# Kernels of critical graph components

This post is motivated by G(N,p), the classical Erdos-Renyi random graph, specifically its critical window, when $p=p(N)=\frac{1}{N}(1+\lambda N^{-1/3})$.

We start with the following observation, which makes no restriction on p. Suppose a component of G(N,p) is a tree. Then, the graph geometry of this component is that of a uniform random tree on the appropriate number of vertices. This is deliberately informal. To be formal, we’d have to say “condition on a particular subset of vertices forming a tree-component” and so on. But the formality is broadly irrelevant, because at the level of metric scaling limits, if we want to describe the structure of a tree component, it doesn’t matter whether it has $\log N$ or $\frac{1}{7}N$ vertices, because in both cases the tree structure is uniform. The only thing that changes is the scaling factor.

In general, when V vertices form a connected component of a graph with E edges, we define the excess to be E-V+1. So the excess is non-negative, and is zero precisely when the component is a tree. I’m reluctant to say that the excess counts the number of cycles in the component, but certainly it quantifies the amount of cyclic structure present. We will sometimes, in a mild abuse of notation, talk about excess edges. But note that for a connected component with positive excess, there is a priori no way to select which edges would be the excess edges. In a graph process, or when there is some underlying exploration of the component, there sometimes might be a canonical way to classify the excess edges, though it’s worth remarking that the risk of size-biasing errors is always extremely high in this sort of situation.

Returning to the random graph process, as so often there are big changes around criticality. In the subcritical regime, the components are small, and most of them, even the largest with high probability, are trees. In the supercritical regime, the giant component has excess $\Theta(N)$, which is qualitatively very different.

It feels like every talk I’ve ever given has begun with an exposition of Aldous’s seminal paper [Al97] giving a distributional scaling limit of the sizes of critical components in the critical window, and a relation between the process on this time-scale and the multiplicative coalescent. And it remains relevant here, because the breadth-first exploration process can also be used to track the number of excess edges.

In a breadth-first exploration, we have a stack of vertices we are waiting to explore. We pick one and look its neighbours restricted to the rest of the graph, that is without the vertices we have already fully explored, and also without the other vertices in the stack. That’s the easiest way to handle the total component size. But we can simultaneously track how many times we would have joined to a neighbour within the stack, which leads to an excess edge, and Aldous derives a joint distributional scaling limit for the sizes of the critical components and their excesses. (Note that in this case, there is a canonical notion of excess edge, but it depends not just on the graph structure, but also on the extra randomness of the ordering within the breadth-first search.)

Roughly speaking, we consider the reflected exploration process, and its scaling limit, which is a reflected parabolically-drifting Brownian motion (though the details of this are not important at this level of exposition, except that it’s a well-behaved non-negative process that hits zero often). The component sizes are given by the widths of the excursions above zero, scaled up in a factor $N^{1/3}$. Then conditional on the shape of the excursion, the excess is Poisson with parameter the area under the excursion, with no rescaling. That is, a critical component has $\Theta(1)$ excess.

So, with Aldous’s result in the background, when we ask about the metric structure of these critical components, we are really asking: “what does a uniformly-chosen connected component with fixed excess look like when the number of vertices grows?”

I’ll try to keep notation light, but let’s say T(n,k) is a uniform choice from connected graphs on n vertices with excess k.

[Note, the separation of N and n is deliberate, because in the critical window, the connected components have size $n = \Theta(N^{2/3})$, so I want to distinguish the two problems.]

In this post, we will mainly address the question: “what does the cycle structure of T(n,k) look like for large n?” When k=0, we have a uniform tree, and the convergence of this to the Brownian CRT is now well-known [CRT2, LeGall]. We hope for results with a similar flavour for positive excess k.

2-cores and kernels

First, we have to give a precise statement of what it means to study just the cycle structure of a connected component. From now on I will assume we are always working with a connected graph.

There are several equivalent definitions of the 2-core C(G) of a graph G:

• When the excess is positive, there are some cycles. The 2-core is the union of all edges which form part of some cycle, and any edges which lie on a path between two edges which both form part of some cycle.
• C(G) is the maximal induced subgraph where all degrees are at least two.
• If you remove all the leaves from the graph, then all the leaves from the remaining graph, and continue, the 2-core is the state you arrive at where there are no leaves.

