Carlos Yoong

Anders Thorin

Mathias Legrand

Mathias Legrand Nonsmooth

Nonsmooth modal analysis for a one-dimensional finite bar subject to unilateral contact using the Wave Finite Element Method

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Extended Abstract

The 4 th Joint International Conference on Multibody System Dynamics May 29 -June 2, 2016, Montréal, Canada Nonsmooth modal analysis for a one-dimensional finite bar subject to unilateral contact using the Wave Finite Element Method Carlos Yoong 1 , Anders Thorin 1 and Mathias Legrand 1 1 Department of Mechanical Engineering, McGill University, carlos.yoong@mail.mcgill.ca, anders.thorin@mcgill.ca, mathias.legrand@mcgill.ca This contribution suggests a numerical procedure, involving the Wave Finite Element Method (WFEM), able to perform nonsmooth modal analysis [START_REF] Nonsmooth | Nonlinear modal analysis of mechanical systems with frictionless contact interfaces[END_REF] of a frictionless unilateral contact problem defined on a finite one-dimensional domain, as depicted in Fig 1.

The WFEM is a shock-fitting method which consists in locating and tracking shock waves propagating in a mechanical system [START_REF] Shorr | The Wave Finite Element Method[END_REF]. In this approach, the space domain is discretized into cells of length x D x iC1 x i and time is discretized into intervals of length t D t nC1 t n , where i is the index of the cell interface and n is the index of the time step. Then, a system of equations is derived for the calculation of the state (defined here by the stress and velocity v) of the cells at each time step t n . The simulation of an elastic wave propagating with a given finite speed c is performed by iterating these relations in discrete time with a prescribed time interval t D x=c. This method accurately captures potentially discontinuous wave fronts propagating at a finite speed. For an unforced system without unilateral constraints, the WFEM yields the typical system of equations

Q .nC1/ D AQ .n/ (1)
where Q .n/ D OE .n/ v .n/ > is the state of the system at time step t n . The matrix A 2 R 2N 2N , where N is the number of cells, embeds the dynamics of the system. Using Eq. ( 1), Q .n/ can be directly expressed in terms of the initial conditions

Q .0/ Q .n/ D A n Q .0/ (2)
where A n is the matrix A to the power of n. Consider the free-fixed bar in Fig. 1: it is subject to a unilateral contact constraint. Once the WFEM discretization is performed, the contact constraints are enforced using the concept of floating boundary conditions [START_REF] Shorr | The Wave Finite Element Method[END_REF], which loosely speaking, can be regarded as a conditional switch between free and fixed boundary conditions when a penetration is detected during the iterative process. From Eq. ( 2), two types of matrices A shall then be distinguished: A f for the free-fixed condition (no contact) and A c for fixed-fixed condition (contact). T -periodic solutions are sought using the condition Q .n T / D Q .0/ where n T t D T . The solution is assumed to be composed by r consecutive steps in the free phase and p consecutive steps in the contact phase, leading to the following periodicity condition

Q .n T / D A p c A r f Q .0/ (3)
where n T D r C p. The durations of free and contact phases corresponding to periodic motions are not known a priori; therefore, the values of r and p leading to solutions are unknowns. Also, note that periodic solutions with more than one contact phase per period are not targeted with the proposed formulation. Periodicity condition in Eq. ( 3) simplifies to

A p c A r f I Q .0/ D 0: (4)
The initial condition Q .0/ satisfying Eq. ( 4) is called a "potential solution". Potential solutions are admissible solutions of the formulation if they satisfy the following additional conditions:

COND1: Point of contact must not penetrate the wall during periodic motion: x

.n/ 1 6 d; n D 0; 1; : : : ; r COND2: Point of contact cannot separate from the wall during contact: x

.n/ 1 D d; n D r C 1; r C 2; : : : ; r C p COND3: Contact interaction must be compressive: .n/ 6 0; n D r C 1; r C 2; : : : ; r C p In the above conditions, x

.n/ 1 is the displacement of the contact node at time step t n , d is the initial gap between the bar tip and the rigid wall, and is the contact force.

