S2 Appendix. Order statistics for Dirichlet density

S2 Appendix. Order statistics for Dirichlet density Let us introduce the following notation for the order integration ∫

∫

∞

I(dθ1 , ..., dθn ) ≡

dθ1 0

∫

θ1

θn−1

dθ2 ... 0

dθn .

(1)

0

Now the average over the order statistics of the Dirichlet density is defined as m ⟨θ(r) ⟩

∑n ∏n I(dθ1 , ..., dθn ) θrm δ( k=1 θk − 1) k=1 θkβ−1 = . ∑n ∏n I(dθ1 , ..., dθn ) δ( k=1 θk − 1) k=1 θkβ−1

(2)

In the numerator of (2) we change variables as θˆk = rθk (r > 0), multiply both sides by e−r , and then integrate both sides over r ∈ [0, ∞): I(dθ1 , ..., dθn ) θrm

δ(

n ∑

θk − 1)

k=1

n ∏

θkβ−1

× Γ[nβ + m] = I(dθˆ1 , ..., dθˆn ) θˆrm

k=1

n ∏

ˆ θˆkβ−1 e−θk .

(3)

k=1

The denominator of (2) is worked out analogously. Let us now define ∏n ˆ Γ[nβ] I(dθˆ1 , ..., dθˆn ) δ(y − θˆr ) k=1 θˆkβ−1 e−θk χr (y; m) = ∏ n Γ[nβ + m] I(dθˆ1 , ..., dθˆn ) k=1 θˆkβ−1 e−θˆk

(4)

so that the following relation holds ∫ m ⟨θ(r) ⟩=

∞

dy y m χr (y; m).

(5)

0

∏n ˆ This is the equation (9) of the main text. Working out I(dθˆ1 , ..., dθˆn ) k=1 θˆkβ−1 e−θk and I(dθˆ1 , ..., dθˆn ) δ(y − ∏ ˆ n θˆr ) k=1 θˆkβ−1 e−θk in (4) via integration by parts (starting from the last integration in I(dθˆ1 , ..., dθˆn )) we obtain equations (7–9) of the main text. If (n − r) ≫ 1 and r ≫ 1 the behavior of χr (y; m) in equations (7) of the main text is determined by the exponential factor e(n−r) ln φ(y)+r ln(1−φ(y)) . Working it out via the saddle-point method we conclude that asymptotically: χr (y; m) ≃

2 1 Γ[nβ] 1 √ e− 2σ (y−y0 ) , Γ[nβ + m] 2πσ

(6)

(n − r)r 1 , 3 ′ n [ φ (y0 ) ]2

(7)

where y0 and σ are defined as follows n−r = φ(y0 ), n

σ=

where ϕ′ (y) = dφ(y)/dy. Hence we get from (5) and (6, 7): ⟨θ(r) ⟩ =

y0 , nβ

2 ⟨θ(r) ⟩ − ⟨θ(r) ⟩2 =

(8) nβσ − y02 1 = [nβ]2 (nβ + 1) [nβ]2 (nβ + 1)

(

) β(n − r)r 1 2 − y 0 . n2 [φ′ (y0 )]2

(9)

The importance of fluctuations is characterized by 2 ⟨θ(r) ⟩ − ⟨θ(r) ⟩2

⟨θ(r) ⟩2

=

1 nβ + 1

=

1 nβ + 1

(

) β(n − r)r 1 − 1 n2 y02 [φ′ (y0 )]2

(

) β(n − r)r 2 −2β 2y0 Γ [β] y e − 1 , 0 n2

(10) (11)

where we employed φ(y) =

1 Γ[β]

∫

y

dx xβ−1 e−x

(12)

0

This is the equation (8) of the main text. Eq. (11) is a good approximation of equations (7-9) of the main text; see Fig. 1 of the main text.

2 ⟨θ(r) ⟩−⟨θ(r) ⟩2 ⟨θ(r) ⟩2

calculated (exactly) from