It’s very helpful to think of the overall structure of the graph as consisting of the 2-core, with pendant trees ‘hanging off’ the 2-core. That is, we can view every vertex of the 2-core as the root of a (possibly size 1) tree. This is particular clear if we remove all the edges of the 2-core from the graph. What remains is a forest, with one tree for each vertex of the 2-core.

In general, the k-core is the maximal induced subgraph where all degrees are at least k. The core is generally taken to be something rather different. For this post (and any immediate sequels) I will never refer to the k-core for k>2, and certainly not to the traditional core. So I write ‘core’ for ‘2-core’.

As you can see in the diagram, the core consists of lots of paths, and topologically, the lengths of these paths are redundant. So we will often consider instead the kernel, K(G), which is constructed by taking the core and contracting all the paths between vertices of degree greater than 2. The resulting graph has minimal degree at least three. So far we’ve made no comment about the simplicity of the original graphs, but certainly the kernel need not be simple. It will regularly have loops and multiple edges. The kernel of the graph and core in the previous diagram is therefore this:

Kernels of critical components

To recap, we can deconstruct a connected graph as follows. It has a kernel, and each edge of the kernel is a path length of some length in the core. The rest of the graph consists of trees hanging off from the core vertices.

For now, we ask about the distribution of the kernel of a T(n,K). You might notice that the case k=1 is slightly awkward, as when the core consists of a single cycle, it’s somewhat ambiguous how to define the kernel. Everything we do is easily fixable for k=1, but rather than carry separate cases, we handle the case $k\ge 2$.

We first observe that fixing k doesn’t confirm the number of vertices or edges in the kernel. For example, both of the following pictures could correspond to k=3:

However, with high probability the kernel is 3-regular, which suddenly makes the previous post relevant. As I said earlier, it can introduce size-biasing errors to add the excess edges one-at-a-time, but these should be constant factor errors, not scaling errors. So imagine the core of a large graph with excess k=2. For the sake of argument, assume the kernel has the dumbbell / handcuffs shape. Now add an extra edge somewhere. It’s asymptotically very unlikely that this is incident to one of the two vertices with degree three in the core. Note it would need to be incident to both to generate the right-hand picture above. Instead, the core will gain two new vertices of degree three.

Roughly equivalently, once the size of the core is fixed (and large) we have to make a uniform choice from connected graphs of this size where almost every vertex has degree 2, and $\Theta(1)$ of the rest have degree 3 or higher. But the sum of the degrees is fixed, because the excess is fixed. If there are n vertices in the core, then there are $\Theta(n)$ more graphs where all the vertices have degree 2 or 3, than graphs where a vertex has degree at least 4. Let’s state this formally.

Proposition: The kernel of a uniform graph with n vertices and excess $k\ge 2$ is, with high probability as $n\rightarrow\infty$, 3-regular.

This proved rather more formally as part of Theorem 7 of [JKLP], essentially as a corollary after some very comprehensive generating function setup; and in [LPW] with a more direct computation.

In the previous post, we introduced the configuration model as a method for constructing regular graphs (or any graphs with fixed degree sequence). We observe that, conditional on the event that the resulting graph is simple, it is in fact uniformly-distributed among simple graphs. When the graph is allowed to be a multigraph, this is no longer true. However, in many circumstances, as remarked in (1.1) of [JKLP], for most applications the configuration model measure on multigraphs is the most natural.

Given a 3-regular labelled multigraph H with 2(k-1) vertices and 3(k-1) edges, and K a uniform choice from the configuration model with these parameters, we have

$\mathbb{P}\left( K \equiv H \right) \propto \left(2^{t(H)} \prod_{e\in E(H)} \mathrm{mult}(e)! \right)^{-1},$

where t(H) is the number of loops in H, and mult(e) the multiplicity of an edge e. This might seem initially counter-intuitive, because it looks we are biasing against graphs with multiple edges, when perhaps our intuition is that because there are more ways to form a set of multiple edges we should bias in favour of it.

