Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble
aa r X i v : . [ m a t h . P R ] A p r NON-INTERSECTING BROWNIAN BRIDGES AND THE LAGUERREORTHOGONAL ENSEMBLE
GIA BAO NGUYEN AND DANIEL REMENIK
Abstract.
We show that the squared maximal height of the top path among N non-intersecting Brownian bridges starting and ending at the origin is distributed as the topeigenvalue of a random matrix drawn from the Laguerre Orthogonal Ensemble. Thisresult can be thought of as a discrete version of K. Johansson’s result that the supremumof the Airy process minus a parabola has the Tracy-Widom GOE distribution, and assuch it provides an explanation for how this distribution arises in models belonging tothe KPZ universality class with flat initial data. The result can be recast in terms of theprobability that the top curve of the stationary Dyson Brownian motion hits an hyperboliccosine barrier. Introduction and Main Results
Motivation and background.
The
Kardar-Parisi-Zhang (KPZ) universality class describes a broad collection of models, including stochastic interface growth on a one-dimensional substrate, polymer chains directed in one dimension and fluctuating transver-sally in the other due to a random potential, driven lattice gas models, reaction-diffusionmodels in two-dimensional random media, and randomly forced Hamilton-Jacobi equa-tions. Although there is no precise definition of the KPZ universality class, it can beidentified at the roughest level by its unusual t / scale of fluctuations (decorrelating ata spatial scale of t / ). The asymptotic distribution of the fluctuations, in the long timelimit t → ∞ , is conjectured to depend only on the initial (or boundary) condition imposedon each particular model.There are three special classes of initial data which stand out because of their scaleinvariance, usually referred to as curved , flat and stationary . Based on exact computationsfor a few models which enjoy a special determinantal structure, the distribution of theasymptotic fluctuations in these three cases is known explicity. One of the most intriguingaspects of the KPZ universality class is that these limiting fluctuations are given in termsof objects coming from random matrix theory (RMT). This is particularly evident in thecases of curved and flat initial data: the asymptotic fluctuations are given, respectively,by the Tracy-Widom GUE and GOE distributions [TW94; TW96]. The first of these twodistributions describes the asymptotic fluctuations of the largest eigenvalue of a randomHermitian matrix with Gaussian entries (the Gaussian Unitary Ensemble), while the sec-ond one is the analog in the real symmetric case (the Gaussian Orthogonal Ensemble);both will be introduced explicitly later on. For more background on this aspect of theKPZ universality class we refer the reader to the reviews [Cor12; QR14]; for some otherperspectives we refer additionally to [Qua11; BP14; QS15].It is very natural in this context to wonder about what lies behind the connection betweenthe KPZ class and RMT. Perhaps the most basic relationship one may seek is to find amodel which lies in the KPZ universality class and which, at the same time, is naturallyexpressed as an object in RMT. As it turns out, in the case of the GUE (corresponding tocurved initial data in the KPZ class) this can be achieved by considering a simple model:non-intersecting Brownian bridges (which we will introduce in detail in Section 1.2). Thismodel is, on the one hand, one of the simplest and most studied models belonging tothe KPZ class, while on the other hand it is equivalent to Dyson Brownian motion, a process which describes the evolution of the eigenvalues of a GUE matrix whose entriesundergo independent Ornstein-Uhlenbeck diffusions. A straightforward consequence of thisequivalence is that the positions of the N non-intersecting Brownian paths at a single timeare distributed as the eigenvalues of a GUE matrix of size N , and this leads directly toanalog statements about their asymptotic fluctuations. Interestingly, the scope of thisrelationship extends also to looking at the entire paths of these processes. For instance, ifone scales the top path of Dyson Brownian motion (or non-intersecting Brownian bridges)appropriately, then in the limit one obtains the Airy process, which is known to describethe spatial fluctuations of models in the KPZ class with curved initial data. Beyond thebasic relationship which we have just described, more recent developments in the areaknown as integrable probability have led to other, arguably deeper, ways of understandingthe connection (see for instance [BP14; BG15]).The situation in the case of GOE, which corresponds to flat initial data in the KPZ class,is much less clear. In fact, essentially no results are available, and it has been a questionof interest for several years now, both for probabilists and for physicists, to understandwhether a relationship similar to the one available for the GUE case is available for GOE,or whether the appearance of the Tracy-Widom GOE distribution in the KPZ class is notmuch more than a coincidence.The fact that the GOE/flat link is much more difficult to understand is actually notsurprising given that, as it is widely accepted, for most (if not all) models both in theKPZ class and in RMT, the GOE/flat case is considerably more difficult to analyze thanthe GUE/curved one. This is because many aspects of the integrability of these modelswhich are present in the latter case, and lead to relatively simple exact formulas, are lostin the former one. It should be noted moreover that, in a certain sense, the GOE/flatconnection is necessarily more tenuous than the GUE/curved one. In fact, if one considersthe GOE version of Dyson Brownian motion then it is natural to expect (as conjecturedin [BFPS07]), by analogy with the GUE case, that the top path would converge, underappropriate scaling, to the Airy process, which is the analog of the Airy process formodels in the KPZ class with flat initial data. Nevertheless, [BFP08] provided convincingnumerical evidence showing that this is not the case.The main goal of this article is to provide an explanation of how the GOE/flat linkarises. We will achieve this by considering the model of non-intersecting Brownian bridgesmentioned above but focusing now on a different quantity, namely the maximal heightattained by the top path. Our main result will show that the distribution of the maximalheight coincides with that of the largest singular value of a large rectangular matrix withGaussian entries, or in other words, with the square root of the largest eigenvalue of amatrix from the Laguerre Orthogonal Ensemble, i.e. a real Wishart matrix. We remarkthat this identity will be established at the pre-asymptotic level (that is, for a finite numberof paths and for a finite matrix), which is interesting in itself as we will explain in Section1.3. The connection with the GOE is established through the known RMT fact that thesquare root of the top eigenvalue of a real Wishart matrix converges under the right scalingto a Tracy-Widom GOE random variable. The way in which this result fits into the contextof the KPZ universality class with flat initial data can be understood in terms of certainvariational problems, and will be explained in Section 1.5.In the next two subsections we will change a bit our perspective to focus in more detailon the model of non-intersecting Brownian bridges, as well as on the Airy process and onsome previous results which relate it to the Tracy-Widom GOE distribution.1.2. Non-intersecting Brownian bridges.