As seen from Eq. (4), a periodic solution is necessarily an element of the kernel of the matrix S T D A p c A r f I, i.e. Q .0/ 2 ker S T . The dimension m of ker S T depends on the combinations of r and p. A non-trivial solution may exist only if det.S T / D 0, hence only the combinations of r and p that correspond to m > 1 are of interest. Given a basis fe 1 ; e 2 ; : : : ; e m g of ker S T , Q .0/ reads

Q .0/ D ˛1e 1 C ˛2e 2 C : : : C ˛me m (5)
Initial state Q .0/ is completely determined by the unknown coefficients ˛1; ˛2; : : : ; ˛m. The state Q .n/ is calculated at each time step through Eq. ( 2); hence this state can be expressed in terms of the coefficients ˛1; ˛2; : : : ; ˛m. A solution is obtained by calculating the coefficients that satisfy the above necessary conditions. It has been observed that when dim.ker S T / D 1, only one solution satisfies all the conditions. Additionally, by changing r and p, it is possible to find other admissible solutions in the vicinity of an existing solution. A continuum of periodic orbits then emerges: it defines a nonsmooth mode of vibration (NSM) [START_REF] Thorin | Nonsmooth modal analysis: Investigation of a 2-dof spring-mass system subject to an elastic impact law[END_REF]. The displacement x 1 is represented in Fig. 2

Fig. 1 :

 1 Fig. 1: Elastodynamic problem with unilateral contact constraints and free-fixed boundary conditions

  (a) for three solutions belonging to the first NSM (continuation of the first linear modeshape of the bar) whose Frequency-Energy branch is shown in Fig.2(b). The periodic solutions are traveling waves with discontinuous wave fronts interacting with each other and the boundaries.

T c T b T a d Time t Displacement x 1 ( a )

 1a Periodic displacements of the contacting node

Fig. 2 :

 2 Fig.2: First nonsmooth mode of the free-fixed bar subjected to unilateral contact constraints. Energy is normalized with respect to the energy of the grazing solution


Benedikt Stufler
email: benedikt.stufler@ens-lyon.fr

Unlabelled Gibbs partitions

Keywords: Gibbs partitions, random partitions of sets, unlabelled structures 1

Introduction

The study of the evolution of shapes of random ensembles, as the total size becomes large, has a long history, and connections to a variety of fields sucht statistical mechanics, representation theory, and combinatorics are known. A sketch of the history of limit shapes may be found in the work by Erlihson and Granovsky [START_REF] Erlihson | Limit shapes of Gibbs distributions on the set of integer partitions: the expansive case[END_REF] on Gibbs partitions in the expansive case, and we refer the reader to this informative summary and references given therein for an adequate treatment of the historical development.

The term Gibbs partitions was coined by Pitman [START_REF] Pitman | Combinatorial stochastic processes[END_REF] in his comprehensive survey on combinatorial stochastic processes. It describes a model of random partitions of sets, where the collection of classes as well as each individual partition class are endowed with a weighted structure. For example, in simply generated random plane forest, each component is endowed with a tree structure carrying a non-negative weight, and the collection of components carries a linear order. Likewise, Gibbs partitions also encompass various of types of random graphs whose vertex sets are partitioned by their connected components.