I think it’s most helpful to look at a diagram of a multigraph as shown, and ask how to assign stubs to edges. At a vertex with degree three, all stub assignments are different, that is 3!=6 possibilities. At the multiple edge, however, we care which stubs match with which stubs, but we don’t care about the order within the multi-edge. Alternatively, there are three choices of how to divide each vertex’s stubs into (2 for the multi-edge, 1 for the rest), and then two choices for how to match up the multi-edge stubs, ie 18 in total = 36/2, and a discount factor of 2.

We mention this because in fact K(T(n,k)) converges in distribution to this uniform configuration model. Once you know that K(T(n,k)) is with high probability 3-regular, then again it’s probably easiest to think about the core, indeed you might as well condition on its total size and number of degree 3 vertices. It’s then not hard to convince yourself that a uniform choice induces a uniform choice of kernel. Again, let’s state that as a proposition.

Proposition: For any H a 3-regular labelled multigraph H with 2(k-1) vertices and 3(k-1) edges as before,

$\lim_{n\rightarrow\infty}\mathbb{P}\left( K(T(n,k)) \equiv H \right) \propto \left(2^{t(H)} \prod_{e\in E(H)} \mathrm{mult}(e)! \right)^{-1}.$

As we said before, the kernel describes the topology of the core. To reconstruct the graph, we need to know the lengths in the core, and then how to glue pendant trees onto the core. But this final stage depends on k only through the total length of paths in the core. Given that information, it’s a combinatorial problem, and while I’m not claiming it’s easy, it’s essentially the same as for the case with k=1, and is worth treating separately.

It is worth clarifying a couple of things first though. Even the outline of methods above relies on the fact that the size of the core diverges as n grows. Again, the heuristic is that up to size-biasing errors, T(n,k) looks like a uniform tree with some uniformly-chosen extra edges. But distances in T(n,k) scale like $n^{1/2}$ (and thus in critical components of G(N,p) scale like $N^{1/3}$). And the core will be roughly the set of edges on paths between the uniformly-chosen pairs of vertices, and so will also have length $\Theta(n^{1/2})$.

Once you have conditioned on the kernel structure, and the (large) number of internal vertices on paths in the core (ie the length of the core), it is natural that the assignment of the degree-2 vertices to core paths / kernel edges is uniform. A consequence of this is that if you record $(Y_1,\ldots,Y_m)$ the lengths of paths in the core, where m=3(k-1), then

$\frac{(Y_1,\ldots,Y_m)}{\sum Y_i} \stackrel{d}\rightarrow \mathrm{Dirichlet}(1,1,\ldots,1).$

This is stated formally as Corollary 7 b) of [ABG09]. It’s worth noting that this confirms that the lengths of core paths are bounded in probability away from zero after the appropriate rescaling. In seeking a metric scaling limit, this is convenient as it means there’s so danger that two of the degree-3 vertices end up in ‘the same place’ in the scaling limit object.

To recap, the only missing ingredients now to give a complete limiting metric description of T(n,k) are 1) a distributional limit of the total core length; 2) some appropriate description of set of pendant trees conditional on the size of the pendant forest. [ABG09] show the first of these. As remarked before, all the content of the second of these is encoded in the unicyclic k=1 case, which I have written about before, albeit slightly sketchily, here. (Note that in that post we get around size-biasing by counting a slightly different object, namely unicyclic graphs with an identified cyclic edge.)

However, [ABG09] also propose an alternative construction, which you can think of as glueing CRTs directly onto the stubs of the kernel (with the same distribution as before). The proof that this construction works isn’t as painful as one might fear, and allows a lot of the other metric distributional results to be read off as corollaries.