The model of non-intersecting Brownianbridges corresponds to considering a collection of N Brownian bridges ( B ( t ) , . . . , B N ( t )),all starting from zero at time t = 0 and ending at zero at time t = 1, and conditioningthem (in the sense of Doob) to not intersect in the region t ∈ (0 , ON-INTERSECTING BROWNIAN BRIDGES AND THE LAGUERRE ORTHOGONAL ENSEMBLE 3 that the paths are ordered so that B ( t ) < · · · < B N ( t ) for t ∈ (0 , watermelons withouta wall ) and variants of it have been studied extensively in the last decade, see for instance[TW04; AM05; BS07; Fei08; KIK08; DKZ11; FV12; Lie12; Joh13] among many others. Themodel can be thought of as a limit of non-intersecting random walks, which in the physicsliterature are known as vicious walkers , and were introduced by Fisher [Fis84] (under anadditional conditioning on the walks staying positive) as a model for wetting and melting.The interest in studying systems of non-intersecting paths, both in the statistical physicsand probability literatures, is due in large part to their intimate connection with RMT andthe KPZ universality class. As an example, it has been shown that as the number ofpaths N → ∞ , and under proper scaling, several variants of these models converge to universal processes, such as the sine, Airy, Pearcey and tacnode processes. Universal heremeans that the same limiting processes arise for a wide class of other models (for more onthis aspect see [Joh13; AFM10] and references therein). The first two of these universalprocesses also arise naturally in RMT. For instance, and as we already mentioned, for fixed t ∈ (0 ,
1) the distribution of ( B ( t ) , . . . , B N ( t )) coincides (modulo some scaling) with thatof the eigenvalues of a random matrix drawn from the Gaussian Unitary Ensemble (GUE)and converge, under suitable scaling at the edge of the GUE spectrum, to the Airy pointprocess.A particular aspect which has been subject of intense research has been the study of themaximal height attained by the highest path of a collection of non-intersecting paths. Inthe physics literature, [SMCRF08; Fei09; RS10; RS11] obtain various expressions for thedistribution of this maximum. As in the case of the limiting processes mentioned above,their main motivation lies in the computation of the asymptotic distribution in the N → ∞ limit, which for many different models is conjectured to be given by the Tracy-Widom GOEdistribution. This was achieved in the physics literature using non-rigorous methods (seefor instance [FMS11], which further establishes connections with Yang-Mills theory). Forthe case of non-intersecting Brownian motions on the half-line (with either absorbing orreflecting boundary condition at zero) this was rigorously proved by Liechty [Lie12].In this paper we will focus on the distribution of the maximal height of a finite numberof non-intersecting Brownian bridges. More precisely, for fixed N we are interested in thedistribution of the random variable M N = max t ∈ [0 , B N ( t ) . (1.1)As we already mentioned, under proper centering and scaling M N should converge indistribution as N → ∞ to a Tracy-Widom GOE random variable. The question in whichwe will be interested here is whether there is a finite N version of this result. Rathersurprisingly, and as we mentioned already above, we will find that the answer is yes. Butbefore stating the result, and in order to provide additional motivation (and in particularexplain why this is in itself a natural question), let us discuss in some detail the GOE resultin the N → ∞ regime.1.3. The Airy process and GOE. The Airy process A was introduced by Pr¨ahoferand Spohn [PS02] in the study of the scaling limit of a discrete polynuclear growth (PNG)model. It is expected to govern the asymptotic spatial fluctuations in a wide variety ofrandom growth models on a one-dimensional substrate with curved initial conditions, andthe point-to-point free energies of directed random polymers in 1 + 1 dimensions. For itsdefinition and a detailed discussion of its properties and relevance we refer the reader to[QR14]; let us just mention that the Airy process is non-Markovian and stationary, withmarginal distributions given by the Tracy-Widom GUE distribution.The Airy process is also known to arise in the setting of (geometric) last passage per-colation . Here one considers a family (cid:8) w ( i, j ) } i,j ∈ Z + of independent geometric random ON-INTERSECTING BROWNIAN BRIDGES AND THE LAGUERRE ORTHOGONAL ENSEMBLE 4 variables with parameter q (i.e. P ( w ( i, j ) = k ) = q (1 − q ) k for k ≥
0) and let Π N bethe collection of up-right paths of length N , that is, paths π = ( π , . . . , π n ) such that π i − π i − ∈ { (1 , , (0 , } . The point-to-point last passage time is defined, for M, N ∈ Z + ,by L point ( M, N ) = max π ∈ Π M + N :(0 , → ( M,N ) M + N X i =0 w ( π i ) , where the maximum is taken over all up-right paths connecting the origin to ( M, N ).Johansson [Joh00] proved that there are explicit constants c and c , depending only on q ,such that P (cid:0) L point ( N, N ) ≤ c N + c N / r (cid:1) −→ F GUE ( r ) as N → ∞ , with F GUE the Tracy-Widom GUE distribution. Next one defines a process t H N ( t ) by linearly interpolatingthe values given by scaling L point ( N, M ) through the relation L point ( N + k, N − k ) = c N + c N / H n ( c N − / k ) , (1.2)where c is another explicit constant which depends only on q . Johansson [Joh03] went onto show that H N ( t ) −→ A ( t ) − t (1.3)in distribution, in the topology of uniform convergence on compact sets. On the otherhand one can define the point-to-line last passage time by L line ( N ) = max k = − N,...,N L point ( N + k, N − k ) . (1.4)From the definition and Johansson’s result (1.3) it follows that c − N − / [ L line ( N ) − c N ] −→ sup t ∈ R {A ( t ) − t } in distribution. But it was known separately [BR01] that the quantity on the left convergesin distribution to a Tracy-Widom GOE random variable, from which Johansson deducedin [Joh03] the remarkable fact that P (cid:18) max t ∈ R ( A ( t ) − t ) ≤ r (cid:19) = F GOE (4 / r ) , (1.5)where F GOE denotes the Tracy-Widom GOE distribution (an explicit formula for F GOE will be given in Section 1.4). A more direct proof of (1.5) was given in [CQR13], basedon formulas for the hitting probabilities for the Airy process. This method has led toseveral other results about the Airy and related processes (see e.g. [MFQR13] or thereview [QR14]) and it is the one we will use in this paper in the context of non-intersectingBrownian bridges.The relation between the Airy process and the study of M N lies in the fact that, suitablyrescaled, the top path of a collection of non-intersecting Brownian bridges converges to theAiry process minus a parabola:2 N / (cid:16) B N (cid:0) (1 + N − / t ) (cid:1) − √ N (cid:17) −→ A ( t ) − t (1.6)in the sense of convergence in distribution in the topology of uniform convergence oncompact sets. This result is well-known in the sense of convergence of finite-dimensionaldistributions; the stronger convergence stated here was proved in [CH14]. In view of thisresult, a similar argument as the one leading to (1.5) together with (1.5) itself gives thefollowing: Theorem 1.1. lim N →∞ P (cid:16) N / (cid:0) max t ∈ [0 , B N ( t ) − √ N (cid:1) ≤ r (cid:17) = F GOE (4 / r ) . (1.7)It is this version of Johansson’s result (1.5) which provided the original motivation for ourpaper. We remark that, as a by-product of our results, we obtain a more direct derivationof (1.7). ON-INTERSECTING BROWNIAN BRIDGES AND THE LAGUERRE ORTHOGONAL ENSEMBLE 5
GOE and LOE.
In this section we will quickly introduce the two ensembles of ran-dom matrices which are most relevant to our results. The first one is the
Gaussian Or-thogonal Ensemble (GOE) . Let N ( a, b ) denote a Gaussian random variable with mean a and variance b . An N × N GOE matrix is a symmetric matrix A such that A ij = N (0 , i > j and A ii = N (0 , N × N real symmetric matrices A with density C N e − tr( A ) for some normalization constant C N . The joint density of theeigenvalues ( λ , . . . , λ N ) of a GOE matrix can be explicitly computed, and is given by1 Z N N Y i =1 e − λ i Y ≤ i