Many structures such as classes of graphs may also be viewed up to symmetry. The symmetric group acts in a canonical way on the collection of composite structures over a fixed set, and its orbits are called unlabelled objects. Sampling such an isomorphism class with probability proportional to its weight is the natural unlabelled version of the Gibbs partition model. This encompasses as a special case the important model of random multisets, which has been studied by Bell, Bender, Cameron and Richmond [START_REF] Bell | Asymptotics for the probability of connectedness and the distribution of number of components[END_REF], and which is also encompassed in the setting by Arratia, Barbour and Tavaré [START_REF] Arratia | Logarithmic combinatorial structures: a probabilistic approach[END_REF] and Barbour and Granovsky [START_REF] Barbour | Random combinatorial structures: the convergent case[END_REF]. The important example of forests of unlabelled trees has been considered by Mutafchiev [START_REF] Mutafchiev | The largest tree in certain models of random forests[END_REF]. General unlabelled Gibbs partitions, however, appear to have not received any attention in the literature so far. This is possibly due to the fact that this model of random ensembles is quite involved, as the symmetries of both the inner and outer structures influence its behaviour. This makes it particularly hard to arrive at general results that characterize the asymptotic behaviour for a wide range of species of structures. Nevertheless, it is natural to consider combinatorial objects up symmetry, and to ask, whether similar regimes such as for example the expansive case [START_REF] Erlihson | Limit shapes of Gibbs distributions on the set of integer partitions: the expansive case[END_REF] or the convergent case [START_REF] Stufler | Gibbs partitions: the convergent case[END_REF] may also be found in the unlabelled setting.

For this reason, the present work aims make a first step in this direction, with the hope that this may incite further research. We study a general setting characterized by the formation of a giant component with a stochastically bounded remainder. This phenomenon may for example be observed for uniformly sampled unordered forests of unlabelled trees as the total number of vertices tends to infinity, regardless whether we consider trees that are rooted or unrooted, ordered or unordered. The small fragments are shown to converge in total variation towards a limit object following a Pólya-Boltzmann distribution, a term coined by Bodirsky, Fusy, Kang and Vigerske [START_REF] Bodirsky | Boltzmann samplers, Pólya theory, and cycle pointing[END_REF], who generalized and further developed the theory of Boltzmann samplers initiated in [START_REF] Duchon | Boltzmann samplers for the random generation of combinatorial structures[END_REF][START_REF] Flajolet | Boltzmann sampling of unlabelled structures[END_REF]. Rather than taking a pure generating function viewpoint, our approach is to use the methods from [START_REF] Bodirsky | Boltzmann samplers, Pólya theory, and cycle pointing[END_REF] to reduce each problem to probabilistic questions. This allows us to prove our results in great generality and economically make use of available results for heavy-tailed and subexponential probability distributions [START_REF] Foss | An introduction to heavytailed and subexponential distributions[END_REF][START_REF] Embrechts | Functions of power series[END_REF][START_REF] Embrechts | The asymptotic behaviour of series and power series with positive coefficients[END_REF][START_REF] Chover | Functions of probability measures[END_REF].

The present work is also the logical continuation of [START_REF] Stufler | Gibbs partitions: the convergent case[END_REF], where a gelation phenomenon was observed for labelled Gibbs partitions. The Pólya-Boltzmann sampler framework of [START_REF] Bodirsky | Boltzmann samplers, Pólya theory, and cycle pointing[END_REF] allows us to pursue a similar overall strategy as in [START_REF] Stufler | Gibbs partitions: the convergent case[END_REF], but our proofs are more involved and technical, as we have to consider objects up to symmetry.

The motivation of this particular line of research stems from the study of random graphs from restricted classes. McDiarmid [START_REF] Mcdiarmid | Random graphs on surfaces[END_REF][START_REF]Random graphs from a minor-closed class[END_REF] showed that the small fragments of a random graph from a minor-closed addable class converge toward a Boltzmann Poisson random graph. In this work, McDiarmid poses the question, if a similar behaviour may be observed for unlabelled graphs. As was shown in [START_REF] Stufler | Gibbs partitions: the convergent case[END_REF], an approach via Gibbs partitions and conditioned Galton-Watson trees is possible in the labelled setting. Hence it is natural to ask, whether a similar strategy also works in the unlabelled setting. The present work provides a first piece to the puzzle, and we hope to pursue this question further in future work.

Plan of the paper

In Section 2 we fix notations and recall some background related to Gibbs partitions, combinatorial species, Pólya-Boltzmann distributions and subexponential sequences. Section 3 presents our main results for unlabelled Gibbs partitions. In Section 4 we collect all proofs.

