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Extreme statistics of non-intersecting Brownian paths. (English) Zbl 1390.60297

Summary: We consider finite collections of \(N\) non-intersecting Brownian paths on the line and on the half-line with both absorbing and reflecting boundary conditions (corresponding to Brownian excursions and reflected Brownian motions) and compute in each case the joint distribution of the maximal height of the top path and the location at which this maximum is attained. The resulting formulas are analogous to the ones obtained in [G. Moreno Flores et al., Commun. Math. Phys. 317, No. 2, 363–380 (2013; Zbl 1257.82117)] for the joint distribution of \(\mathcal{M} =\max_{x\in \mathbb{R}}\{\mathcal{A}_2(x)-x^2\}\) and \(\mathcal{T} =\operatorname{argmax}_{x\in \mathbb{R}}\{\mathcal{A}_2(x)-x^2\}\), where \(\mathcal{A}_2\) is the \(\text{Airy}_2\) process, and we use them to show that in the three cases the joint distribution converges, as \(N\rightarrow \infty \), to the joint distribution of \(\mathcal{M}\) and \(\mathcal{T}\). In the case of non-intersecting Brownian bridges on the line, we also establish small deviation inequalities for the argmax which match the tail behavior of \(\mathcal{T}\). Our proofs are based on the method introduced in [I. Corwin et al., ibid. 317, No. 2, 347–362 (2013; Zbl 1257.82112); A. Borodin et al., Ann. Inst. Henri PoincarĂ©, Probab. Stat. 51, No. 1, 28–58 (2015; Zbl 1357.60012)] for obtaining formulas for the probability that the top line of these line ensembles stays below a given curve, which are given in terms of the Fredholm determinant of certain “path-integral” kernels.

MSC:

60J65 Brownian motion
60B20 Random matrices (probabilistic aspects)
15B52 Random matrices (algebraic aspects)
82C23 Exactly solvable dynamic models in time-dependent statistical mechanics
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