Corrigendum for the paper
The geometry of random minimal factorizations of a long cycle via biconditioned bitype random trees
Ann. Henri Lebesgue 1, pp. 149-226, 2018.
Valentin Féray and Igor Kortchemski June 28, 2021
The last display in the proof of Lemma 4.12 is incorrect; the authors are grateful to Victor Dubach for pointing out this mistake. As a consequence, Propositions 4.8, Lemmas 4.12, Lemma 4.13, Corollary 4.14 and Proposition 4.15 are incorrect as stated in the article due to a missing constant. We explain here how to correct them.
Most importantly, we emphasize that the corrections do not affect the proof of Theorem 1.3 (iii).
The correct statements are the following.
PROPOSITION4.8.Assume thatK√nn → ∞and Knn →γ∈[0,1)asn→ ∞. SetD◦n=p
(σ◦n)2Kn. Then
Bnbu(n−K
n)c( ˜Tn) q1 +α2γ·Dn◦
: 0≤u≤1
−→(d) n→∞ e
for a certain constantαγ ≥0. In addition, ifγ >0, then
sup
0≤u≤1
Hnbu(n−K
n)c( ˜Tn) Kn
−u
(P)
n→∞−→ 0 and 1 n max
1≤i≤n`•,ni −→(P)
n→∞ 0.
The last display in the proof of Lemma 4.12 should read
Hb1(n),Bnn−K
n−n−KK n
n+1Hb1(n)D•n D◦n
!
n→∞−→ (W1, X1),
whereW1,X1are independent standard Gaussian random variables. Accordingly, one needs to modify the defini- tion ofBb(n)in the following way (the factorDn•was forgotten in the article):
Bbu(n)= Bnbu(n−Kn)c−n−KK n
n+1Dn•Hbu(n)
D◦n .
The statements of Lemma 4.12 and Lemma 4.13 are then correct with this modified definition ofBb(n). However, the statement of Corollary 4.14 should be modified as follows.
COROLLARY 4.14. Assume that, asn → ∞, we have K√nn → ∞and Knn → γ withγin[0,1). There is a constantαγ ≥0 such that conditionally given the event{Hnn−Kn = Kn+ 1, Bnn−Kn =−1}, the following convergence in distribution holds jointly
Bnbu(n−K
n)c
Dn◦
!
0≤u≤1
−→(d)
n→∞ Xbr+αγWbr, sup
0≤u≤1
Hnbu(n−K
n)c
Kn
−u
n→∞−→ 0.
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Proof. We have
Bnbu(n−Kn)c
D◦n =Bb(n)u +(n−Kn)D•n (Kn+ 1)Dn◦ Hbu(n) Whenγ >0, we have
D•n(n−Kn)
D◦n(Kn+ 1) ∼1−γ γ3/2
σn• σn◦.
In the regimeγ >0, the quantitiesσ•nandσn◦ have positive limits asntends to+∞, so that DDn•◦(n−Kn)
n(Kn+1) tends to a positive limit denoted byαγ. Whenγ= 0, DDn•◦(n−Kn)
n(Kn+1) →α0:= 0. The first part of Corollary 4.14 (convergence of B
n bu(n−Kn)c
Dn◦ ) then follows from Lemma 4.13. The proof of the convergence ofsup0≤u≤1
Hnbu(n−Kn)c Kn −u
is unchanged.
Using the fact thatXbr+αγ·Wbrhas the same distribution asq
1 +α2γ·Xbr, Proposition 4.15 is modified as follows.
PROPOSITION4.15. Assume that K√nn → ∞and Knn → γ ∈[0,1)asn → ∞. There is a constantαγ ≥0 such that the following convergences hold jointly in distribution
Bbu(n−Kn)c(Tn) q1 +α2γ·Dn◦
: 0≤u≤1
−→(d)
n→∞ e, sup
0≤u≤1
Hbu(n−Kn)c(Tn) Kn
−u
n→∞−→ 0.
The multiplicative constantq
1 +α2γ does not affect the proof of Theorem 1.3 (ii).
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