Wishart matrix ). Inapplications to statistics, one thinks of the rows of X as containing n independent samplesof an N -variate Gaussian population (with covariance matrix given by the identity), so that n M corresponds to the sample covariance matrix. The joint density of the eigenvalues of M is also explicit in this case, and is given by1 Z N Y ≤ i 2. The weights λ ai e − λ i / appearing inthis case are the ones associated to the (generalized) Laguerre polynomials, which explainsthe name of this family of random matrices. By the Marˇcenko-Pastur law [MP67] theeigenvalues of M are concentrated on the interval [0 , N ]. Under our scaling, if a =( n − N − / N ) then the fluctuations at the soft edge N are of order N / , and have the same limiting distribution as in the GOE case [Joh01]:denoting by λ LOE ( N ) the largest eigenvalue of the LOE matrix, we havelim N →∞ P (cid:0) λ LOE ( N ) ≤ N + 2 / N / r (cid:1) = F GOE ( r ) . (1.11)The scaling at the hard edge at the origin is different and gives rise to a different limitdistribution, but we will not need it in this paper.In all that follows we will be interested exclusively in the case a = 0.1.5. Main results. We are ready now to state the main result of this paper. Let M bean LOE matrix with a = 0, that is, M = X T X with X an ( N + 1) × N matrix withindependent N (0 , 1) entries. For this choice of a we will denote by F LOE ,N the distributionfunction of the largest eigenvalue of M , F LOE ,N ( r ) = P (cid:0) λ LOE ( N ) ≤ r ) . (1.12)Recall the definition in (1.1) of M N as the maximum height of a collection of N non-intersecting Brownian bridges. Theorem 1.2. Let B ( t ) < · · · < B N ( t ) be a collection of non-intersecting Brownianbridges as above. Then for all r ≥ we have P (cid:18) max t ∈ [0 , √ B N ( t ) ≤ r (cid:19) = F LOE ,N (2 r ) . (1.13) In other words, M N is distributed as the largest eigenvalue of an LOE matrix or, alter-natively, M N is distributed as the largest singular value of the ( N + 1) × N matrix X introduced above. Let us quickly verify that the scaling in this result is consistent with the one in Theorem1.1 and (1.11). Theorem 1.2 says that M N (d) = p λ LOE ( N ) / 4. By (1.11), this implies that M N = q N + 2 − / N / ζ GOE + o ( N / ) = √ N + 2 − / N − / ζ GOE + o ( N − / ) , where ζ GOE is a Tracy-Widom GOE random variable. In other words, 2 N / (cid:0) M N −√ N (cid:1) =4 − / ζ GOE + o (1), which is exactly the content of Theorem 1.1. In particular, Theorem 1.1follows as a corollary of (1.11) and (1.13).We take a brief pause now and go back to an issue left open at the end of Section 1.1,which is the question of why M N should be interpreted as a flat initial data object in theKPZ universality class. In a way, the convergence of M N a Tracy-Widom GOE randomvariable should be taken, in itself, as enough evidence of this fact. But the connectiongoes a bit further, and can be understood in terms of certain variational formulas. Forexample, in the context of last passage percolation (LPP), the point-to-line last passagetimes (1.4) leading to GOE fluctuations are defined in terms of the maximum of point-to-point last passage times (1.2), which in turn lead to GUE fluctuations. The parallelwith (1.13) is direct. The exact same relationship can be established at the level of manyother polymer models (of which LPP is a zero-temperature version), and at the level ofthe totally asymmetric exclusion process (which can be mapped to LPP).This straightforward way of expressing flat initial data quantities in terms of their curvedinitial data analogues is not as explicit in the case of some other models, such as the partiallyasymmetric exclusion process, which have less (or at least a more complicated) algebraicstructure, but it is interesting to note that it does hold at the level of another of the ON-INTERSECTING BROWNIAN BRIDGES AND THE LAGUERRE ORTHOGONAL ENSEMBLE 7 most important members of the KPZ universality class, the KPZ equation. Without goinginto much detail, the KPZ equation can be understood by studying the stochastic heatequation (SHE), which is linear. It turns out that the flat initial data for the KPZ equationcorresponds to starting the SHE with initial condition Z ≡ 1, and thus by linearity theflat solution can be obtained by convolving the constant function 1 with the solution ofthe SHE starting with Z = δ , which corresponds to curved initial data. Note that therelationship in this last case is not written directly in terms of a variational problem asdescribed before, but one can check that (at least conjecturally, by essentially appealingto a version of Laplace’s method) one recovers a variational problem as time t → ∞ . Formuch more on this see [QR14; CQR15; QR15].Coming back to the description of our main results, Theorem 1.2 is equivalent to a state-ment about the probability that the top path of Dyson Brownian motion hits an hyperboliccosine barrier, and it is that version of the result which we will prove. Consider an N × N random matrix drawn from the Gaussian Unitary Ensemble , that is, a (complex-valued)Hermitian matrix A such that A ij = N (0 , / 4) + i N (0 , / 4) for i > j and A ii = N (0 , / A evolve by letting each Gaussian variable in the construction diffuse accordingto independent copies of the Ornstein-Uhlenbeck process X t defined as the solution of theSDE dX t = − X t dt + σ dW t , where W t is a standard Brownian motion and σ = √ for off-diagonal entries and σ = 1on the diagonal. We write the eigenvalues of this matrix at time t as ( λ ( t ) , . . . , λ N ( t )),with λ i ( t ) increasing with i . This eigenvalue diffusion is known as the stationary (GUE)Dyson Brownian motion , and it defines an ensemble of almost surely non-intersecting curvesindexed by R . It can alternatively be written as the solution of a certain N -dimensionalSDE, and it is not hard to check that it is stationary, with marginals at any time t givenby the eigenvalue distribution of an N × N GUE matrix. Theorem 1.3. Let ( λ ( t ) , . . . , λ N ( t )) be the stationary Dyson Brownian motion definedabove and let F LOE ,N be defined as in (1.12) . Then P ( λ N ( t ) ≤ r cosh( t ) ∀ t ∈ R ) = F LOE ,N (2 r ) . (1.14)The equivalence between the two results is due to the fact that non-intersecting Brownianbridges can be mapped into the stationary Dyson Brownian motion in such a way that theprobabilities on the left-hand side of (1.13) and (1.14) coincide. We will explain this inmore detail in Section 2.The proof of Theorem 1.3 has two steps. The first one consists in obtaining an ex-plicit formula for the probability on the left-hand side of (1.14). By the mapping betweennon-intersecting Brownian bridges and the stationary Dyson Brownian motion alluded toabove, this is equivalent to finding a formula for the distribution of M N . As we alreadymentioned, there are formulas in the literature for the distribution of the maximal heightof several models related to non-intersecting Brownian bridges, which can be obtainedthrough a direct application of the Karlin-McGregor/Lindstr¨om-Gessel-Viennot formula[KM59; Lin73; GV85]. For completeness, let us state the formula in the case of M N (see ON-INTERSECTING BROWNIAN BRIDGES AND THE LAGUERRE ORTHOGONAL ENSEMBLE 8 [SMCRF08]) : P ( M N ≤ r ) = 2 N (2 π ) N/ r N Q Nj =1 j ! Z [0 , ∞ ) N d~y e − P j y j / r (cid:18) det h y j − i cos( y i + jπ ) i Ni,j =1 (cid:19) . (1.15)By using the Cauchy-Binet identity, the right-hand side can be turned into a single N × N determinant with entries involving Hermite polynomials, see (102)–(103) in [RS11]. Theresulting formula is reminiscent of some of the formulas we will obtain below, see (3.6)together with (1.17), but it is not clear how to use it directly to obtain a proof of Theorem1.1 (nor of (1.13)). Moreover, as we will explain next, while the structure of the Fredholmdeterminant formula for the distribution of M N which we will obtain in this paper (seeProposition 1.4) makes very apparent a connection with Johansson’s result (1.5) — thiswas an important clue for us in the discovery of (1.13) — from the formula appearing in[RS11] such a connection is not at all clear. It is worth mentioning that in the case ofBrownian excursions, for which the analog of (1.15) turns out to be slightly simpler, theanalog of Theorem 1.1 (with the same limit) was proved by Liechty [Lie12] by appealingto a Riemann-Hilbert analysis of a certain system of discrete orthogonal polynomials.Here we follow a different strategy, leading to an arguably simpler formula which alsohas some intrinsic interest. Working at the level of Dyson Brownian motion, we appeal toa result of [BCR15] in order to obtain an expression for P ( λ N ( t ) ≤ r cosh( t ) ∀ t ∈ [ − L, L ]),for fixed L > L → ∞ in the resulting formula leadsto the following result. Let ϕ n be the harmonic oscillator functions (which we will referto as Hermite functions ), defined by ϕ n ( x ) = e − x / p n ( x ), with p n the n -th normalizedHermite polynomial (i.e., so that k ϕ k = 1), and define the Hermite kernel as K Herm ,N ( x, y ) = N − X n =0 ϕ n ( x ) ϕ n ( y ) . (1.16)We introduce also the reflection operator ̺ r on L ( R ), defined by ̺ r f ( x ) = f (2 r − x ) . Proposition 1.4. For any r ≥ , P ( λ N ( t ) ≤ r cosh( t ) ∀ t ∈ R ) = det( I − K Herm ,N ̺ r K Herm ,N ) L ( R ) . (1.17) The same formula holds for P (cid:0) max t ∈ [0 , √ B N ( t ) ≤ r (cid:1) . This result will be proved in Section 2.The expression on the right-hand side of (1.17) is a close analog of the formula for F GOE appearing in (1.9). To see this we introduce the Airy kernel , defined as K Ai ( x, y ) = Z ∞ dλ Ai( x + λ ) Ai( y + λ ) . This kernel is closely related to GUE, as it is the limiting correlation kernel of the GUEeigenvalues near the edge of the spectrum. It is related to the Tracy-Widom GOE distri-bution because of the identity R ∞−∞ dλ Ai( a + λ ) Ai( b − λ ) = 2 − / Ai(2 − / ( a + b )), which(since K Ai = B P B , with B defined in (1.10)) implies that K Ai ̺ r K Ai = B P e B r P B (1.18) This formula was derived in [SMCRF08] using path-integral techniques. Although we are not awareof a derivation in the literature based on the Karlin-McGregor formula, for the case of non-intersectingBrownian excursions (corresponding to imposing an absorbing boundary at zero) the analog formula, alsoderived in [SMCRF08], was rederived in this way in [KT07]. ON-INTERSECTING BROWNIAN BRIDGES AND THE LAGUERRE ORTHOGONAL ENSEMBLE 9 with e B r ( x, y ) = 2 − / Ai(2 − / ( x + y + 2 r )). Since B = I (this identity is related to the factthat the family of functions (cid:8) Ai( x + λ ) (cid:9) λ ∈ R constitutes a generalized eigenbasis of L ( R )),the cyclic property of the determinant and (1.9) allow us to conclude that F GOE (4 / r ) = det( I − K Ai ̺ r K Ai ) L ( R ) . (1.19)We point out that there does not appear to be a direct analog of (1.18) for K Herm ,N (although one can obtain explicit formulas for K Herm ,N ̺ r K Herm ,N involving no integrals,see for instance (A.4) and (A.6)).We can actually push the analogy between (1.17) and (1.19) a bit further and use it toprovide a simple proof of Theorem 1.1. Indeed, a simple scaling argument on the right-hand side of (1.17) leads to P (cid:16) N / ( M N − √ N ) ≤ r (cid:17) = det (cid:16) I − e K Herm ,N ̺ r e K Herm ,N (cid:17) with e K Herm ,N ( x, y ) = κ N K Herm ,N ( κ N x + √ N , κ N y + √ N ), where κ N = 2 − / N − / . On theother hand, it is well known that e K Herm ,N converges to K Ai as N → ∞ , where the conver-gence is strong enough to imply the convergence of the associated Fredholm determinants.In view of (1.19), and omitting the details, this implies Theorem 1.1.A related observation is that, in a sense, Proposition 1.4 serves as a generalization ofJohansson’s result for the Airy process, (1.5). In fact, the scaling argument used in thelast paragraph leads to det (cid:0) I − e K Herm ,N ̺ r e K Herm ,N (cid:1) = P (cid:0) λ N ( t ) ≤ ( κ N r + √ N ) cosh( t ) ∀ t ∈ R (cid:1) . On the other hand, it is known that e λ N ( t ) = κ − N ( λ N ( N − / t ) − √ N ) convergesto A ( t ) (this is just a restatement of (1.6) in view of the mapping between the twomodels). If we knew that the convergence is strong enough to imply the convergence of P (cid:16)e λ N ( t ) ≤ a ∀ t ∈ R (cid:17) with some control on a , then (1.5) would follow, because by theargument sketched in the last paragraph the determinant would go to F GOE (4 / r ), while κ − N (cid:2) ( κ N r + √ N ) cosh( N − / t ) − √ N (cid:3) = r + t + O ( N − / ).The second step in the proof of Theorem 1.3 consists in showing that the right-handside of (1.17), i.e. det( I − K Herm ,N ̺ r K Herm ,N ) L ( R ) , equals F LOE ,N (2 r ). This is proved inSection 3. We remark that, together with the preceding discussion, this identity providesan alternative proof of the result of [Joh01] in the case a = 0.2. Hitting probabilities for Dyson Brownian motion Recall the stationary Dyson Brownian motion introduced in Section 1.5. As we men-tioned, this model is intimately related to non-intersecting Brownian bridges. The basicrelation is that if one considers the non-stationary version of Dyson Brownian motion(where the Gaussian variables making up the entries of a GUE matrix evolve according toa plain Brownian motion), then the dynamics of the eigenvalues of this evolving matrixcoincide with those of a collection of Brownian motions conditioned to never intersect.The analogous relation in our setting goes through a time-change, and is given explicitlyin [TW07, Section 2.2.2]: if B ( t ) < · · · < B N ( t ), t ∈ [0 , λ ( t ) < · · · < λ N ( t ), t ∈ R , are defined as a stationary Dyson Brownianmotion, then (cid:0) B i ( t ) (cid:1) i =1 ,...,N (d) = (cid:16)p t (1 − t ) λ i ( log( t/ (1 − t ))) (cid:17) i =1 ,...,N as processes defined for t ∈ [0 , t e s / (1 + e s ) leads tomax t ∈ [0 , B N ( t ) (d) = max t ∈ [0 , p t (1 − t ) λ N ( log( t/ (1 − t ))) = sup s ∈ R λ N ( s ) √ s ) , ON-INTERSECTING BROWNIAN BRIDGES AND THE LAGUERRE ORTHOGONAL ENSEMBLE 10 which shows that Theorems 1.2 and 1.3 are equivalent . The rest of this section will thusbe devoted to computing P (cid:0) λ N ( t ) ≤ r cosh( t ) ∀ t ∈ R (cid:1) .2.1. Path-integral kernel. The finite-dimensional distributions of the stationary (GUE)Dyson Brownian motion are classically expressed through a Fredholm determinant in termsof the extended Hermite kernel K extHerm ,N K extHerm ,N ( s, x ; t, y ) = (P N − n =0 e n ( s − t ) ϕ n ( x ) ϕ n ( y ) if s ≥ t, − P ∞ n = N e n ( s − t ) ϕ n ( x ) ϕ n ( y ) if s < t where ϕ n ( x ) = e − x / p n ( x ) and p n is the n -th normalized Hermite polynomial. Explicitly,if −∞ < t < t < . . . < t n < ∞ and r , . . . , r n ∈ R , then P (cid:0) λ N ( t j ) ≤ r j , j = 1 , . . . , n (cid:1) = det (cid:0) I − f K extHerm ,N f (cid:1) L ( { t ,...,t n }× R ) , (2.1)where we have counting measure on { t , . . . , t n } and Lebesgue measure on R , and f isdefined on { t , . . . , t n } × R by f( t j , x ) = x ∈ ( r j , ∞ ) (for more details see [TW07]).The first step in our derivation is to obtain a formula for the probability that λ N ( t )stays below r cosh( t ) on a finite interval [ − L, L ]. To that end, we need to consider a finitemesh t < · · · < t n of [ − L, L ], let r i = r cosh( t i ), and then take a limit of the correspondingprobabilities as given in (2.1) as the mesh size goes to zero. But these probabilities becomeincreasingly cumbersome as n increases, due to the n -dependence in the L space on whichthe operators act. The way to overcome this problem is to first manipulate the right-handside of (2.1) into a Fredholm determinant of some other kernel acting on L ( R ). Such aformula was first stated, in the context of the Airy process, in [PS02] (see also [PS11]),and the resulting formula was used in [CQR13] to obtain a formula for the probabilitythat A ( t ) stays below a given function g ( t ) on a finite interval. Later on, the procedurethat converts the extended kernel formula into a formula with a Fredholm determinantacting on L ( R ) was generalized in [BCR15] (see also [QR13]) to a wide class of processesthat includes the stationary Dyson Brownian motion, and from the resulting formula theyobtained a continuum statistics formula for Dyson Brownian motion in a similar way as in[CQR13]. In order to state the formula we need to introduce some operators.First, recall the definition of the Hermite kernel K Herm ,N , given in (1.16), and note that K extHerm ,N ( t, x ; t, y ) = K Herm ,N ( x, y ) for any t . Next we introduce the differential operator D = − (∆ − x + 1)(∆ is the Laplacian on R ). D and K Herm ,N are related: D ϕ n = nϕ n , so that K Herm ,N is the projection operator onto the space span { ϕ , . . . , ϕ N − } associated to the first N eigenvalues of D . In particular, even though e t D is well-defined in general only for t ≤ e t D K Herm ,N is well defined for all t , and its integral kernel is given by e t D K Herm ,N ( x, y ) = N − X n =0 e tn ϕ n ( x ) ϕ n ( y ) . (2.2)Now fix ℓ < ℓ and consider a function g ∈ H ([ ℓ , ℓ ]) (i.e. both g and its derivativeare in L ( R )). We introduce an operator Θ g [ ℓ ,ℓ ] acting on L ( R ) as follows: Θ g [ ℓ ,ℓ ] f ( x ) = A similar argument, together with the fact [TW04] that √ N / ( λ N ( N − / t ) − √ N ) converges to A ( t ) in the sense of finite-dimensional distributions, provides a justification for a version of (1.6) in thisweaker sense. ON-INTERSECTING BROWNIAN BRIDGES AND THE LAGUERRE ORTHOGONAL ENSEMBLE 11 u ( ℓ , x ), where u ( ℓ , · ) is the solution at time ℓ of the boundary value problem ∂ t u + D u = 0 for x < g ( t ) , t ∈ ( ℓ , ℓ ) u ( ℓ , x ) = f ( x ) x