# Reunion probability of vicious walkers: typical and large fluctuations for large

###### Abstract

We consider three different models of non-intersecting Brownian motions on a line segment with absorbing (model A), periodic (model B) and reflecting (model C) boundary conditions. In these three cases we study a properly normalized reunion probability, which, in model A, can also be interpreted as the maximal height of non-intersecting Brownian excursions (called "watermelons" with a wall) on the unit time interval. We provide a detailed derivation of the exact formula for these reunion probabilities for finite using a Fermionic path integral technique. We then analyse the asymptotic behavior of this reunion probability for large using two complementary techniques: (i) a saddle point analysis of the underlying Coulomb gas and (ii) orthogonal polynomial method. These two methods are complementary in the sense that they work in two different regimes, respectively for and . A striking feature of the large limit of the reunion probability in the three models is that it exhibits a third-order phase transition when the system size crosses a critical value . This transition is akin to the Douglas-Kazakov transition in two-dimensional continuum Yang-Mills theory. While the central part of the reunion probability, for , is described in terms of the Tracy-Widom distributions (associated to GOE and GUE depending on the model), the emphasis of the present study is on the large deviations of these reunion probabilities, both in the right [] and the left [] tails. In particular, for model B, we find that the matching between the different regimes corresponding to typical and atypical fluctuations in the right tail is rather unconventional, compared to the usual behavior found for the distribution of the largest eigenvalue of GUE random matrices. This paper is an extended version of [G. Schehr, S. N. Majumdar, A. Comtet, J. Randon-Furling, Phys. Rev. Lett. 101, 150601 (2008)] and [P. J. Forrester, S. N. Majumdar, G. Schehr, Nucl. Phys. B 844, 500-526 (2011)].

∎

## 1 Introduction

Non-intersecting random walkers, first introduced by de Gennes deG68 (), followed by Fisher Fisher84 () (who called them "vicious walkers"), have been studied extensively in statistical physics as they appear in a variety of physical contexts ranging from wetting and melting all the way to polymers and vortex lines in superconductors. Lattice versions of such walkers also have beautiful combinatorial properties KGV2000 (). Non-intersecting Brownian motions, defined in continuous space and time, have also recently appeared in a number of contexts. In particular their connection to the random matrix theory has been noted in a variety of situations Ba00 (); Fo01 (); Jo03 (); Na03 (); KT2004 (); FP2006 (); TW2007 (); DaKu07 (); SMCR08 (); NM09 (); No09 (); RS10 (); BoKu10 (); BDK11 (); AMV12 (). Rather recently, we have unveiled an unexpected connection between vicious walkers and two-dimensional continuum Yang-Mills theory on the sphere with a given gauge group FMS11 (), which depends on the boundary conditions in the vicious walkers problem (a different type of connection between the Yang-Mills theory and vicious walkers problems had also been noticed in Ref. HT04 ()).

Specifically, we consider a set of non-intersecting Brownian motions on a finite segment of the real line with different boundary conditions. Assuming that all the walkers start from the vicinity of the origin, we then define the reunion probability as the probability that the walkers reunite at the origin after a fixed interval of time which can be set to unity without any loss of generality. Within the time interval , the walkers stay non-intersecting. Next we ‘normalize’ this reunion probability in a precise way to be defined shortly. In one case, namely when both boundaries at and are absorbing, one can relate this ‘normalized’ reunion probability to the probability distribution of the maximal height of non-intersecting Brownian excursions. In Ref. FMS11 () it was shown that this normalized reunion probability in the Brownian motion models maps onto the exactly solvable partition function (up to a multiplicative factor) of two-dimensional Yang-Mills theory on a sphere. The boundary conditions at the edges and select the gauge group of the associated Yang-Mills theory. We consider three different boundary conditions: absorbing (model A), periodic (model B), and reflecting (model C) which correspond respectively to the following gauge groups in the Yang-Mills theory: (A) absorbing , (B) periodic and (C) reflecting . As a consequence of this connection, in each of these Brownian motion models, as one varies the system size , a third order phase transition occurs at a critical value in the large limit. It was shown in Ref. NM11 () that a similar third order phase transition also occurs for the probability distribution function of the largest eigenvalue of random matrices belonging to the standard Gaussian ensembles. Furthermore, third order phase transitions in the large deviation function of appropriate variables have also been found recently in a variety of other problems, such as in the distribution of conductance through a mesoscopic cavity such as a quantum dot VMB08 (); VMB10 (); DMTV11 () and in the distribution of the entanglement entropy in a bipartite random pure state NMV10 (); NMV11 ().

