matches Theorem 4 from [3]. In the general conjugacy class case, ϕ may be described implicitly as
follows. For t ≥ 0 and z ∈ [0, 1], let G
t
z = expt P
J j=2
jc
j
z
j −1
− 1, and let ρt be the smallest solution of the equation in z
∈ [0, 1]: G
t
z = z. Then ϕ is defined by ϕt =
1 K
Z
t J
X
j=2
c
j
jρs − ρs
j
ds, 6
where K := P
J j=2
c
j
j − 1. It is a fact that ρt 1 if and only if t t
c
, hence ϕt = t for t ≤ t
c
and ϕt t for t t
c
. As we will see later, ρt is the extinction probability of the Gatson-Watson
process with generating function G
t
z.
1.3 Heuristics
The k-cycle random walk is a simple generalization of the random transpositions random walk on S
n
, for which the phase transition in Theorem 3 was proved in [3]. Observe that any k-cycle x
1
, . . . , x
k
may always be written as the product of k − 1 transpositions: x
1
, . . . , x
k
= x
1
, x
2
. . . x
k −1
, x
k
This suggests that, qualitatively speaking, the k-cycle random walk should behave as “random trans- positions speed up by a factor of k
− 1, and thus one might expect that phase transitions occur at a time that is inversely proportional to k. This is for instance what happens with the mixing time
t
mix
= 1
k n log n
7 for the total variation distance. This was recently proved in [4] and was previously known [17] for
k ≤ 6, the particular case k = 2 being the celebrated Diaconis-Shahshahani theorem [9]. See [14]
and [8] for an excellent introduction to the general theory of mixing times, and [18] in particular for mixing times of random walks on groups. The comparison to mixing times is not fortuitous: it
is explained in [4] how the value of the mixing time is connected to Schramm’s result [19] about Poisson-Dirichlet structure within the giant component. It may therefore come as a surprise that
t
c
= 1[kk − 1] rather than t
c
= 1k although note that, of course, mixing occurs on a different time scale as specified by 7. As it emerges from the proof, the reason for this fact is as follows. We
introduce a coupling of σ
t
, t ≥ 0 with a random hypergraph process H
t
, t ≥ 0 on V = {1, . . . , n},
which is the analogue of the coupling between random transpositions and Erd˝ os-Renyi random
graphs introduced in [3]. As we will see in more details, hypergraphs are graphs where edges or rather hyperedges may connect several vertices at the same time. In this coupling, every time a cycle
x
1
, . . . , x
k
is performed in the random walk, H
t
gains a hyperedge connecting x
1
, . . . , x
k
. This is essentially the same as adding the complete graph K
k
on {x
1
, . . . , x
k
} in the graph H
t
. Thus the degree of a typical vertex grows at a speed which is kk
− 12 faster than in the standard Erd˝os- Renyi random graph. This results in a giant component occurring kk
− 12 faster as well. This explains the formula t
−1 c
= kk − 1, and an easy generalisation leads to 3.
Organisation of the paper: The rest of the paper is organised as follows. We first give the proof of Theorem 1. In the following section we introduce the coupling between
σ
t
, t ≥ 0 and the
random hypergraph process H
t
, t ≥ 0. In cases where the conjugacy class is particularly simple
156
e.g. random k-cycles, a combinatorial treatment analogous to the classical analysis of the Erd˝ os-
Renyi random graph is possible, leading to exact formulae. In cases where the conjugacy class is arbitrary, our method is more probabilistic in nature relying primarily on martingale methods and
the formulae take a different form H
t
is then closer to the Molly and Reed model of random graphs with prescribed degree distribution, [15] and [16]. The proof is thus slightly different in these
two cases respectively dealt with in Section 3 and 4, even though conceptually there are no major differences between the two cases.
2 Emergence of giant cycles in random transpositions
In this section we give a full proof of Theorem 1. As the reader will observe, the proof is really elementary and is based on well-known and easy results on random graphs. Consider the random
graph process G
t
, t ≥ 0 on V = {1, . . . , n} obtained by putting an edge between i and j if the
transposition i, j has occurred prior to time t. Then every edge is independent and has probability p
t
= 1 − e
−t
n 2
, so G
t
is a realisation of the Erd˝ os-Renyi random graph Gn, p
t
. For t
≥ 0 and i ∈ V , let C
i
denote the cycle that contains i. Recall that if C
i
= C
j
then a transposition i, j yields a fragmentation of C
i
= C
j
into two cycles, while if C
i
6= C
j
then the transposition i, j yields a coagulation of C
i
and C
j
. It follows from this observation that every cycle of σ
t
is a subset of one of the connected components of G
t
. Thus let N t be the number of cycles of σ
t
and let ¯ N t
denote the number of components of G
t
. Then we obtain N t
≥ ¯ N t,
t ≥ 0.