Let α = e ℓ , β = e ℓ , and denote by Θ g [ ℓ ,ℓ ] ( x, y ) the integral kernelof Θ g [ ℓ ,ℓ ] . Then Θ g [ ℓ ,ℓ ] ( x, y ) = e ( y − x )+ ℓ e − ( e ℓ x − e ℓ y ) / (4( β − α )) p π ( β − α ) × P ˆ b ( α )= e ℓ x, ˆ b ( β )= e ℓ y (cid:16) ˆ b ( t ) ≤ √ t g (cid:0) log(4 t ) (cid:1) ∀ t ∈ [ α, β ] (cid:17) , (2.5) where the probability is computed with respect to a Brownian bridge ˆ b ( t ) from e ℓ x at time α to e ℓ y at time β and with diffusion coefficient 2.Proof. Let u ( t, x ) be the solution to the boundary value PDE (2.3) and consider the trans-formation u ( t, x ) = e x / t v ( τ, z ) with τ = e t , z = e t x . It is not hard to check then that v ( τ, z ) satisfies the following boundary value problem associated to the heat equation: ∂ τ v − ∂ z v = 0 for z < √ τ g (cid:0) log(4 τ ) / (cid:1) , τ ∈ ( α, β ) v ( α, z ) = e − z / (8 α ) − log(4 α ) / f (cid:0) z/ √ α (cid:1) { z< √ α g (log(4 α ) / } v ( τ, z ) = 0 for z > √ τ g (cid:0) log(4 τ ) / (cid:1) , where α = e ℓ , β = e ℓ . This boundary value PDE can be solved explicitly in termsof Brownian motion by using the Feynman-Kac formula: letting ˆ b ( s ) denote a Brownianbridge with diffusion coefficient 2, we have v ( β, z ) = Z √ α g (log(4 α ) / −∞ dx e − x / (8 α ) − log(4 α ) / f (cid:16) x √ α (cid:17) e − ( x − z ) / (4( β − α )) p π ( β − α ) · P ˆ b ( α )= x, ˆ b ( β )= z (cid:16) ˆ b ( τ ) ≤ √ τ g (cid:0) log(4 τ ) / (cid:1) on [ α, β ] (cid:17) . Now using the fact that u ( ℓ , y ) = e y / ℓ v ( e ℓ / , e ℓ y ) and recalling that α = e ℓ weimmediately obtain u ( ℓ , y ) = Z e ℓ g ( ℓ ) −∞ dx e y − e − ℓ x + ℓ − ℓ e − ( x − e ℓ y ) / (4( β − α )) p π ( β − α ) f ( e − ℓ x ) · P ˆ b ( α )= x, ˆ b ( β )= e ℓ y (cid:16) ˆ b ( τ ) ≤ √ τ g (cid:0) log(4 τ ) / (cid:1) on [ α, β ] (cid:17) . Changing variables x e ℓ x in the integral, the formula for Θ g [ ℓ ,ℓ ] ( x, y ) readily follows. (cid:3) ON-INTERSECTING BROWNIAN BRIDGES AND THE LAGUERRE ORTHOGONAL ENSEMBLE 12 Hyperbolic cosine barrier. Observe now the key fact that, in our case g ( t ) = r cosh( t ), the probability appearing in (2.5) is reduced to the probability of a Brownianbridge staying below the linear function 2 rt + r , which can be computed explicitly. In fact,assuming that x ≤ e − ℓ (2 rα + r/ 2) = r cosh( ℓ ) and y ≤ e − ℓ (2 rβ + r/ 2) = r cosh( ℓ ) (notethat the probability below is obviously zero if either condition is not met), the Cameron-Martin-Girsanov formula yields P ˆ b ( α )= e ℓ x ˆ b ( β )= e ℓ y (cid:16) ˆ b ( t ) ≤ rt + r on [ α, β ] (cid:17) = 1 − e − r ( e ℓ y − e ℓ x )+ r ( β − α ) e − ( eℓ y − rβ − eℓ x +2 rα )24( β − α ) e − ( eℓ y − eℓ x )24( β − α ) P ˆ b ( α )= e ℓ x − rα ˆ b ( β )= e ℓ y − rβ (cid:18) max t ∈ [ α,β ] ˆ b ( t ) > r (cid:19) . The last probability can be computed easily using the reflection principle, and it equals e − ( e ℓ x − rα − r/ e ℓ y − rβ − r/ / ( β − α ) (see for instance page 67 in [BS02]). Putting this backin our formula (2.5) for Θ g ( t )[ ℓ ,ℓ ] with g ( t ) = r cosh( t ), which for simplicity we will denotefrom now on as Θ ( r )[ ℓ ,ℓ ] , givesΘ ( r )[ ℓ ,ℓ ] ( x, y ) = x ≤ r cosh( ℓ ) , y ≤ r cosh( ℓ ) e ( y − x )+ ℓ p π ( β − α ) × (cid:16) e − ( e ℓ x − e ℓ y ) / (4( β − α )) − e − r ( e ℓ y − e ℓ x )+ r ( β − α ) − ( e ℓ x + e ℓ y − r ( α + β ) − r ) / (4( β − α )) (cid:17) . (2.6)The above expression splits into two terms. Note that if we disregard the indicator function,then by the above derivation the first term corresponds to the solution of (2.3) with g = ∞ ,and thus it is nothing but e − ( ℓ − ℓ ) D ( x, y ). As a consequence, we deduce thatΘ ( r )[ ℓ ,ℓ ] = ¯ P r cosh( ℓ ) (cid:16) e − ( ℓ − ℓ ) D − R ( r )[ ℓ ,ℓ ] (cid:17) ¯ P r cosh( ℓ ) , (2.7)where ¯ P a f ( x ) = ( I − P a ) f ( x ) = f ( x ) x ≤ a and R ( r )[ ℓ ,ℓ ] is the reflection term R ( r )[ ℓ ,ℓ ] ( x, y ) = √ π ( β − α ) e ( y − x )+ ℓ − r ( e ℓ y − e ℓ x )+ r ( β − α ) − ( e ℓ x + e ℓ y − r ( α + β ) − r ) / (4( β − α )) (2.8)and, we recall, α = e ℓ , β = e ℓ .Now we set − ℓ = ℓ = L , so that by Proposition 2.1 we have P ( λ N ( t ) ≤ r cosh( t ) ∀ t ∈ R ) = lim L →∞ det (cid:16) I − K Herm ,N + Θ ( r )[ − L,L ] e L D K Herm ,N (cid:17) . Using now the cyclic property of the Fredholm determinant and the identities e L D K Herm ,N =( e L D K Herm ,N ) and e − L D K Herm ,N e L D K Herm ,N = e L D K Herm ,N e − L D K Herm ,N = K Herm ,N (whichfollow directly from (2.2) and the orthonormality of the ϕ n ’s) we may rewrite the last iden-tity as P ( λ N ( t ) ≤ r cosh( t ) ∀ t ∈ R ) = lim L →∞ det (cid:16) I − K Herm ,N + e L D K Herm ,N Θ ( r )[ − L,L ] e L D K Herm ,N (cid:17) . (2.9)Note that s Θ ( r )[ − L,s ] is a semigroup, so that Θ ( r )[ − L,L ] = Θ ( r )[ − L, Θ ( r )[0 ,L ] , and thus in viewof (2.7) we may writeΘ ( r )[ − L,L ] = ¯ P r cosh( L ) (cid:0) e − L D − R ( r )[ − L, (cid:1) ¯ P r (cid:0) e − L D − R ( r )[0 ,L ] (cid:1) ¯ P r cosh( L ) . Following [CQR13], we decompose Θ ( r )[ − L,L ] in the following way:Θ ( r )[ − L,L ] = (cid:0) e − L D − R ( r )[ − L, (cid:1) ¯ P r (cid:0) e − L D − R ( r )[0 ,L ] (cid:1) − Ω ( r ) L , (2.10) ON-INTERSECTING BROWNIAN BRIDGES AND THE LAGUERRE ORTHOGONAL ENSEMBLE 13 whereΩ ( r ) L = (cid:0) e − L D − R ( r )[ − L, (cid:1) ¯ P r (cid:0) e − L D − R ( r )[0 ,L ] (cid:1) − ¯ P r cosh( L ) (cid:0) e − L D − R ( r )[ − L, (cid:1) ¯ P r (cid:0) e − L D − R ( r )[0 ,L ] (cid:1) ¯ P r cosh( L ) . (2.11)The idea is that Ω ( r ) L is an error term which goes to 0 as L → ∞ . This is the content ofthe next result, whose proof we defer to Appendix B: Lemma 2.3. Assume r > . Then e Ω ( r ) L := e L D K Herm ,N Ω ( r ) L e L D K Herm ,N −−−−→ L →∞ in tracenorm. Since the mapping A det( I + A ) is continuous with respect to the trace norm, thelemma together with (2.9) and (2.10) show that ifΛ := lim L →∞ h K Herm ,N − e L D K Herm ,N (cid:0) e − L D − R ( r )[ − L, (cid:1) ¯ P r (cid:0) e − L D − R ( r )[0 ,L ] (cid:1) e L D K Herm ,N i (2.12)exists in the trace class topology, then P ( λ N ( t ) ≤ r cosh( t ) ∀ t ∈ R ) = det (cid:0) I − Λ (cid:1) . (2.13)But, as we will see next, the operator inside the brackets in (2.12) in fact does not dependon L (the analogous property was proved in [CQR13] in the setting of the Airy process).The key step is the following result: Lemma 2.4. For all L > , e L D K Herm ,N R ( r )[ − L, = K Herm ,N ̺ r and R ( r )[0 ,L ] e L D K Herm ,N = ̺ r K Herm ,N , where ̺ r is the reflection operator ̺ r f ( x ) = f (2 r − x ) . Using this lemma and the fact that e L D K Herm ,N e − L D = K Herm ,N we get K Herm ,N − e L D K Herm ,N (cid:0) e − L D − R ( r )[ − L, (cid:1) ¯ P r (cid:0) e − L D − R ( r )[0 ,L ] (cid:1) e L D K Herm ,N = K Herm ,N − K Herm ,N ( I − ̺ r )¯ P r ( I − ̺ r ) K Herm ,N = K Herm ,N ̺ r K Herm ,N , where the second equality follows from the identity ( I − ̺ r )¯ P r ( I − ̺ r ) = I − ̺ r . In view of(2.12) and (2.13), this yields Proposition 1.4. All that is left to prove then is Lemma 2.4. Proof of Lemma 2.4. We will only provide the proof of the first formula, the second one isvery similar. Using (2.8) we write the kernel of the operator R ( r )[ − L, as R ( r )[ − L, ( x, y ) = √ π (1 − e − L ) e − ax + b y x + c y with a = e − L − e − L ) , b y = e − L (2 r − y )1 − e − L and c y = − (1+ e − L )(2 r − y ) − e − L ) . This formula together with (2.2) and the contour integral representation of the Hermitefunction ϕ n ( x ), ϕ n ( x ) = (2 n n ! √ π ) − / e − x / n !2 πi I dt e tx − t t n +1 (where the contour of integration encircles the origin), gives us e L D K Herm ,N R ( r )[ − L, ( x, y ) = Z R dz N − X n =0 e Ln ϕ n ( x ) ϕ n ( z ) R ( r )[ − L, ( z, y )= √ π (1 − e − L ) N − X n =0 e Ln ϕ n ( x )(2 n n ! √ π ) − / n !2 πi I dt e − t t n +1 Z R dz e − z / tz − az + b y z + c y . ON-INTERSECTING BROWNIAN BRIDGES AND THE LAGUERRE ORTHOGONAL ENSEMBLE 14 The z integral is just a Gaussian integral, and computing it the last expression becomes N − X n =0 e Ln ϕ n ( x )(2 n n ! √ π ) − / n !2 πi I dt e − e − L t +2 e − L t (2 r − y ) − (2 r − y ) / t n +1 = N − X n =0 e Ln ϕ n ( x )(2 n n ! √ π ) − / n !2 πi I dt e − t +2 t (2 r − y ) − (2 r − y ) / t n +1 e Ln , where we have performed the change of variables t t e L . The last integral and its prefac-tor are nothing but ϕ n (2 r − y ), so this yields e L D K Herm ,N R ( r )[ − L, ( x, y ) = K Herm ,N ( x, r − y )as needed. (cid:3) Connection