Close to the critical point, these reunion probabilities in the Brownian motion models, properly shifted and scaled, can be related to the Tracy-Widom (TW) distributions. Let us briefly remind the readers about the Tracy-Widom distributions. Tracy-Widom distribution describes the limiting form of the distribution of the scaled largest eigenvalue in the three classical Gaussian random matrix ensembles TW94a (). These limiting distributions are usually denoted by for the Gaussian orthogonal ensemble (GOE), by for the Gaussian unitary ensemble (GUE) and by for the Gaussian symplectic ensemble (GSE). For example, in the GUE case where one considers the set of complex Hermitian matrices with measure proportional to and denote the largest eigenvalue, the typical fluctuations around its mean value have a limiting distribution TW94a ()

(1) |

known as the Tracy-Widom distribution. In Eq. (1), satisfies Painlevé II (PII) differential equation

(2) |

with the asymptotic behavior

(3) |

where is the Airy function. One can indeed show that this asymptotic behavior (3) determines a unique solution of PII (2), known as the Hastings-McLeod solution. Similarly, the distribution concerns real symmetric matrices. Explicitly, with the GOE specified as the set of real symmetric matrices with measure proportional to , and denoting the largest eigenvalue, one has TW96 ()

(4) | |||||

known as the Tracy-Widom distribution. Because of their relevance in many fundamental problems in mathematics and physics, these TW distributions have been widely studied in the literature. In particular, their asymptotic behaviors are given, to leading order by

(5) |

where and correspond respectively to the GOE and the GUE case.

Our study of constrained vicious walkers problems started in Ref. SMCR08 () where we derived, using a Fermionic path integral method, an exact expression for the ratio of reunion probability, for model A and for any finite number of walkers. This calculation, in this specific case, was motivated by the interpretation of this ratio in terms of an extreme value quantity, namely the maximal height of non-intersecting Brownian excursions, which is recalled below in section 2 (see also Fig. 1). We showed later in Ref. FMS11 () that this ratio, and its extension to other boundary conditions in model B and C, is actually equal, up to a multiplicative prefactor, to the partition function of Yang-Mills theory on the sphere with a given gauge group , which depends on the boundary conditions in the vicious walkers problem (see Table 1). Following the pioneering works of Refs. DK93 (); GM94 () where the large analysis of Yang-Mills theory on the sphere with the gauge group , and extended to other classical Lie groups, including or , in Ref. CNS96 (), we could also perform in Ref. FMS11 () the large analysis of these reunion probabilities. However, most of these results were announced in Refs. SMCR08 (); FMS11 () without any detail. The purpose of the present paper is to give a self-contained derivation of these results, both for finite and large . For large we characterize not only the distribution of typical fluctuations, which can be expressed in terms of and but also provide a detailed analysis of the large deviations of these reunion probabilities, characterizing atypical fluctuations. A special emphasis is put on the matching between different regimes (typical and atypical) of the reunion probability as a function of . In particular, for the case of periodic boundary conditions, we find that the matching found in this vicious walker problem is quite different from the corresponding matching observed in the distribution of the largest eigenvalue of Gaussian random matrices in the unitary ensemble NM09 (); DM06 (); VMB07 (); DM08 (); MV09 (); BEMN10 (); Fo12 ().

The paper is organized as follows. In section 2 we describe the three models A, B and C and summarize the main results for the ratio of reunion probabilities in the large limit. In section 3 we give the details of the path integral method which allows us to obtain an exact expression of these ratios in each of the three models for any finite and . In section 4, we compute the large limit of this ratio for model A and analyze in detail the typical fluctuations (namely the central part of the distribution) as well as the large deviations: both for the left and for the right tails. The corresponding large analysis for model B and C are performed in section 5 and section 6, respectively, before we conclude in section 7. Some details have been relegated to Appendix A.

## 2 Models and main results

We consider three different models of non-intersecting Brownian walkers on a
one-dimensional line segment and label their positions at time by
. These three models, denoted by model A, B and C, differ by the
boundary conditions which are imposed at and :

in model A, we
consider absorbing boundary conditions both at and ,

in model B, we
study periodic boundary conditions, which amounts to consider non-intersecting Brownian
motions on a circle of radius ,

and in model C we consider reflecting
boundary conditions both at and .

