Abstract
We consider three different models of N non-intersecting Brownian motions on a line segment [0,L] 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 N 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 N using a Fermionic path integral technique. We then analyze the asymptotic behavior of this reunion probability for large N 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 \(L\ll O(\sqrt{N})\) and \(L\geq O(\sqrt{N})\). A striking feature of the large N limit of the reunion probability in the three models is that it exhibits a third-order phase transition when the system size L crosses a critical value \(L=L_{c}(N)\sim\sqrt{N}\). 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 L∼L c (N), 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 [L≫L c (N)] and the left [L≪L c (N)] tails. In particular, for model B, we find that the matching between the different regimes corresponding to typical L∼L c (N) and atypical fluctuations in the right tail L≫L c (N) 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 (Schehr et al. in Phys. Rev. Lett. 101:150601, 2008) and (Forrester et al. in Nucl. Phys. B 844:500–526, 2011).
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This research was partially supported by ANR grant 2011-BS04-013-01 WALKMAT and in part by the Indo-French Centre for the Promotion of Advanced Research under Project 4604-3.
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Appendix: Details About the Large Deviation Regime
Appendix: Details About the Large Deviation Regime
In this appendix, we give some details concerning the calculation of \(\tilde{F}_{N}(L)\) in the large deviation regime.
1.1 A.1 Derivation of the Formula Given in Eq. (141)
We first provide a derivation of the formula given in Eq. (141) starting from (61) and (97). Indeed, from the definition of R k (α) one has
and more generally
Therefore, Ω N (α) in Eq. (96) can be rewritten as
which is the formula given in Eq. (141).
1.2 A.2 Derivation of the Formula Given in Eq. (142)
In this appendix we give a detailed derivation, starting from the exact expression of \(\tilde{F}_{N}(L)\) in Eqs. (61), (62), of the asymptotic estimate for \(\ln\tilde{F}_{N}(L)\), valid in the limit \(L \gg\sqrt{2N} \gg1\):
in terms of the variable ξ=π 2/α=2L 2 and the functions G 2N (ξ) given in Eq. (127) and I 2N (ξ) given by
This formula (224) was given in Ref. [31] [see their Eq. (28)] without any detail: that is the purpose of this appendix to fill this gap by providing a detailed derivation of it.
To study the formula for \(\tilde{F}_{N}(L)\) starting from Eqs. (61), (62), we first write the product of the gamma functions in the denominator as
Note that the product of factorials can be written as
Similarly, the product of Γ(k+1/2) in Eq. (226) can be written as
where we have used
Finally, using Eqs. (227) and (228) we write Eq. (226) as
where we have used \(\varGamma(1/2) = \sqrt{\pi}\).
Using the ansatz for R k (α) given in Eq. (116) where we consider \(c_{k}(\alpha) e^{-\pi^{2}/\alpha} \ll1\), with α=π 2/2L 2, we write,
and for k≥1
so that, using the formula in Eq. (230) one obtains that \(\tilde{F}_{N}(L)\) in Eqs. (61), (62) can be written as
We now perform the expansion of this expression (233), considering \(c_{k}(\alpha) e^{-\pi^{2}/\alpha} \ll1\). It is more convenient to expand its logarithm \(\ln\tilde{F}_{N}(L)\) which using the explicit expression for c 1(α) in Eq. (117) together with the expression of c k (α) in terms of G k (ξ) in Eq. (120), can be written as a function of ξ=π 2/α as
where we have used G 1(ξ)=1. It is then possible to write these two sums in Eq. (234) as
Finally, using the following identity satisfied by the polynomials G k (ξ) [30]
which can be shown, for instance, by using their explicit expression of G k (ξ) (127), one obtains finally
as given in the text in Eq. (142).
1.3 A.3 An Integral Representation for I 2N (ξ) in Eq. (143)
We start with the integral representation for G k (ξ) in Eq. (126) to express I 2N (ξ) in Eq. (143) as
Performing an integration by part one obtains
By performing the sum over k in (239) and dropping an N-independent constant term, one finally arrives at the expression given in the text in Eq. (144).
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Schehr, G., Majumdar, S.N., Comtet, A. et al. Reunion Probability of N Vicious Walkers: Typical and Large Fluctuations for Large N . J Stat Phys 150, 491–530 (2013). https://doi.org/10.1007/s10955-012-0614-7
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DOI: https://doi.org/10.1007/s10955-012-0614-7