References

[ABG09] – Addario-Berry, Broutin, Goldschmidt – Critical random graphs: limiting constructions and distributional properties

[CRT2] – Aldous – The continuum random tree: II

[Al97] – Aldous – Brownian excursions, critical random graphs and the multiplicative coalescent

[JKLP] – Janson, Knuth, Luczak, Pittel – The birth of the giant component

[LeGall] – Le Gall – Random trees and applications

[LPW] – Luczak, Pittel, Wierman – The structure of a random graph at the point of the phase transition

# Random Mappings for Cycle Deletion

In previous posts here and here, I’ve talked about attempts to describe a cycle deleting process. We amend the dynamics of the standard random graph process by demanding that whenever a cycle is formed in the graph we delete all the edges that lie on the cycle. The aim of this is to prevent the system growing giant components, and perhaps give a system that displays the characteristics of self-organised criticality. In the posts linked to, we discuss the difficulties caused by the fact that the tree structure of components in such a process is not necessarily uniform.

Today we look in the opposite direction. It gives a perfectly reasonable model to take a multiplicative coalescent with quadratic fragmentation (this corresponds to cycle deletion, since there are $O(n^2)$ edges which would give a cycle if added to a tree on n vertices) and a fragmentation kernel corresponding to adding an extra edge to a uniform spanning tree on n vertices then deleting the edges of the unique cycle. The focus of the rest of this post, we consider this fragmentation mechanism, in particular thinking about how we would sample from it most practically. Not least, without going through Prufer codes or some other clever machinery, it is not trivial to sample a uniform spanning tree.

First, we count the number of unicyclic graphs on n labelled vertices. If we know that the vertices on the cycle are $v_1,\ldots,v_k$, then the number of cycles with an identified edge is

$u_1=1,\quad u_k=\frac{k!}{2},\, k\ge 2.$

If we know that the tree coming off the cycle from vertex v_i has size m, say, then each of the possible rooted labelled trees with size m is equally likely. So taking $w_j=j^{j-1}$, the number of rooted trees on j labelled vertices, we get $B_n(u_\bullet,w_\bullet)$ for the number of such unicyclic graphs on [n]. Recall $B_n$ is the nth Bell polynomial, which gives the size of a compound combinatorial structure, where we have some structure on blocks and some other structure within blocks. Then the random partition of [n] given by the tree sizes has the distribution $\text{Gibbs}_n(u_\bullet,w_\bullet)$.

Consider now a related object, the so-called random mapping digraph. What follows is taken from Chapter 9 of Combinatorial Stochastic Processes. We can view any mapping $M_n:[n]\rightarrow[n]$ as a digraph where every vertex has out-degree 1. Each such digraph contains a collection of directed cycles, supported on those elements x for which $M_n^k(x)=x$ for some k. Such an element x is called a cyclic point. Each cyclic point can be viewed as the root of a labelled tree.

In an identical manner to the unicyclic graph, the sizes of these directed trees in the digraph decomposition of a uniform random mapping is distributed as $\text{Gibbs}_n(\bullet !,w_\bullet)$. So this is exactly the same as the cycle deletion kernel, apart from in the probability that the partition has precisely one block. In practice, for large n, the probability of this event is very small in both cases. And if we wanted to sample the cycle deletion kernel exactly, we could choose the trivial partition with some probability p, and otherwise sample from the random mapping kernel, where p is chosen such that

$p+\frac{1-p}{B_n(\bullet !, w_\bullet)}=\frac{1}{B_n(u_\bullet,w_\bullet)}.$

At least we know from the initial definition of a random mapping, that $B_n(\bullet !,w_\bullet)=n^n$. The number of unicyclic graphs with an identified edge is less clear. It turns out that the partition induced by the random mapping has a nice limit, after rescaling, as the lengths of excursions away from 0 in the standard Brownian bridge on [0,1].

The time for a fuller discussion of this sort of phenomenon is in the context of Poisson-Dirichlet distributions, as the above exchangeable partition turns out to be PD(1/2,1/2). However, for now we remark that the jumps of a subordinator give a partition after rescaling. The case of a stable subordinator is particularly convenient, as calculations are made easier by the Levy-Khintchine formula.

A notable example is the stable-1/2 subordinator, which can be realised as the inverse of the local time process at zero of a Brownian motion. The jumps of this process are then the excursion lengths of the original Brownian motion. A calculation involving the tail of the w_j’s indicates that 1/2 is the correct parameter for a subordinator to describe the random mappings. Note that the number of blocks in the partition corresponds to the local time at zero of the Brownian motion. (This is certainly not obvious, but it should at least be intuitively clear why a larger local time roughly indicates more excursions which indicates more blocks.)