Model A | Model B | Model C | |
---|---|---|---|

Boundary conditions at and | absorbing | periodic | reflecting |

Corresponding affine Weyl chamber | |||

Gauge group of the associated YM theory |

Model A: In the first model the domain is the line segment with absorbing boundary conditions at both boundaries and . This corresponds to -dimensional Brownian motion in an affine Weyl chamber of type Gra99 (); Gra02 (). The non-intersecting Brownian motions start initially at the positions, say, in the vicinity of the origin where eventually we will take the limit for all , as shown later. We define the reunion probability , where the superscript ’A’ refers to model A, as the probability that the walkers return to their initial positions after a fixed time (staying non-intersecting over the time interval ). We define the normalized reunion probability

(6) |

such that . This ratio becomes independent of the starting positions ’s in the limit when for all , as shown later. Hence, depends only on and .

This ratio in Model A has also a different probabilistic interpretation. Consider the same model but now on the semi-infinite line with still absorbing boundary condition at . The walkers, as usual, start in the vicinity of the origin and are conditioned to return to the origin exactly at (see Fig. 1). If one plots the space-time trajectories of the walkers, a typical configuration looks like half of a watermelon (see Fig. 1), or a watermelon in presence of an absorbing wall. Such configurations of Brownian motions are known as non-intersecting Brownian excursions TW2007 () and their statistical properties have been studied quite extensively in the recent past. A particular observable that has generated some recent interests is the so-called ‘height’ of the watermelon SMCR08 (); BM2003 (); Fulmek2007 (); KIK2008 (); KIK08 (); Fe08 (); RS11 (); liechty () defined as follows (see also Ref. BFPSW09 () for a related quantity in the context of Dyson’s Brownian motion). Let denote the maximal displacement of the rightmost walker in this time interval , i.e., the maximal height of the topmost path in half-watermelon configuration (see Fig. 1), i.e., . This global maximal height is a random variable which fluctuates from one configuration of half-watermelon to another. What is the probability distribution of ? For the distribution of is easy to compute and already for it is somewhat complicated KIK2008 (). In Ref. SMCR08 () an exact formula for the distribution of , valid for all , was derived using Fermionic path integral method, which we remind below for consistency (see also KIK2008 (); KIK08 (); Fe08 () for a derivation using different methods). The distribution of , in the large limit, is quite interesting as it gives, in the proper scaling limit, the distribution of the maximum (on the real line) of the Airy process minus a parabola Jo03 (); PS99 (); PS04 (). This latter process describes the universality class of the Kardar-Parisi-Zhang (KPZ) equation in the so-called "droplet" geometry. It was known rather indirectly from the work of Ref. Jo03 () on the Airy process that the limiting distribution of should then be given by . This was one of the main result of Ref. FMS11 () to show this result by a direct computation of the limiting distribution of . This result, for the vicious walkers problem, was then recently proved rigorously in liechty () using Riemann-Hilbert techniques. In Ref. MQR11 () a direct and rigorous proof was established that the distribution of the maximum of Airy minus a parabola is indeed given by . We refer the reader to Refs. Sc12 (); QR12 (); BLS12 () for more recent results on the extreme statistics of the Airy process minus a parabola. We also mention that the limiting distribution of was measured in recent experiments on liquid crystals, and a very good agreement with was indeed found TS12 ().

To relate the distribution of in the semi-infinite system defined above to the ratio of reunion probabilities in the finite segment defined in (6), it is useful to consider the cumulative probability in the semi-infinite geometry, where now is just a variable. To compute this cumulative probability, we need to calculate the fraction of half-watermelon configurations (out of all possible half-watermelon configurations) that never cross the level , i.e., whose heights stay below over the time interval (see Fig. 1). This fraction can be computed by putting an absorbing boundary at (thus killing all configurations that touch/cross the level ). It is then clear that is nothing but the normalized reunion probability defined in (6). As mentioned above, this cumulative probability distribution of the maximal height was computed exactly in Ref. SMCR08 ()