with LOE This section is devoted to the proof of the following result: Proposition 3.1. For r ≥ , det( I − K Herm ,N ̺ r K Herm ,N ) L ( R ) = F LOE ,N (2 r ) . Together with Proposition 1.4, this proposition implies Theorem 1.3.Let us start by introducing an explicit formula for F LOE ,N . To that end, we will utilizethe ensemble ¯ λ (1) < ¯ λ (2) < · · · < ¯ λ ( N ) obtained as the result of superimposing theeigenvalues of two independent copies of our LOE matrices, writing them in increasingorder, and then keeping only the even labelled coordinates (i.e. keeping the largest, 3 rd largest, 5 th largest, and so on). Observe that if λ LOE ( N ) denotes the largest eigenvalue ofan LOE matrix as in Section 1.4, then P ( λ LOE ( N ) ≤ r ) = P (¯ λ ( N ) ≤ r ) . (3.1)The advantage of this representation is that the superimposed ensemble (¯ λ ( i )) i =1 ,...,N is adeterminantal process with a simple correlation kernel e K Lague ,N (see [FR04]). The kernel e K Lague ,N is given as follows. For n ∈ N , introduce the Laguerre function ψ n ( x ) = e − x/ L n ( x ) , (3.2)where L n ( x ) is the n -th normalized Laguerre polynomial (so that k ψ n k = 1), and thendefine the Laguerre kernel as K Lague ,N ( x, y ) = N − X n =0 ψ n ( x ) ψ n ( y ) . Then e K Lague ,N ( x, y ) = − ∂∂x Z y du K Lague ,N ( x, u ) . The determinantal structure of the superimposed ensemble leads directly to a formula forthe distribution of ¯ λ ( N ) (see [Joh03] or [QR14, (1.36)]): P (¯ λ ( N ) ≤ r ) = det (cid:16) I − P r e K Lague ,N P r (cid:17) L ( R ) . (3.3)Observe that e K Lague ,N is a finite rank operator, and thus the last determinant can berepresented as the determinant of a finite matrix. More precisely, if we factor our op-erator as e K Herm ,N = K K with K : ℓ ( { , . . . , N − } ) −→ L ( R ) and K : L ( R ) −→ ℓ ( { , . . . , N − } ) defined by the kernels K ( x, n ) = − ψ ′ n ( x ) and K ( n, y ) = R y du ψ n ( u ), ON-INTERSECTING BROWNIAN BRIDGES AND THE LAGUERRE ORTHOGONAL ENSEMBLE 15 then the cyclic property of the Fredholm determinant implies that the determinant in (3.3)equals det( I − K K ), so thatdet (cid:0) I − P r e K Lague ,N P r (cid:1) L ( R ) = det h δ jk + Z ∞ r dx ψ ′ j ( x ) Z x du ψ k ( u ) i N − j,k =0 = det h δ jk − Z ∞ r dx ψ j ( x ) ψ k ( x ) − ψ j (2 r ) Z r du ψ k ( u ) i N − j,k =0 , where in the second equality we have integrated by parts. Defining now a symmetricmatrix L ∈ R N × N and two column vectors R , R ∈ R N by L jk = Z ∞ r dx ψ j ( x ) ψ k ( x ) , ( R ) j = ψ j (2 r ) and ( R ) j = Z r du ψ j ( u ) , (3.4)for j, k ∈ { , . . . , N − } , we deduce by the last identity, (3.1) and (3.3), that F LOE ,N (2 r ) = det( I − L − R ⊗ R ) . (3.5)Similarly, we have a version of (1.17) in terms of the determinant of a finite matrix (whichcan be obtained by conjugating the kernel inside the Fredholm determinant in (1.17) bythe operator G : L ( R ) −→ ℓ ( { , . . . , N − } ) with kernel G ( n, x ) = ϕ n ( x )):det (cid:0) I − K Herm ,N ̺ r K Herm ,N (cid:1) L ( R ) = det( I − H ) , (3.6)where the symmetric matrix H has entries given by H jk = Z R dx ϕ j ( x ) ϕ k (2 r − x ) . (3.7)Therefore, and in view of (3.5), we see that, in order to prove Proposition 3.1, we have toestablish that det( I − H ) = det( I − L − R ⊗ R ) . (3.8)At this point the main difficulty in proving (3.8) lies in the fact that the two sides ofthe identity are given in terms of objects related to two different families of orthogonalpolynomials, which makes it hard to relate one to the other. So the first step in our proofof the identity consists in replacing the matrix H on the left-hand side by a matrix definedin terms of Laguerre polynomials.To this end, let us introduce the following N × N (real) matrix e H : e H ij = ( − N (cid:0) ψ i + j − N (2 r ) − ψ i + j − N +1 (2 r ) (cid:1) for i, j ∈ { , . . . , N − } . (3.9)Here ψ n is the Laguerre function introduced in (3.2) for n ≥ 0, while we set ψ n ≡ n < 0. Note in particular that e H is zero above the anti-diagonal (i.e. e H ij = 0 if i + j < N − e H is conjugate to H , so that det( I − H ) = det( I − e H ). Moreover, we will see thatthe matrices L and R ⊗ R are also intimately related to e H . In order to state the lemmawe introduce a matrix Q ∈ R N × N and two column vectors u, v ∈ R N by u = ( − N − , v i = ( − i i = 0 , . . . , N − ,Q ij = i < j, − r for i = j, − r for i > j, i, j = 0 , . . . , N − denotes the constant vector with 1 in each entry).Note that the matrices and vectors introduced in this section are always indexed by { , . . . , N − } , and they generally depend on N and r ; we have omitted this dependencefrom the notation for simplicity. As a notational guide, note that while we have used sans-serif fonts to denote operators acting on aHilbert space and their associated kernels, we are using regular fonts to denote (finite) matrices. ON-INTERSECTING BROWNIAN BRIDGES AND THE LAGUERRE ORTHOGONAL ENSEMBLE 16 Lemma 3.2. Let H , e H , L , R , R , Q , u and v be defined as in (3.4) , (3.7) , (3.9) and (3.10) . Then the following properties hold: (i) e H is conjugate to H , i.e. there exists an invertible matrix S ∈ R N × N such that e H = SHS − . (ii) e H = L . (iii) R = e Hu and R = ( I − e H ) v . (iv) ∂∂r e H = Q e H . (v) ∂∂r ( I + e H ) − = ( I − e H ) − e HQ + ( I − e H ) − E ( I + e H ) − , where E = 4 r e Hu ⊗ u . This lemma contains all the key identities which will be needed in the proof of (3.8). Letus thus postpone the proof of the lemma until the end of this section and proceed directlyto the proof of the main result of this section. Proof of Proposition 3.1. As we already explained, all we need to do is prove (3.8). Thestructure of the proof is inspired in that of the proof of (1.9) in [FS05]. Note that bothsides of (3.8) are zero if r = 0 (this is equivalent to the fact that both sides of (3.1) vanishwhen r = 0, which is clear). Therefore we will assume throughout this proof that r > I − H ) and det( I − L − R ⊗ R ) are strictlypositive.We start by using (ii) and (iii) in Lemma 3.2 to rewrite the determinant on the left-handside of (3.8) asdet( I − L − R ⊗ R ) = det (cid:0) I − e H − e Huv T ( I − e H ) (cid:1) = det( I + e H ) det (cid:0) I − ( I + e H ) − e Huv T (cid:1) det( I − e H )= det( I − e H ) det( I + e H ) (cid:0) − h u, ( I + e H ) − e Hv i (cid:1) , where in the third equality we used the fact that ( I + e H ) − e Huv T is rank one. By Lemma3.2(i), we have det( I − H ) = det( I − e H ), and thus (3.8) will follow if we prove thatdet( I − e H ) = det( I + e H ) (cid:0) − h u, ( I + e H ) − e Hv i (cid:1) . (3.11)Note that, by the discussion in the last paragraph, since r > 0, the left-hand side and thetwo factors on the right-hand side are strictly positive.Consider the second factor on the right-hand side of (3.11). Since h u, v i = 0 if N is evenand h u, v i = 2 if N is odd, we can write 1 = h u, v i + ( − N , so that1 − h u, ( I + e H ) − e Hv i = ( − N + h u, v i − h u, ( I + e H ) − e Hv i = ( − N + h u, ( I + e H ) − v i . Taking now logarithm on both sides we see that (3.11) is equivalent tolog det( I − e H ) = log det( I + e H ) + log (cid:0) ( − N + h u, ( I + e H ) − v i (cid:1) . (3.12)We will prove that the derivatives in r of both sides are equal, that is, − Tr (cid:16) ( I − e H ) − ∂∂r e H (cid:17) = Tr (cid:16) ( I + e H ) − ∂∂r e H (cid:17) + h u, ∂∂r ( I + e H ) − v i ( − N + h u, ( I + e H ) − v i , (3.13)where we used the fact that ∂∂r log(det( A )) = Tr (cid:0) A − ∂∂r A (cid:1) if A is a square matrix dependingsmoothly on r . As a consequence, the two sides of (3.12) differ at most by a constant. But,since e H −→ r → ∞ , both sides of (3.12) go to 0 as r → ∞ , so the two sides are equal.Therefore our proof will be ready once we show that (3.13) holds.Since ( I − e H ) − + ( I + e H ) − = 2( I − e H ) − , (3.13) is equivalent to − (cid:16) ( I − e H ) − ∂∂r e H (cid:17)h ( − N + h u, ( I + e H ) − v i i = h u, ∂∂r ( I + e H ) − v i . (3.14) ON-INTERSECTING BROWNIAN BRIDGES AND THE LAGUERRE ORTHOGONAL ENSEMBLE 17 At this stage we use Lemma 3.2(iv) and then the cyclicity of the trace to obtain − (cid:16) ( I − e H ) − ∂∂r e H (cid:17) = − (cid:16) ( I − e H ) − Q e H (cid:17) = − (cid:16) Q e H ( I − e H ) − (cid:17) . Now note that if A is an N × N real symmetric matrix then Tr( QA ) = − r P N − i,j =0 A ij = − r h u, Au i , and thus4 r h u, ( I − e H ) − e Hu i = − (cid:16) ( I − e H ) − ∂∂r e H (cid:17) . (3.15)On the other hand, on the right-hand side of (3.14) we may apply Lemma 3.2(v) and usethe simple identity Qv = ( − N ru to get h u, ∂∂r ( I + e H ) − v i = h u, ( I − e H ) − e HQv + 4 r ( I − e H ) − ( e Hu ⊗ u )( I + e H ) − v i = 4 r h u, ( I − e H ) − e Hu i h ( − N + h u, ( I + e H ) − v i i . (3.16)Using (3.15) in (3.16) we get (3.14), which finishes the proof. (cid:3) Proof of Lemma 3.2. (i) Fix N ∈ N and r > 0, and define a upper triangular matrix S ∈ R N × N as follows: S ij = c j (cid:0) N − − ij − i (cid:1) ( − N − j j ≥ i with c k = r N − − k (cid:16) N − − k k !