Random permutations and giant cycles

The other day, Nathanael Berestycki presented some very nice results on random permutations. Start with the identity permutation of {\mathfrak{S}_N} and compose with random transpositions: in short, at time {k} we consider the random permutation {\sigma_k = \tau_k \circ \ldots \tau_{2} \circ \tau_1} where {\tau_i} is a transposition chosen uniformly at random among the {\frac{N(N-1)}{2}} transpositions of {\mathfrak{S}_N}. How long do we have to wait before seeing a permutation that has a cycle of length greater than {\epsilon N}, where {\epsilon>0} is a fixed constant ?

The answer has been known for quite a long time (O. Schramm,2005), but no simple proof was available: the highlight of Nathanael’s talk was a short and elegant proof of the fact that we only have to wait a little bit more than {N/2} steps to see this giant cycle! The first half of the proof is such a beauty that I cannot resist in sharing it here!

1. Visual representation

First, this is more visual to represent a permutation by its cycle decomposition. Consider for example the permutation {\sigma = (1,3,4)(5,6,8,9,10,11)(2)(7) \in \mathfrak{S}_{10}} that sends {1 \rightarrow 3 \rightarrow 4 \rightarrow 1} etc… It can be graphically represented by the following pictures:

cycle structure - visual representation

cycle structure - visual representation

If this permutation is composed with the transposition {(4,6)} , this gives the following cycle structure:

two cycles merge

two cycles merge

If  it is composed with {(8,11)} this gives,

a cycles breaks down into two

a cycles breaks down into two

It is thus pretty clear that the cycle structure of {\sigma_{k+1}} can be obtained from the cycle structure of {\sigma_k} in only 2 ways:

  • or two cycles of {\sigma_k} merge into one,
  • or one cycle of {\sigma_k} breaks into two cycles.

In short, the process {\{\sigma_k\}_k} behaves like a fragmentation-coalescence processes. At the beginning, the cycle structure of {\sigma_0=\text{Id}} is trivial: there are {N} trivial cycles of length one! Then, as times goes by, they begin to merge, and sometimes they also break down. At each step two integers {i} and {j} are chosen uniformly at random (with {i \neq j}): if {i} and {j} belong to the same cycle, this cycle breaks into two parts. Otherwise the cycle that contains {i} merges with the cycle that contains {j}. Of course, since at the beginning all the cycles are extremely small, the probability that a cycles breaks into two part is extremely small: a cycle of size {k} breaks into two part with probability roughly equal to {\Big(\frac{k}{N}\Big)^2}.

2. Comparison with the Erdos-Renyi random Graph

Since the fragmentation part of the process can be neglected at the beginning, this is natural to compare the cycle structure at time {k} with the rangom graph {G(N,p_k)} where {\frac{N(N-1)}{2}p_k = k}: this graph has {N} vertices and roughly {k} edges since each vertex is open with probability {p_k}. A coupling argument makes this idea more precise:

  1. start with {\sigma_0=\textrm{Id}} and the graph {G_0} that has {N} disconnected vertices {\{1,2, \ldots, N\}}
  2. choose two different integers {i \neq j}
  3. define {\sigma_{k+1}=(i,j) \circ \sigma_k}
  4. to define {G_{k+1}}, take {G_k} and join the vertices {i} and {j} (never mind if they were already joined)
  5. go to step {2}.

The fundamental remark is that each cycle of {\sigma_k} is included in a connected component of {G_k}: this is true at the beginning, and this property obviously remains true at each step. The Erdos-Renyi random graph {G(n,p)} is one of the most studied object: it is well-known that for {c<1}, with probability tending to {1}, the size of the largest connected component of {G(N,\frac{c}{N})} is of order {\ln(N)}. Since {p_k \sim \frac{2k}{N^2}}, this immediately shows that for {k \approx \alpha N}, with {\alpha < \frac{1}{2}} the length of the longuest cycle in {\sigma_k} is of order {\ln(N)} or less. This is one half of the result. It remains to work a little bit to show that if we wait a little bit more than {\frac{N}{2}}, the first cycle of order {N} appears.

For a permutation {\sigma \in \mathfrak{S}_N} with cycle decomposition {\sigma = \prod_{j=1}^r c_i} (cycles of length {1} included), one can consider the quantity

\displaystyle  T(\sigma) = \sum_{j=1}^R |c_j|^2.

This is a quantity between {N} (for the identity) and {N^2} that measures in some extends the sizes of the cycles. In order to normalize everything and to take into account that the cycle structure abruptly changes near the critical value {\frac{N}{2}}, one can also observe this quantity on a logarithmic scale:

\displaystyle  s(\sigma) = \frac{ \ln\Big( T(\sigma)/N^2 \Big) }{\ln(N)}.

This quantity {s(\sigma)} lies between {0} and {1}, and evolves quite smoothly with time: look at what it looks like, once the time has been rescaled by a factor {N}:

fluid limit ?

fluid limit ?

The plot represented the evolution of the quantity {s\Big(\sigma(\alpha N)\Big)} between time k=0 and k=\frac{N}{2} for dimensions N=10000,20000,30000.

Question: does the random quantity s(\alpha,N)={s\Big(\sigma(\alpha N)\Big)} converge (in probability) towards a non trivial constant {S(\alpha) \in (0,1)} ? It must be the analogue quantity for the random graph {G(N,\frac{2\alpha}{N})}, so I bet it is a known result …