤ π/E N can be bounded by CN s/2 |t|≥N 1/3 e −Ct 2 dt, which goes to 0 as N → ∞. When |t| ≤ N 1/3 we have Hence, for |t| ≤ N 1/3 we have the bound and we see that the part of the integral in (6.9) where |t| ≤ N 1/3 has a uniform bound for ξ, η in a compact set. This proves claim (2) in lemma 3.15 for r ≤ s. It also follows from (6.7), (6.8), (6.10), (6.11), (6.12), (6.15), (6.16) and the dominated convergence theorem that the integral in (6.9) converges to Then we should integrate v along the line Re v = 2 from minus to plus infinity, call this contour Γ ′ . We obtain the integral Expand (v − u) −1 as a geometric series. This turns the expression into and we recognize the classical integral representations of the Hermite polynomials. The expression now becomes which proves claim (1) in the lemma in the case r ≤ s. We now turn our attention to the case r > s. Deforming the w-contour through the z-contour in (3.3), we get the same integral as above save for a residue that we pick up at z = w. This is We see that the argument above goes through for the remaining integral also when r > s until (6.18). From there the terms k = 0, . . . , s + 1 give (6.19) as before. In terms k = s + 2, . . . , r + 1 we instead evaluate the v-integral using the formula which is valid for n ≥ 1 and will be proved below. That accounts for the terms j = −r, . . . , s + 1 in definition 1.2.
Using the well known formula It is readily solved as With our rescaling, x = aN + ξ √ 2bN and y = aN + η √ 2bN and the factors g(r, ξ, N )/g(s, η, N ) we see that the integral in (6.20) is where H is the Heaviside function. As N → ∞ we get the limit at least for ξ = η. The case ξ = η is a set of measure zero and is not important. Together with the result for the double integral this completes the proof of claim (1). It remains to show the estimate in claim (2) in this case. But this is easy. The expression in (6.22) is the exact solution of integral (6.20), and since this is bounded in N for ξ, η in a compact set, claim (2) follows. It remains to show the formula (6.21). Observe that, by repeated partial integration, if η > 0 and v ∈ Γ ′ . So in this case the left hand side of our formula can be written Here we can change the order of integration and evaluating the Gaussian integral gives the right hand side of (6.21).
When η < 0, make a change of variables v → −v. The left hand side of (6.21) becomes where Γ ′′ is parameterised v = −2 + it, t = −∞ → ∞. By deforming the contour Γ ′′ into Γ ′ we get where γ is a circle around the origin. The right term can be evaluated using the the result for η > 0 above. The left term can be rewritten using the equality between the two integral formulas for the Hermite polynomials mentioned above and we obtain which proves formula (6.21) for η < 0.
Proof of lemma 4.2. Assume first that r ≤ s. By proposition 4.1 we have to consider the integral where γ r is a circle around −1 with radius r oriented anticlockwise, 1 < r 1 < r 2 < 2 and f (z) = 1 2 ln z − ln(1 + z).
It follows from (6.31), (6.35), (6.36), (6.37) and (6.38) that the part of the integral (6.33) where the t-integration is restricted to N 1/3 ≤ |t| ≤ π/E N can be bounded by which tends to 0 as N → ∞. When |t| ≤ N 1/3 and u ∈ γ, we have where C depends on s, r and η but is independent of N .
Hence, we see that the part of the integral in (6.33) where |t| ≤ N 1/3 has a uniform bound for ξ and η in a compact set. This proves claim (2) for r ≤ s.
It also follows from (6.31), (6.32), (6.34), (6.35), (6.37), (6.38), (6.39), (6.40) and the dominated convergence theorem that the integral in (6.33) converges to √ 2 s−r e η 2 −ξ 2 i 2π 2 γ du R dt 1 (2 − it) − u e (2−it) 2 −2(2−it)η e −u 2 +2uξ (2 − it) s u r , which is exactly the integral in (6.17). This proves claim (1) in the lemma in the case r ≤ s. For r > s we can deform the contours one through the other to get the same integral as we solved above. On the way we pick up the residue of a pole at z = w. It is One then sees that the i = k term dominates when N is large. (6.42) Keeping only the i = k term and plugging in our rescaling and the factors g(r, ξ, N )/g(s, η, N ), we see that the integral in (6.41) is (6.43) √ e η 2 −ξ 2 N s−r N/2 2 r−s β N/2 + 2(r − s) − 1 r − s − 1 H(ξ − η) for ξ ≥ η. When ξ = η this tends to √ e η 2 −ξ 2 2 r−s (ξ − η) r−s−1 (r − s − 1)! H(ξ − η) as N → ∞ which together with the corresponding result for the double integral settles claim (1) in the case r > s. Claim (2) in this case follows from the corresponding result for the double integral,(6.42) and the boundedness of the expression in (6.43).