( N − (cid:17) / for i, j ∈ { , . . . , N − } . We claim that S is invertible, with inverse given by S − ij = c i (cid:0) N − − ij − i (cid:1) ( − N − j j ≥ i . To check this, note first that, since both S and S − (as given above) are upper triangular,we have ( SS − ) ij = 0 for i > j , while ( SS − ) ii = S ii S − ii = 1. Thus it remains to showthat ( SS − ) ij = 0 when i < j . But ( SS − ) ij = ( N − − i )!( N − − j )! ( − N − j P jk = i ( − k ( k − i )!( j − k )! = ( − i + j ( N − − i )!( N − − j )!( j − i )! P j − ik =0 ( − k ( j − i )! k !( j − i − k )! , and by the binomial theorem the last sum on theright-hand side is simply ( − j − i = 0.Now Lemma A.2 in Appendix A allows us to rewrite the symmetric matrix H in termsof Laguerre functions: for j ≥ i , H ji = H ij = c j c i j X k = i (cid:18) j − ik − i (cid:19) ( − k ψ k (2 r ) . (3.17)We will use this representation to show that S − e HS = H . We have( S − e HS ) ij = c j c i X k = i,...,N − ℓ =0 ,...,j (cid:18) N − − ik − i (cid:19)(cid:18) N − − ℓj − ℓ (cid:19) ( − N − j + k e H kℓ . Note that the value of e H kℓ depends only on k + ℓ . Letting e ψ n = ψ n − (2 r ) − ψ n (2 r ), sothat e H kℓ = ( − N e ψ k + ℓ − N +1 , and recalling that, by convention, e ψ n = 0 for n < 0, we maywrite( S − e HS ) ij = c j c i j X n =0 " X k = i,...,N − , ℓ =0 ,...,jk + ℓ − N +1= n (cid:18) N − − ik − i (cid:19)(cid:18) N − − ℓN − − j (cid:19) ( − j + k + N ψ n . (3.18)Performing the change of variables k k + i , ℓ N − − ℓ , and introducing the conventionthat (cid:0) nm (cid:1) = 0 if m > n ≥ 0, the sum in the square brackets turns into X k ≥ , ≤ ℓ ≤ N − k − ℓ = n − i (cid:18) N − − ik (cid:19)(cid:18) ℓN − − j (cid:19) ( − i + j + k + N . ON-INTERSECTING BROWNIAN BRIDGES AND THE LAGUERRE ORTHOGONAL ENSEMBLE 18 We claim now that the sum in 0 ≤ ℓ ≤ N − ℓ ≥ 0. In fact, we mayassume that k ≤ N − − i , since otherwise the first binomial coefficient vanishes. Since ℓ is constrained to be ℓ = k + i − n ≤ N − − n ≤ N − 1, adding the terms with ℓ ≥ N doesnot really contribute to the sum. In view of this, and using Lemma A.1 from Appendix A,our sum can be rewritten as X k ≥ , ℓ ≥ ℓ − k = i − n (cid:18) N − − ik (cid:19)(cid:18) ℓN − − j (cid:19) ( − i + j + k + N = ( ( − j +1 (cid:0) i − ni − j (cid:1) i ≥ j ≥ n for i ≥ n, ( − n +1 (cid:0) j − i − n − i − (cid:1) j ≥ n for i < n. Now we substitute this formula into (3.18) and consider three separate cases: • If i = j , then ( S − e HS ) ii = P in =0 (cid:0) i − n (cid:1) ( − i +1 e ψ n = ( − i ψ i (2 r ). • If i < j , then ( S − e HS ) ij = c j c i P jn = i +1 ( − n +1 (cid:0) j − i − n − i − (cid:1) e ψ n = c j c i P jn = i (cid:0) j − in − i (cid:1) ( − n ψ n (2 r ),where the second identity follows by summation by parts. • If i > j , then proceeding as for i < j we get ( S − e HS ) ij = c j c i ( − j P jn =0 (cid:0) i − n − i − j − (cid:1) ψ n (2 r ).Applying Lemma A.3 from Appendix A we deduce that the last sum equals c j c i i !(2 r ) j j !(2 r ) i X j ≤ n ≤ i (cid:0) i − jn − j (cid:1) ( − n ψ n (2 r ) = c i c j X j ≤ n ≤ i (cid:0) i − jn − j (cid:1) ( − n ψ n (2 r ) . In each case, the expression for ( S − e HS ) ij coincides with the formula for H ij in (3.17),which completes the proof of (i).(ii) We will use the contour integral representation of the Laguerre function ψ n ( x ), ψ n ( x ) = e − x/ π i I dt e − xt − t t n +1 (1 − t ) , (3.19)where the integration is along a small circle around the origin (note that by Cauchy’stheorem this formula is consistent with our convention ψ n ≡ n < L , (3.19) leads to L jk = Z ∞ r dx ψ j ( x ) ψ k ( x ) = 1(2 π i) Z ∞ r dx I I du dv e − x − xu − u − xv − v u j +1 (1 − u ) v k +1 (1 − v ) (3.20)= 1(2 π i) I I du dv e − r ( u − u + v − v ) u j +1 v k +1 (1 − uv ) . On the other hand, from the definition of e H we get( e H ) jk = ( − N N − X n =0 (cid:0) ψ j + n − N (2 r ) − ψ j + n − N +1 (2 r ) (cid:1) (cid:0) ψ n + k − N (2 r ) − ψ n + k − N +1 (2 r ) (cid:1) = 1(2 π i) I I du dv N − X n =0 e − r − r u − u − r v − v (1 − u )(1 − v ) (cid:16) u j + n − N +1 − u j + n − N +2 (cid:17)(cid:16) v n + k − N +1 − v n + k − N +2 (cid:17) = 1(2 π i) I I du dv e − r ( u − u + v − v )(1 − ( uv ) N ) u j +1 v k +1 (1 − uv ) . (3.21)The difference between (3.20) and (3.21) is then given by L jk − ( e H ) jk = 1(2 π i) I I du dv e − r ( u − u + v − v )( uv ) N u j +1 v k +1 (1 − uv ) . Since 0 ≤ j, k ≤ N − 1, the integrand has no poles in u and v inside the chosen contours,and hence the whole integral vanishes. ON-INTERSECTING BROWNIAN BRIDGES AND THE LAGUERRE ORTHOGONAL ENSEMBLE 19 (iii) For the first formula, we compute directly e Hu to get( e Hu ) i = ( − N − N − X k =0 (cid:2) ψ i + k − N (2 r ) − ψ i + k − N +1 (2 r ) (cid:3) = ψ i (2 r ) − ψ i − N (2 r ) = ( R ) i , where the last identity follows because ψ i − N (2 r ) = 0 (since i < N ). For the secondone, we use the property ∂∂x ( L n ( x ) − L n +1 ( x )) = L n ( x ) of Laguerre polynomials to obtain ∂∂x ( ψ n ( x ) − ψ n +1 ( x )) = ( ψ n ( x ) + ψ n +1 ( x )), which, together with the fact that L n (0) = 1for all n ∈ N , gives 12 Z r dx [ ψ n ( x ) + ψ n +1 ( x )] = ψ n (2 r ) − ψ n +1 (2 r )for all n ∈ N . Hence we can write the entries of e H as e H ij = i + j < N − , ( − N +1 e − r for i + j = N − , ( − N (cid:0) Ψ i + j − N (2 r ) + Ψ i + j − N +1 (2 r ) (cid:1) for i + j > N − , (3.22)with Ψ n ( s ) = R s dx ψ n ( x ) (note that ψ ( x ) = e − x/ ). Now we can compute(( I − e H ) v ) i = N − X k =0 ( δ ik − e H ik )( − k − i − − N − i e − r − N − X k = N − i ( − k e H ik = 2( − i (1 − e − r ) − i − X k =0 ( − k + N − i ( − N k (2 r ) + Ψ k +1 (2 r ))= 2( − i (1 − e − r ) − ( − i Ψ (2 r ) + Ψ i (2 r ) = ( R ) i . (iv) From (3.22) we get ∂∂r e H ij = i + j < N − , ( − N r e − r for i + j = N − , ( − N r (cid:0) ψ i + j − N (2 r ) + ψ i + j − N +1 (2 r ) (cid:1) for i + j > N − . This expression coincides with the i, j entry, for all i, j ∈ { , . . . , N − } , of the matrix Q e H , which is given by( Q e H ) ij = − r i − X k =0 e H kj − r e H ij = ( − N r (cid:0) ψ i + j − N (2 r ) + ψ i + j − N +1 (2 r ) (cid:1) . (v) For i, j ∈ { , . . . , N − } we have( Q e H ) ij + ( e HQ ) ij = − r (cid:16) e H ij + i − X k =0 e H kj + N − X k = j +1 e H ik (cid:17) . Since P i − k =0 e H kj = P j − k =0 e H ik , the right-hand side of the last identity equals − r P N − k =0 e H ik ,which coincides with − (4 r e Hu ⊗ u ) ij . Thus, recalling the notation E = 4 r e Hu ⊗ u , we have Q e H = − e HQ − E. (3.23)Now recall that if A is a square matrix which depends smoothly on a parameter r , then ∂ r A − = − A − ∂ r AA − . Then, in view of (iv) and the last identity, we have ∂ r ( I + e H ) − = − ( I + e H ) − Q e H ( I + e H ) − = ( I − e H ) − ( I − e H )( e HQ + E )( I + e H ) − . ON-INTERSECTING BROWNIAN BRIDGES AND THE LAGUERRE ORTHOGONAL ENSEMBLE 20 Comparing this with the right-hand side of the identity we seek to prove, we see that itis enough to check that e HQ ( I + e H ) + E = ( I − e H )( e H Q + E ), which follows easily from(3.23). (cid:3) Appendix A. Some formulas for Hermite and Laguerre polynomials We begin with a combinatorial result which was used in the proof of Lemma 3.2(i) andwhich will also be used later in this appendix.Throughout this appendix we adopt the convention that, for k ∈ N and ℓ ∈ Z , (cid:0) kℓ (cid:1) = 0 if ℓ < ℓ > k (this can be justified, for instance, by replacing the factorials with Gammafunctions). Lemma A.1. Let n, m ∈ N and a ∈ Z . Then X i,j ≥ j − i = a (cid:18) ni (cid:19)(cid:18) jm (cid:19) ( − i = ( − n (cid:18) am − n (cid:19) for a ≥ , ( − m − a (cid:18) n − m − n − m + a (cid:19) { n ≥ m − a } for a < .Proof. Assume first that a ≥ 0. Then the formula we seek to prove can be