So it turns out, after checking some of the technicalities, that it will suffice to show that the rescaled number of blocks in the random mapping partition $\frac{|\Pi_n|}{\sqrt{n}}$ converges to the Raleigh density, which is a size-biased Normal random variable (hence effectively first conditioned to be positive), and which also is the distribution of the local time of the standard Brownian bridge.

After that very approximate description, we conclude by showing that the distribution of the number of blocks does indeed converge as we require. Recall Cayley’s formula $kn^{n-k-1}$ for the number of labelled forests on [n] with a specified set of k roots. We also need to know how many labelled forests there are with any set of roots. Suppose we introduce an extra vertex, labelled 0, and connect it only to the roots of a rooted labelled forest on [n]. This gives a bijection between unlabelled trees on {0,1,…,n} and labelled forests with a specified set of roots on [n]. So we can use Cayley’s original formula to conclude there are $(n+1)^{n-1}$ such forests. We can do a quick sanity check that these are the same, which is equivalent to showing

$\sum_{k=1}^n k n^{-k-1}\binom{n}{k}=\frac{1}{n}(1+\frac{1}{n})^{n-1}.$

This odd way of writing it is well-motivated. The form of the LHS is reminiscent of a generating function, and the additional k suggests taking a derivative. Indeed, the LHS is the derivative

$\frac{d}{dx}(1+x)^n,$

evaluated at $\frac{1}{n}$. This is clearly the same as the RHS.

That said, having established that the random mapping partition is essentially the same, it is computationally more convenient to consider that instead. By the digraph analogy, we again need to count forests with k roots on n vertices, and multiply by the number of permutations of the roots. This gives:

$\mathbb{P}(|\Pi_n|=k)=\frac{kn^{n-k-1}\cdot k! \binom{n}{k}}{n^n}=\frac{k}{n}\prod_{i=1}^{k-1}\left(1-\frac{i}{n}\right).$

Now we can consider the limit. Being a bit casual with notation, we get:

$\lim \mathbb{P}(\frac{|\Pi_n|}{\sqrt{n}}\in dl)\approx \sqrt{n}dl \mathbb{P}(|\Pi_n|=l\sqrt{n}).$

Since the Raleigh distribution has density $l\exp(-\frac12 l^2)dl$, it suffices for this informal verification to check that

$\prod_{i=1}^{l\sqrt{n}}(1-\frac{i}{n})\approx \exp(-\frac12 l^2).$ (*)

We take logs, so the LHS becomes:

$\log(1-\frac{1}{n})+\log(1-\frac{2}{n})+\ldots+\log(1-\frac{l\sqrt{n}}{n}).$

If we view this as a function of l and differentiate, we get

$d(LHS)=\sqrt{n}dl \log (1-\frac{l}{\sqrt{n}})\approx \sqrt{n}dl \left[-\frac{l}{\sqrt{n}}-\frac{l^2}{2n}\right]\approx -ldl.$

When l is zero, the LHS should be zero, so we can obtain the desired result (*) by integrating then taking an exponential.

# Bell Polynomials

Trees with a single cycle

When counting combinatorial objects, it is often the case that we have two types of structure present at different levels. The aim of this post is to introduce the Bell polynomials, which provides the most natural notation for describing this sort of situation, and to mention some of the results that become easier to derive in this framework. This post is based on material and exercises from Chapter 1 of Jim Pitman’s book Combinatorial Stochastic Processes, which is great, and also available online here.

The structures that Bell polynomials enumerate are called composite structures in this account. Rather than give a definition right away, I shall give an example. An object I have been thinking about in the past few weeks are graphs on n vertices containing precisely one cycle. Some of the background for this has been explained in recent posts.