(7) |

where

(8) |

is the Vandermonde determinant and where the amplitude is given by

(9) |

This quantity was also computed in Refs. KIK08 (); Fe08 () using different methods. In Ref. KIK08 () the authors studied the maximal height of non-intersecting Brownian excursions and used the Karlin-McGregor formula karlin_mcgregor (). In Ref. Fe08 () the author computed the cumulative distribution of the maximal height of -non intersecting discrete lattice paths (of discrete steps) in presence of a wall, using the Lindström-Gessel-Viennot formula LGV1 (); LGV2 (), and then he considered the asymptotic limit of a large number of steps . The formulas obtained using these two similar methods are actually different from the one given in Eq. (2) obtained by using a fermionic path integral and it is a non-trivial task to check, as expected, that they are indeed equivalent to (2) KIK08 (). A derivation of the formula given above in Eqs. (2, 9), which turns out to be the most convenient expression of in view of a large asymptotic analysis, is given in section 3.

Remarkably, if one denotes by

(10) |

the partition function of the two-dimensional (continuum) Yang-Mills theory on the sphere (denoted as ) with gauge group and area it was shown in Ref. FMS11 () that is related to with the gauge group via the relation

(11) |

In Ref. DK93 (); CNS96 (), it was shown that for large , exhibits a third order phase transition at the critical value separating a weak coupling regime for and a strong coupling regime for . This is the so called Douglas-Kazakov phase transition DK93 (), which is the counterpart in continuum space-time, of the Gross-Witten-Wadia transition GW80 (); Wadia80 () found in two-dimensional lattice quantum chromo-dynamics (QCD), which is also of third order. Using the correspondence , we then find that , considered as a function of with large but fixed, also exhibits a third order phase transition at the critical value . Furthermore, the weak coupling regime () corresponds to and thus describes the right tail of , while the strong coupling regime corresponds to and describes instead the left tail of (see Fig. 2). The critical regime around is the so called "double scaling" limit in the matrix model and has width of order . It corresponds to the region of width around where , correctly shifted and scaled, is described by the Tracy-Widom distribution in Eq. (4).

Although the occurrence of the Painlevé transcendent (2) in this double scaling limit was known since the work of Periwal and Shevitz PS90a (), its probabilistic interpretation in relation to the Tracy-Widom distribution was one of the main achievements of Ref. FMS11 (). In this paper, we provide a detailed analysis of the three regimes: the left tail, the central part and the right tail of . Our results can be summarized as follows

(12) |

where is the TW distribution for GOE, whose explicit expression is given in (4) and its asymptotic behaviors are given in Eq. (5). The rate functions can be computed exactly (see later): of particular interest are their asymptotic behaviors when from below (left tail) and from above (right tail), which are given by

(13) | |||||

The different behavior of in Eq. (12) for and leads, in the limit to a phase transition at the critical point in the following sense. Indeed if one scales by , keeping the ratio fixed, and take the limit one obtains

(14) |

If one interprets in Eq. (2) as the partition function of a discrete Coulomb gas, its logarithm can be interpreted as its free energy. Since when approaches from below, then the third derivative of the free energy at the critical point is discontinuous, which can then be interpreted as a third order phase transition.

On the other hand, comparing the asymptotic behavior of in Eq. (5) with the ones of the rate functions (13) we can check that the expansion of the large deviation functions around the transition point coincides with the tail behaviors of the central region Tracy-Widom scaling function. This property holds here both for the left and the right tails. For instance, consider first the left tail in Eq. (12), i.e., when . When from below, we can substitute the asymptotic behavior of the rate function from Eq. (13) in the first line of Eq. (12). This gives

(15) |

On the other hand, consider now the second line of Eq. (12) that describes the central typical fluctuations. When the deviation from the mean is large, i.e., (compared to the typical scale ), we can substitute in the second line of Eq. (12) the left tail asymptotic behavior of the Tracy-Widom function as described in the first line of Eq. (5) (with ). This gives,

(16) |

which, after a trivial rearrangement, is identical to the expression in Eq. (15). Thus the left tail of the central region matches smoothly with the left large deviation function. Similarly, on the right side, using the behavior of in Eq. (13), one finds from Eq. (12), that

(17) |

which matches perfectly with the right tail of the central part described by (5, 12). Hence in this case of model A, the matching between the different regimes is similar to the one found in previous studies of large deviation formulas associated with the largest eigenvalue of random matrices NM09 (); DM06 (); VMB07 (); DM08 (); MV09 (); BEMN10 ().