rewritten as X i ≥ (cid:18) ni (cid:19)(cid:18) i + am (cid:19) ( − n − i = (cid:18) am − n (cid:19) . (A.1)For x ∈ R and with our convention, we have (using Newton’s generalized binomial theorem) X m ≥ X i ≥ (cid:18) ni (cid:19)(cid:18) i + am (cid:19) ( − n − i x m = X i ≥ (cid:18) ni (cid:19) ( − n − i (1 + x ) i + a = (1 + x ) a ( − x )) n = (1 + x ) a x n = x n X ℓ ∈ Z (cid:18) aℓ (cid:19) x ℓ . By equating the coefficient in front of x m on both sides, we obtain (A.1) .When a < 0, we first let b = − a > X j ≥ (cid:18) nj + b (cid:19)(cid:18) jm (cid:19) ( − j − m = (cid:18) n − m − n − m − b (cid:19) { n ≥ m + b } . (A.2)Pick x ∈ R such that | x | < (cid:12)(cid:12) x − x (cid:12)(cid:12) < 1. Using three times the identity x k (1 − x ) k +1 = ∞ X n ≥ k (cid:18) nk (cid:19) x n (A.3)(which is a straightforward consequence of Newton’s generalized binomial theorem) to-gether with our convention we have X n ≥ X j ≥ (cid:18) nj + b (cid:19)(cid:18) jm (cid:19) ( − j − m x n = X j ≥ (cid:18) jm (cid:19) x j + b (1 − x ) j + b +1 ( − j − m = x b (1 − x ) b +1 ( x/ (1 − x )) m (1 + x/ (1 − x )) m +1 = x b + m (1 − x ) b = x m +1 X ℓ ≥ b − (cid:18) ℓb − (cid:19) x ℓ . (A.2) now follows from the fact that the coefficient of x n on the right-hand side is givenby (cid:0) n − m − n − m − b (cid:1) when n ≥ m + b and equals 0 when n < m + b . (cid:3) ON-INTERSECTING BROWNIAN BRIDGES AND THE LAGUERRE ORTHOGONAL ENSEMBLE 21 Lemma A.2. For n, m ∈ N with n ≥ m and any r ∈ R \ { } , the following relation holds: Z R dx ϕ n ( x ) ϕ m (2 r − x ) = r m − n (cid:18) m n !2 n m ! (cid:19) / n X k = m (cid:18) n − mk − m (cid:19) ( − k ψ k (2 r ) . (A.4) Similarly, for the case r = 0 we have Z R dx ϕ n ( x ) ϕ m ( − x ) = ( − n m = n . (A.5) Proof. Recall that the Hermite polynomials have a simple generating function, namely ∞ X n =0 H n ( x ) n ! t n = e xt − t . We write the convolution of Hermite functions in (A.4) as Z R dx ϕ n ( x ) ϕ m (2 r − x ) = 1 √ n + m πn ! m ! Z R dx H n ( x ) e − x / H m (2 r − x ) e − (2 r − x ) / and then use the above generating function to evaluate the sum ∞ X n,m =0 t n t m n ! m ! Z R dx H n ( x ) e − x / H m (2 r − x ) e − (2 r − x ) / = Z R dx e xt − t − x / r − x ) t − t − (2 r − x ) / = √ πe − r +2 r ( t + t ) − t t . By equating the coefficient of t n t m on each side, we obtain an explicit formula for theleft-hand side of (A.4): Z R dx ϕ n ( x ) ϕ m (2 r − x ) = 1 √ n + m n ! m ! e − r ∂ nt ∂ mt e r ( t + t ) − t t (cid:12)(cid:12)(cid:12) t = t =0 = 1 √ n + m n ! m ! e − r m X ℓ =0 ( − ℓ ℓ ! (cid:18) nℓ (cid:19)(cid:18) mℓ (cid:19) (2 r ) n + m − ℓ . (A.6)In particular, we get (A.5), so from now on we will assume r = 0.Turning to the right-hand side of (A.4), we use the explicit power series expansion ofthe Laguerre polynomials, L k ( x ) = k X ℓ =0 (cid:18) kℓ (cid:19) ( − ℓ ℓ ! x ℓ , (A.7)to rewrite it as r m − n (cid:18) m n !2 n m ! (cid:19) / n X k = m k X ℓ =0 (cid:18) n − mk − m (cid:19) ( − k e − r (cid:18) kℓ (cid:19) ( − ℓ ℓ ! (2 r ) ℓ . We need to show that this expression equals the right-hand side of (A.6) or, equivalently,that n − m X k =0 k + m X ℓ =0 (cid:18) n − mk (cid:19)(cid:18) k + mℓ (cid:19) ( − k + m ( − r ) ℓ ℓ ! = n X ℓ = n − m (cid:18) mn − ℓ (cid:19) ( − n ( − r ) ℓ ℓ ! , where we have performed the changes of variables k k + m on the left-hand side and ℓ n − ℓ on the right-hand side. Using our convention, this is equivalent to ∞ X ℓ =0 " ∞ X k =0 (cid:18) n − mk (cid:19)(cid:18) k + mℓ (cid:19) ( − n − m − k ( − r ) ℓ ℓ ! = ∞ X ℓ =0 (cid:18) mn − ℓ (cid:19) ( − r ) ℓ ℓ ! . It follows from Lemma A.1 (or, more specifically, from (A.1)) that the coefficients of ( − r ) ℓ ℓ ! on both sides of this identity coincide, and this finshes the proof. (cid:3) ON-INTERSECTING BROWNIAN BRIDGES AND THE LAGUERRE ORTHOGONAL ENSEMBLE 22 Lemma A.3. For any n, m ∈ N with n ≥ m and any x ∈ R , the following relation holds: ( − m x n n ! m X k =0 (cid:18) n − k − n − m − (cid:19) L k ( x ) = x m m ! n X k = m (cid:18) n − mk − m (cid:19) ( − k L k ( x ) . (A.8) Proof. We will use (A.7) in order to extract the coefficients of x ℓ in the polynomials ap-pearing on both sides. The coefficient of x ℓ on the left-hand side of (A.8) is clearly 0 if ℓ < n , while for n ≤ ℓ ≤ n + m it is given by( − ℓ + m − n n !( ℓ − n )! m X k =0 (cid:18) n − k − n − m − (cid:19)(cid:18) kℓ − n (cid:19) = ( − ℓ + m − n n !( ℓ − n )! (cid:18) nℓ − m (cid:19) , (A.9)where we have used a variant of Vandermonde’s identity which can be obtained by equatingthe coefficient of x n in the expansion of both sides of the identity x x n − m − (1 − x ) n − m x ℓ − n (1 − x ) ℓ − n +1 = x ℓ − m (1 − x ) ℓ − m +1 obtained by using (A.3). On the other hand, for ℓ < m the coefficient of x ℓ onthe right-hand side of (A.8) is clearly zero, while for m ≤ ℓ ≤ n + m it is given by( − ℓ − m m !( ℓ − m )! n − m X k =0 (cid:18) n − mk (cid:19)(cid:18) k + mℓ − m (cid:19) ( − k + m = ( − ℓ + n − m m !( ℓ − m )! (cid:18) mℓ − n (cid:19) , (A.10)where we used the change of variables k k + m and Lemma A.1. Notice that (A.10)equals 0 by our convention if m ≤ ℓ < n and it clearly equals (A.9) if n ≤ ℓ ≤ n + m (recallthat we are assuming n ≥ m ). The proof is thus complete. (cid:3) Appendix B. Proof of Lemma 2.3 Throughout the proof we will use c and c to denote positive constants which do notdepend on L and whose value may change from line to line. We will denote by k · k and k · k the trace class and Hilbert-Schmidt norms of operators on L ( R ). We recall that k AB k ≤ k A k k B k and k A k = Z dx dy A ( x, y ) (B.1)if A has integral kernel A ( x, y ); for more details see [QR14, Section 2] or [Sim05].In view of (2.11) we write e Ω ( r ) L = e Ω ( r, L + e Ω ( r, L , where e Ω ( r, L = e L D K Herm ,N P r cosh( L ) (cid:0) e − L D − R ( r )[ − L, (cid:1) ¯ P r (cid:0) e − L D − R ( r )[0 ,L ] (cid:1) ¯ P r cosh( L ) e L D K Herm ,N , e Ω ( r, L = e L D K Herm ,N (cid:0) e − L D − R ( r )[ − L, (cid:1) ¯ P r (cid:0) e − L D − R ( r )[0 ,L ] (cid:1) P r cosh( L ) e L D K Herm ,N . We will focus on e Ω ( r, L and show that it goes to zero in trace norm, the proof for e Ω ( r, L isvery similar so we will omit it.We factor e Ω ( r, L as e Ω ( r, L = Υ Υ with Υ = e L D K Herm ,N P r cosh( L ) (cid:0) e − L D − R ( r )[ − L, )¯ P r and Υ = ¯ P r (cid:0) e − L D − R ( r )[0 ,L ] (cid:1) ¯ P r cosh( L ) e L D K Herm ,N . By (B.1), it is enough to show that k Υ k k Υ k −→ L → ∞ . We start with Υ ,which is made of two terms which we will bound separately. By (B.1) and the fact that ON-INTERSECTING BROWNIAN BRIDGES AND THE LAGUERRE ORTHOGONAL ENSEMBLE 23 the family ( ϕ n ) n ∈ N is orthonormal we have k e L D K Herm ,N P r cosh( L ) e − L D ¯ P r k = N − X n,n ′ =0 Z ∞−∞ dx Z r −∞ dy e L ( n + n ′ ) ϕ n ( x ) ϕ n ′ ( x ) × Z [ r cosh( L ) , ∞ ) dzdz ′ ϕ n ( z ) ϕ n ′ ( z ′ ) e − L D ( z, y ) e − L D ( z ′ , y )= N − X n =0 e nL Z r −∞ dy Z ∞ r cosh( L ) dz ϕ n ( z ) e − L D ( z, y ) ! ≤ N e N − L Z ∞ r cosh( L ) dz Z ∞−∞ dy ( e − L D ( z, y )) , where we have used the Cauchy-Schwarz inequality. Using the formula for the kernel of e − L D which is implicit in (2.6) and (2.7) we see that that the y integral is just a Gaussianintegral, and computing it gives k e L D K Herm ,N P r cosh( L ) e − L D ¯ P r k ≤ N e N − L coth( L ) − √ π coth( L ) Z ∞ r cosh( L ) dz e L − z tanh( L ) . The last integral is bounded by c e L − r cosh( L ) tanh( L ) for all L > 0, and thus, recallingthat we are assuming r > k e L D K Herm ,N P r cosh( L ) e − L D ¯ P r k ≤ c e NL − c e L . for sufficiently large L . The estimate for the other term appearing in Υ is very similarand leads to the same type of bound. We deduce that k Υ k ≤ c e NL − c e L (B.2)for large enough L . On the other hand it is easy to check that the same calculation asabove leads to k Υ k ≤ c e NL (B.3)(note that in this case the projection P r cosh( L ) appearing in Υ is replaced by ¯ P r cosh( L ) ;this accounts for the fact that the factor e − c e L disappears from the upper bound). Bycombining (B.2) and (B.3) together we immediately get k e Ω ( r, L k ≤ c e NL − c e L −−−−→ L →∞ . Acknowledgements. The authors would like to thank Jeremy Quastel for valuable com-ments on this manuscript. They also thank Gregory Schehr for pointing out to us formulas(102)–(103) in [RS11]. 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