In a recent post on Prufer codes, I gave the classical argument showing that the number of trees on n vertices is $n^{n-2}$. We might consider a unicyclic graph to be a tree with an extra edge. But if we consider the number of ways to add a further vertex to a tree, we get

$n^{n-2}\left[\binom{n}{2}-(n-1)\right]=n^{n-2}\binom{n-1}{2}.$

Obviously, we have overcounted. If the single cycle in a graph has length k, then the graph has been counted exactly k times in this enumeration. But it is not obvious how many graphs have a single cycle of length k.

Instead, we stop worrying about exactly how many of these there are, as there might not be a simple expression anyway. As soon as we start using them in any actual argument, it will be useful to know various properties about the graphs, but probably not exactly how many there are.

Let’s focus on this single cycle of length k say. If we remove the edges of the cycle, we are left with a collection of trees. Why? Well if there was a cycle in the remaining graph, then the original graph would have had at least two cycles. So we have a collection of trees, unsurprisingly called a forest. Remembering that some of the trees may in fact be a single vertex (on the cycle), it is clear that there is a bijection between these trees and the vertices of the cycle in the obvious way. We can think of the graph as a k-cycle, dressed with trees.

Alternatively, once we have specified its size, we can forget about the k-cycle altogether. The graph is precisely defined by a forest of k trees on n vertices, with a specified root in each tree indicating which vertex lies on the cycle, and a permutation specifying the cyclic ordering of the trees. We can write this as

$N_{n,k}=(k-1)!\sum_{(A_1,\ldots,A_k)\in\mathcal{P}^k(n)}a_1^{a_1-1}\cdot\ldots\cdot a_k^{a_k-1},\quad \text{for }a_i=|A_i|,$

where $\mathcal{P}^k(n)$ is the number of partitions of [n] with k blocks. Remember that the blocks in a partition are necessarily unordered. This makes sense in this setting as the cyclic permutation chosen from the (k-1)! possibilities specifies the order on the cycle.

Bell Polynomials

The key point about this description is that there are two types of combinatorial structure present. We have the rooted trees, and also a cyclic ordering of the rooted trees. Bell polynomials generalise this idea. It is helpful to be less specific and think of partitions of [n] into blocks. There are $w_j$ arrangements of any block of size j, and there are $v_k$ ways to arrange the blocks, if there are k of them. Note that we assume $v_k$ is independent of the arrangements within the collection of blocks. So in the previous example, $w_j=j^{j-2}$, and $v_k=(k-1)!$. Pitman denotes these sequences by $v_\bullet,w_\bullet$. Then the (n,k)th partial Bell polynomial, $B_{n,k}(w_\bullet)$ gives the number of divisions into k blocks:

$B_{n,k}(w_\bullet):=\sum_{(A_1,\ldots,A_k)\in\mathcal{P}^k(n)}\prod_{i=1}^k w_{a_i}.$

The total number of arrangements is given by the Bell polynomial

$B_n(v_\bullet,w_\bullet):=\sum_{k=1}^n v_k B_{n,k}(w_\bullet).$

Here are some other examples of Bell polynomials. The Stirling numbers of the first kind $c_{n,k}$ give the number of permutations of [n] with k cycles. Since we don’t want to impose any combinatorial structure on the set of cycles, we don’t need to consider $v_\bullet$, and the number of ways to make a j-cycle from a j-block is $w_j=(j-1)!$, so $c_{n,k}:=B_{n,k}((\bullet-1)!)$. Similarly, the Stirling numbers of the second kind $S_{n,k}$ give the number of permutations of [n] into k blocks. Almost by definition, $S_{n,k}:=B_{n,k}(1^\bullet)$, where $1^\bullet$ is defined to be the sequence containing all 1s.

Applications

So far, this is just a definition that gives an abbreviated description for the sizes of several interesting sets of discrete objects. Having clean notation is always important, but there are further advantages of using Bell polynomials. I don’t want to reproduce the entirety of the chapter I’ve read, so my aim for this final section is to give a very vague outline of why this is a useful formulation.