Model B: In the second model we consider periodic boundary conditions on the line segment . Alternatively, one can think of the domain as a circle of circumference (of radius ). This corresponds to -dimensional Brownian motion in an affine Weyl chamber of type Gra99 (); Gra02 (). All walkers start initially in the vicinity of a point on the circle which we call the origin. We can label the positions of the walkers by their angles (see section 3 for details). Let the initial angles be denoted by where ’s are small. Eventually we will take the limit . We denote by the reunion probability after time (note that the walkers, in a bunch, may wind the circle multiple times), i.e, the probability that the walkers return to their initial positions after time (staying non-intersecting over the time interval ). Evidently depends on and also on the starting angles . To avoid this additional dependence on the ’s, let us introduce the normalized reunion probability defined as the ratio

(18) |

where we assume that we have taken the limit. One can actually check that the ’s dependence actually cancels out between the numerator and the denominator: this is the main motivation for studying this ratio of reunion probabilities (18) in this case. Although in the case of model A it is rather natural to expect that this limit is well defined, given the interpretation of as the cumulative distribution of the maximal height of non-intersecting excursions, this is not so obvious for where such an interpretation does not exist. Nevertheless one can show that this property also holds in this case, yielding (see Ref. FMS11 () and also section 3)

(19) |

where is the Vandermonde determinant (8) and the prefactor

(20) |

ensures that . In Ref. FMS11 () it was shown that this normalized reunion probability is, up to a prefactor, exactly identical to the partition function of the -d Yang-Mills theory on a sphere with gauge group :

(21) |

This partition function exhibits the Douglas-Kazakov third order phase transition for which means that in that case the transition between the right tail and the left tail behavior of occurs for . In this paper, we provide a detailed analysis of the various regimes, right tail, central part and left tail of . Our results can be summarized as follows

(22) |

where is the TW distribution for GUE, whose explicit expression is given in (1) and its asymptotic behaviors are given in Eq. (5). The rate functions can be computed exactly (see later). Of particular interest are their asymptotic behaviors when from below (left tail) and from above (right tail), which are given by

(23) | |||||

Notice also the oscillating sign in the right tail of in Eq. (22) which is not problematic here as does not have the meaning of a cumulative distribution. As explained before for model A in Eq. (14), the cubic behavior of when approaches from below is again the signature of a third order phase transition in this model. On the other hand, by comparing the asymptotic behavior of in Eq. (5) with the one of the rate functions (23) we can check, in this case of model B, that the expansion of the large deviation functions around the transition point coincides with the tail behavior of the TW scaling function, only for the left tail.

Indeed, using the behavior of in Eq. (23), one finds from Eq. (22) that

(24) |

which matches perfectly with the left tail of the central region (5). However, we find that this property does not hold for the right tail. Indeed, using the asymptotic behavior of in Eq. (23) one finds from (22)

(25) |

with an oscillating sign . On the other hand the right tail of the central region, described by in Eq. (5), yields for

(26) |

without any oscillating sign and where the argument of the exponential is twice larger than the one in (25). This mismatching is the sign of an interesting crossover which happens in this case and which we study in detail below. It can be summarized as follows. Close to , with , a careful computation beyond leading order shows that

(27) |

where is the Hastings-McLeod solution of PII (2, 3). In the large limit, these two competing terms and in Eq. (27) behave like (3, 5)

(28) |

Therefore what happens when one increases from the critical region towards the large deviation regime in the right tail is the following (see Fig. 3): the amplitude of the second term in the right hand side of Eq. (27) which is oscillating with , increases relatively to the amplitude of the first term. At some crossover value it becomes larger than the first one and in the large deviation regime it becomes the leading term, still oscillating with (22). Balancing these two terms and making use of the above asymptotic behaviors (28) one obtains an estimate of as

(29) |

Note that such a peculiar crossover is absent in the distribution of the largest eigenvalue of GUE random matrices and it is thus a specific feature of this vicious walkers problem.