8 Now it is a classical and easy fact that the number ¯
N t has a phase transition at time n 2 cor-
responding to the emergence of a giant component at this time. More precisely, let θ c be the
asymptotic fraction of vertices in the giant component at time cn, so θ c is the survival probability
of a Poisson Galton-Watson process with mean offspring 2c in particular θ c = 0 if c 12.
Let c 12 and fix an interval of time [t
1
, t
2
] such that t
2
= cn and t
1
= t
2
− n
3 4
. Our goal will be to prove that a cycle of size at least
δn appears during the interval I = [t
1
, t
2
], where δ = θ c
2
8.
Lemma 5. For any c 12, with t
1
and t
2
as above, ¯
N t
1
− ¯ N t
2
∼ t
2
− t
1
[1 − θ
2
c] in the sense that the ratio of these two quantities tends to 1 in probability as n
→ ∞. Proof. This lemma follows easily from the following observation. The total number of edges that
are added during I is a Poisson random variable with mean t
2
− t
1
. Now, each time an edge is added to G
t
, this changes the number of components by -1 if and only if the two endpoints are in distinct components otherwise the change is 0. Thus if a transposition is applied at time s
∈ [t
1
, t
2
], then given
F
s
where F
s
denotes the σ-field generated by the entire process up to time s, the
conditional probability that it will change the number of components is 1 −
P
i
x
2 i
s, where x
i
s denotes the rescaled component sizes at time s, ordered in decreasing sizes. We claim that this
converges in probability to 1 − θ c
2
uniformly over s ∈ [t
1
, t
2
]. Indeed observe first that this quantity is monotone in s, so it suffices to establish the claim for s = t
1
and s = t
2
. We only treat s = t
2
for simplicity. It is clear that P
i
x
2 i
≥ x
2 1
→ θ c
2
. Moreover, E P
i
x
2 i
→ θ c
2
as the 157
second largest component has rescaled size smaller than β log nn with high probability see, e.g.,
Theorem 2.3.2 in [10]. Hence P
i ≥2
x
2 i
→ 0 in expectation and hence in probability, so the claim is proved. The law of large numbers now concludes the proof: fix
ǫ 0 and consider the event E that P
i
x
2 i
∈ [θ
2
− ǫ, θ
2
+ ǫ] for all s ∈ [t
1
, t
2
], so PE → 1. On this event, the number of times that G
t
changes in which case it changes by -1 is stochastically dominated above and below by a Poisson random variable with mean t
2
− t
1
1 − θ
2
c ± ǫ. The lemma follows easily.
Lemma 6.
E
sup
t ≤cn
|Nt − ¯ N t
|
≤ 4c + 1n
1 2
. Proof. We already know that N t
≥ ¯ N t for all t
≥ 0. It thus suffices to show that the excess number of cycles is never more than 4c + 1n
1 2
in expectation. Call a cycle good if it never experienced a fragmentation and let N
g o
t be the number of good cycles at time t. Call it excess otherwise, and let N
e x
t be the number of such cycles at time t. It is clear that points in a standard cycle are part of the same connected component in G
t
, which does not contain any additional vertex. Thus every good cycle may be associated to one connected component of G
t
in such a way that no two good cycles are associated to the same component of G
t
. Hence N t = N
g o
t + N
e x
t ≤ ¯
N t + N
e x
t, so that |Nt − ¯ N t
| ≤ N
e x
t. Note also that there can never be more than n
1 2
cycles of size greater than n
1 2
. Thus it suffices to count the number N
e x ↓
t of excess cycles of size ≤ n
1 2
: |Nt − ¯
N t | ≤ N
e x ↓
t + n
1 2
. These excess cycles of size
≤ n
1 2
at time t must have been generated by a fragmentation at some time s
≤ t where one of the two pieces was smaller than n
1 2
. But at each step, the probability of making such a fragmentation is smaller than 2n
−12
. Indeed, given the position of the first marker i, there are at most 4n
1 2
possible choices for j which result in a fragmentation where one of the two pieces is of size smaller than n
1 2
. To see this, note that if a transposition i, j is applied to a permutation
σ, and C
i
= C
j
, so σ
k
i = j for some 1 ≤ k ≤ |C|, then the two pieces are precisely given by
σ i, . . . , σ
k −1
i and σ j, . . . , σ
|C|−k−1
j. Thus to obtain two pieces of size k and
|C| − k there are at most two possible choices, which are σ
k
i and σ
−k
i. Thus E
F
↓
cn ≤ cn · 4n
−12
, where F
↓
cn is the total number of fragmentation events where one of the pieces is smaller than n
1 2
by time cn. Since sup
t ≤cn
N
e x ↓
t ≤ F
↓
cn this finishes the proof.