Bell polynomials can be treated rather nicely via generating functions. The key to this is to take a sum not over partitions, but rather over ordered partitions, which are exactly the same, except now we also care about the order of the blocks. This has the advantage that there is a correspondence between ordered partitions with k blocks and compositions with k terms. If the composition is $n_1+\ldots+n_k=n$, it is clear why there are $\binom{n}{n_1,\ldots,n_k}$ ordered partitions encoding this structure. This multinomial coefficient can be written as a product of factorials of $n_i$s over i, and so we can write:

$B_{n,k}(w_\bullet)=\frac{n!}{k!}\sum_{(n_1,\ldots,n_k)}\prod_{i=1}^k \frac{w_{n_i}}{n_i!}.$

This motivates considering the exponential generating function given by

$w(\xi)=\sum_{j=1}^\infty w_j\frac{\xi_j}{j!},$

as this leads to the neat expressions:

$B_{n,k}(w_\bullet)=n![\xi^n]\frac{w(\xi)^k}{k!},\quad B_n(v_\bullet,w_\bullet)=n![\xi^n]v(w(\xi)).$

The Bell polynomial $B_n(v_\bullet,w_\bullet)$ counts the number of partitions of [n] subject to some extra structure. If we choose uniformly from this set, we get a distribution on this combinatorial object, for which the Bell polynomial provides the normalising constant. If we then ignore the extra structure, the sequences $v_\bullet,w_\bullet$ induce a probability distribution on the set of partitions of n. This distribution is known as a Gibbs partition. It is interesting to consider when and whether it is possible to define a splitting mechanism such that the Gibbs partitions can be coupled to form a fragmentation process. This is the opposite of a coalescence process. Here, we have a sequence of masses, and at each integer time we have rules to determine which mass to pick, and a rule for how to break it into two pieces. It is certainly not the case that for an arbitrary splitting rule and sequences $v_\bullet,w_\bullet$, the one-step fragmentation of the Gibbs partition on n gives the corresponding Gibbs partition on (n-1).

CLT for random permutations

For the final demonstration of the use of Bell polynomials, I am going to sketch the outline of a solution to exercise 1.5.4. which shows that the number of cycles in a uniformly chosen permutation has a CLT. This is not at all obvious, since the number of permutations of [n] with k cycles is given by $B_{n,k}((\bullet-1)!)$ and there is certainly no simple form for this, so the possibility of doing a technical limiting argument seems slim.

For ease of notation, we copy Pitman and write $c_{n,k}:=B_{n,k}((\bullet-1)!)$ as before. First we show exercise 1.2.3. which asserts that

$x(x+1)\ldots(x+(n-1))=\sum_{k=1}^n c_{n,k}x^k.$

We argue combinatorially. The RHS is the number of ways to choose $\sigma\in S_n$ and a colouring of [n] with k colours such that the orbits of $\sigma$ are monochromatic. We prove that the LHS also has this property by induction on the number of vertices. We claim there is a 1-to-(x+n) map from configurations on n vertices to configurations on (n+1) vertices. Given $\sigma\in S_n$ and colouring, for any $a\in[n]$, we construct $\sigma_a\in S_{n+1}$ by $\sigma_a(a)=n+1$, $\sigma_a(n)=\sigma(a)$ and for all other x, $\sigma_a(x)=\sigma(x)$. We give n+1 the same colour as a. This gives us n possibilities. Alternatively, we can map (n+1) to itself and give it any colour we want. This gives us x possibilities. A slightly more careful argument shows that this is indeed a 1-to-(x+n) map, which is exactly what we require.

So the polynomial

$A_n(z)=\sum_{k=0}^nc_{n,k}z^k,$

has n real zeros, which allows us to write

$\frac{c_{n,k}}{A_n(1)}=\mathbb{P}(X_1+\ldots+X_n=k),$

where the Xs are independent but not identically distributed Bernoulli trials. The number of cycles is then given by this sum, and so becomes a simple matter to verify the CLT by checking a that the variances grows appropriately. As both mean and variance are asymptotically log n, we can conclude that:

$\frac{K_n - \log n}{\sqrt{\log n}}\stackrel{d}{\rightarrow} N(0,1).$

In a future post, I want to give a quick outline of section 1.3. which details how the Bell polynomials can be surprisingly useful to find the moments of infinitely divisible distributions.