Extremes of N Vicious Walkers for Large N: Application to the Directed Polymer and KPZ Interfaces

Abstract

We compute the joint probability density function (jpdf) P _N (M,τ _M ) of the maximum M and its position τ _M for N non-intersecting Brownian excursions, on the unit time interval, in the large N limit. For N→∞, this jpdf is peaked around $M = \sqrt{2N}$ and τ _M =1/2, while the typical fluctuations behave for large N like $M - \sqrt{2N} \propto s N^{-1/6}$ and τ _M −1/2∝wN ^−1/3 where s and w are correlated random variables. One obtains an explicit expression of the limiting jpdf P(s,w) in terms of the Tracy-Widom distribution for the Gaussian Orthogonal Ensemble (GOE) of Random Matrix Theory and a psi-function for the Hastings-McLeod solution to the Painlevé II equation. Our result yields, up to a rescaling of the random variables s and w, an expression for the jpdf of the maximum and its position for the Airy₂ process minus a parabola. This latter describes the fluctuations in many different physical systems belonging to the Kardar-Parisi-Zhang (KPZ) universality class in 1+1 dimensions. In particular, the marginal probability density function (pdf) P(w) yields, up to a model dependent length scale, the distribution of the endpoint of the directed polymer in a random medium with one free end, at zero temperature. In the large w limit one shows the asymptotic behavior logP(w)∼−w ³/12.

Appendix: Asymptotic Behavior of the Marginal Probability Density Function P(w)

In this appendix, we study the asymptotic behavior of the (marginal) pdf P(w) of the position of the maximum. The starting point of our analysis is the formula given in the text in Eq. (134), which we recall here

(137)

We first obtain the leading behaviors of f(x,w) and f(x,−w) (125) by estimating the small and large ζ behavior, respectively, of Φ ₂(ζ,x).

One can obtain the small ζ behavior of Φ ₂(ζ,x) by analyzing the coupled equations for Φ ₁(ζ,x),Φ ₂(ζ,x) in (112), (113) and using that Φ ₁(−ζ,x)=Φ ₁(ζ,x) while Φ ₂(−ζ,x)=−Φ ₂(ζ,x). One obtains

(138)

(139)

where at this stage the constants C and D remain undetermined. The large w>0 behavior of f(x,w) is then given by

(140)

The constants C and D can then be obtained by extracting the large x behavior of the above expression (140) and matching it with the large w expansion of the expression given in Eq. (104), assuming that the limits w→∞ and x→∞ do commute. When x→∞ the integral over z in Eq. (140) is diverging and it is easy to see that one has

(141)

Therefore, to leading order in x, for x→∞, one obtains from Eqs. (140) and (141)

(142)

while the expression in Eq. (104) yields

(143)

which, by comparison, yields immediately

(144)

The computation of D is slightly more involved but by comparing Eq. (140) in the large x limit one obtains

(145)

Given that q(y)>0, for all y real, one sees immediately on that expression (145) that D>0.

To compute the asymptotic behavior of f(x,−w) for w→∞ we notice that, in this case, the behavior of the integral over ζ in Eq. (125) is instead dominated by the region ζ→∞ where, to leading order in w, Φ ₂(ζ,x) can thus be replaced by its asymptotic behavior given in Eq. (118). Performing the integral over ζ one thus arrives at the expression obtained previously in Eq. (104), with the substitution w→−w from which one has to extract carefully the large w behavior. One obtains

(146)

These asymptotic behaviors in Eqs. (140), (146) suggest to separate the integral over s in Eq. (137) into two parts,

(147)

(148)

(149)

Let us first analyze P ₊(w) in Eq. (148). There, because of the exponential term e ^{−w  x/2} coming from (146) one can perform, in the integral over x, the change of variable z=wx to obtain

(150)

where $\tilde{c}$ can be read from Eqs. (140), (146):

(151)

where we have used the explicit expression of C in Eq. (145). Finally, from Eq. (150) one obtains the large w behavior of P ₊(w), to leading order as

(152)

Let us now analyze P ₋(w) in Eq. (149), whose analysis turns out to be more complicated. We first notice that ${\mathcal{F}}_{1}(s)$ is bounded for s<0 and in addition, its asymptotic behavior for s→−∞ is given by [75, 76]

(153)

so that there exists a constant K such that

(154)

Therefore one has

(155)

For large w, because of the exponential term e ^−wx/2 in Eq. (146), the integral over s in Eq. (155) is dominated by the region of large negative s. On the other hand, for large negative s, the integral over x is also dominated by the region where x is large and negative. Now using the behavior of q(s) (see e.g. Ref. [75])

(156)

one obtains that, for large negative s, one has

(157)

where d>0 is a constant which can be read from Eqs. (140), (146). Using this last estimate (157), one immediately sees that the large w behavior I ₋(w) in Eq. (155) is given by

(158)

Finally, using the estimates in Eqs. (152) and (158) together with the inequality in Eq. (155) one obtains

(159)

as given in the text (135).