Model C: We consider a third model of non-intersecting Brownian motions where the walkers move again on a finite line segment , but this time with reflecting boundary conditions at both boundaries and . This corresponds to -dimensional Brownian motion in an affine Weyl chamber of type Gra99 (); Gra02 (). Again the walkers start in the vicinity of the origin at time and we consider the reunion probability that they reunite at time at the origin. Following Models A and B, we define the normalized reunion probability

(30) |

that is independent of the starting positions in the limit when all the ’s tend to zero and hence depends only on and . As shown in Ref. FMS11 (), see also in section 3, can be computed exactly as

(31) |

where is the Vandermonde determinant (58) and the prefactor is given by

(32) |

which ensures that . In Ref. FMS11 () we showed that , up to a prefactor, is exactly identical to the partition function of the -d Yang-Mills theory on a sphere with gauge group :

(33) |

This partition function exhibits the Douglas-Kazakov third order phase transition for which means that in that case the transition between the right tail and the left tail behavior of occurs for . In this paper, we provide a detailed analysis of the various regimes, right tail, central part and left tail of . Our results can be summarized as follows

(34) |

where is the TW distribution for GUE and the TW distribution for GOE, whose explicit expressions are given in (1, 4) and their asymptotic behaviors are given in Eq. (5). The rate functions can be computed exactly (see below): their asymptotic behaviors are given in Eq. (13). As explained before for model A in Eq. (14), the cubic behavior of when approaches from below is again the signature of a third order phase transition in this model. Similarly, as in the case of model A, we can check that the expansion of the large deviation functions around the transition point coincides with the tail behaviors of the central region both in the left tail

(35) |

as well as in the right tail

(36) |

where, here again, the minus sign is not problematic as does not have the interpretation of a cumulative probability distribution.

## 3 Derivation of the formula for the reunion probabilities

In this section, we derive the expressions of the normalized reunion probabilities given in Eqs. (2, 19, 31). The derivations are based on a Fermionic path integral method. For model A, this result was first reported in Ref. SMCR08 ().

### 3.1 Model A: absorbing boundary conditions at and

We start by the computation of defined in Eq. (6). It then follows that

(37) |

where we use the notation and where is the probability that the Brownian paths, with diffusion coefficient , starting at at come back to the same points at without crossing each other and staying within the interval , with absorbing boundary conditions at and .

To proceed, let us first consider the simple case of independent and free Brownian walkers on a line, each with diffusion constant , over the unit time interval , but without the non-intersection constraint. The probability measure of an assembly of trajectories, that start at and also end at , would then be simply proportional to the propagator

(38) |

If, in addition, we subject the walkers to stay within the box during the time interval , this is equivalent to putting an infinite potential at the two ends of the box . Then one can use path integral techniques to write as the propagator

(39) |

where is a confining potential with

(40) |

If one denotes by the eigenvalues of (39) and the corresponding eigenvectors, one has

(41) |

So far, we have not implemented the non-intersection constraint. The important observation that we make is that this constraint can be incorporated within the path integral framework by simply insisting that the many body wave function must be Fermionic, i.e. it vanishes if any of the two coordinates are equal. This many-body antisymmetric wave function is thus constructed from the one-body eigenfunctions of by forming the associated Slater determinant. In the case of model A, we note that the single particle wave-functions vanishing at and are

(42) |

such that the eigenfunctions and eigenvalues in Eq. (41) are given here by

(43) |

Hence one has

(44) |

We now have to study the limit of in Eq. (44) when (). One can then check that to leading order, one has

(45) |

where is a numerical constant independent of ’s and ’s. Therefore one obtains

(46) |

with . To compute , starting from Eq. (44) one first notices that the product of determinants can be replaced by the same limiting behavior as in the limit given in Eq. (45) as this product is a function of the variables ’s only. One can then perform the remaining multiple sums over ’s in (44) by noticing that they can be replaced by integrals in the limit – as done below in Eq. (50). Finally, one obtains that

(47) |

where is a constant independent of . Therefore, combining these results (46, 47), and using the symmetry of the summand in Eq. (46) under the transformation , one arrives at

(48) |

as announced before (2), where the constant remains however undetermined. It can be computed using the normalization condition of which follows directly from its definition (6)

(49) |

Indeed, as we mentioned it above, in the limit when , the discrete variables which naturally enter into the expression of in Eq. (48) become continuous variables. Therefore the discrete sums in Eq. (48) can be replaced by integrals in the limit, giving

(50) |

If one restricts the integrals over and performs the change of variable one obtains

(51) |

where the integral can now be evaluated using a limiting case of the Selberg integral Fo10 ():