Proof of Theorem 1. Applying Markov’s inequality in Lemma 6, we see that since n
1 2
≪ n
3 4
= t
2
− t
1
, we also have N t
1
− Nt
2
∼ t
2
− t
1
1 − θ
2
c in probability, by Lemma 5. On the other hand, N t changes by -1 in the case of a coalescence and
by +1 in the case of a fragmentations. Hence N t
1
− Nt
2
= Poissont
2
− t
1
− 2FI, where FI is the total number of fragmentations during the interval I. We therefore obtain by the law of large
numbers for Poisson random variables: F I
∼ 1
2 t
2
− t
1
θ c
2
. 158
But observe that if F I is large, it cannot be the case that all cycles are small - otherwise we would very rarely pick i and j in the same cycle. Hence consider the decreasing events E
s
= {τ
δ
s} for s
∈ [t
1
, t
2
] and let E = E
t
2
. On E
s
, the maximal cycle size up to time s is no more than δn. Hence at
the next transposition, the probability of making a fragmentation is no more than δ. We deduce that
on the event E, F I is stochastically dominated by a Poisson random variable with mean t
2
− t
1
δ. Thus, on E
∩ E
′
where PE
′
→ 1, FI ≤ 2δt
2
− t
1
. Since 2δ = θ c
2
4, it follows immediately that PE
→ 0 as n → ∞. This completes the proof.
Remark 7. This proof is partly inspired by the calculations in Lemma 8 of [2].
3 Random hypergraphs and Theorem 3.
We now start the proof of Theorem 3 in the case of random k-cycles. We first review some relevant definitions and results from random hypergraphs.
A hypergraph is a graph where edges can connect several vertices at the same time. Formally:
Definition 1. A hypergraph H = V, E is given by a set V of vertices and a subset E of P V , where
P V denotes the set of all subsets of V . The elements of E are called hyperedges. A d-regular hyper- graph is a hypergraph where all edges connect d vertices, i.e. for all e
∈ E, |e| = d. For a given d
≥ 2 and 0 p 1, we call G
d
n, p the probability distribution on d-regular hyper- graphs on V =
{1, . . . , n} where each hyperedge on d vertices is present independently of the other hyperedges with probability p. Observe that when d = 2 this is just the usual Erd˝
os-Renyi random graph case, since a hyperedge connecting two vertices is nothing else than a usual edge. For basic
facts on Erd˝ os-Renyi random graphs, see e.g. [5].
The notion of a hypertree needs to be carefully formulated in what follows. We start with the d-regular case. The excess e xH of a given d-regular hypergraph H is defined to be
e xH = d − 1h − r
9 where r =
|H| and h is the number of edges in H. Observe that if H is connected then e xH
≥ −1 as can be easily seen by induction on h each new hyperedge adds at most d
− 1 new vertices to H.
Definition 2. We call a connected d-regular hypergraph H a hypertree if e xH = −1.
Likewise if e xH = 0 and H is connected we will say that H is unicyclic and if the excess is positive we will say that the component is complex.
Remark 8. This is the definition used by Karo´ nski and Luczak in [13], but differs from the definition in
their older paper [11] where a hypertree is a connected hypergraph such that removing any hyperedge would make it disconnected.
In the case where H is not necessarily regular, the excess of a connected hypergraph H made up of the hyperedges h
1
, . . . , h
n
is defined to be e xH = P
n i=1
|h
i
| − 1 − |H|, where |h
i
| denotes the size of the hyperedge h
i
and |H| is the cardinality of the vertex set of H. Then exH ≥ −1 and H is said
to be a hypertree if e xH = −1.
159
3.1 Critical point for random hypergraphs