Preliminaries

Notation

We use the notation

N = {1, 2, . . .}, N 0 = {0} ∪ N, [n] = {1, 2, . . . , n}, n ∈ N 0 ,
and let R >0 and R ≥0 denote the sets of positive and non-negative real numbers, respectively. Throughout, we assume that all considered random variables are defined on a common probability space (Ω, F , P). All unspecified limits are taken as n becomes large, possibly along an infinite subset of N.

A function h : R >0 → R >0 is termed slowly varying, if for any fixed t > 0 it holds that

lim x→∞ h(tx) h(x) = 1.
For any power series

f (z) = n f n z n , we let [z n ]f (z) = f n denote the coeffi- cient of z n . A sequence of R-valued random variables (X n ) n≥1 is stochastically bounded, if for each > 0 there is a constant M > 0 with lim sup n→∞ P(|X n | ≥ M ) ≤ .
The total variation distance between two random variables X and Y with values in a countable state space S is defined by

d TV (X, Y ) = sup E⊂S |P(X ∈ E) -P(Y ∈ E)|.

Weighted combinatorial species and cycle index sums

The present section recalls the necessary species-theory following Joyal [START_REF]Une théorie combinatoire des séries formelles[END_REF].

A species of combinatorial structures F ω with non-negative weights is a functor that produces for each finite set U a finite set F[U ] of F-structures and a map

ω U : F[U ] → R ≥0 .
We will often write ω(F ) instead of ω U (F ) for the weight of a structure

F ∈ F[U ].
If no weighting is specified explicitly, we assume that any structure receives weight 1. We refer to the set U as the set of labels or atoms of the structure. For any F-object

F ∈ F[U ] we let |F | := |U | ∈ N 0 denote its size.
The species F is further required to produce for each bijection σ : U → V a corresponding bijection

F[σ] : F[U ] → F[V ]
that preserves the ω-weights. In other words, the following diagram must commute.

F[U ] F [σ] / / ω U # # F[V ] ω V R ≥0
Species are also subject to the usual functoriality requirements: the identity map id U on U gets mapped to the identity map

F[id U ] = id F [U ] on the set F[U ].
For any bijections σ : U → V and τ : V → W the diagram

F[U ] F [σ] / / F [τ σ] # # F[V ] F [τ ] F[W ]
commutes. We further assume that F[U ] ∩ F[V ] = ∅ whenever U = V . This is not much of a restriction, as we may always replace

F[U ] by {U } × F[U ]
for all sets U , to make sure that it is satisfied. Two weighted species F ω and H γ are said to be structurally equivalent or isomorphic, denoted by F ω H γ , if there is a family of weight-preserving bijections (α U : F[U ] → H[U ]) U with U ranging over all finite sets, such the following diagram commutes for each bijection σ : U → V of finite sets.

F[U ] α U F [σ] / / F[V ] α V H[U ] G[σ] / / H[V ]
For any finite set U , the symmetric group S U acts on the set U via

σ.F = F[σ](F )
for all F ∈ F[U ] and σ ∈ S U . A bijection σ with σ.F = F is termed an automorphism of F . We let F[U ] denote the orbits of this group action. All F-objects of an orbit F have the same size and same ω-weight, which we denote by | F | and ω( F ). It will be convenient to use the notation

U (F) = k≥0 U k (F) with U k (F) = F[k].
Formally, an unlabelled F-object is defined as an isomorphism class of Fobjects. We may also identify the unlabelled objects of a given size n with the orbits of the action of the symmetric group on any n-sized set. In particular, the collection of unlabelled F-objects may be identified with the set U (F). By abuse of notation, we treat unlabelled objects as if they were regular F-objects. The power series

Fω (z) = F ∈U (F ) ω( F )z | F | .
is the ordinary generating series of the species.

To any species F we may assign the corresponding species Sym(F) of F-symmetries such that

Sym(F)[U ] = {(F, σ) | F ∈ F[U ], σ ∈ U , σ.F = F }.
In other words, a symmetry is a pair of an F-object and an automorphism. The transport along a bijection γ :

U → V is given by Sym(F)[γ](F, σ) = (F[γ](F ), γσγ -1 ).
For any permutation σ we let σ i denote its number of i-cycles. The cycle index series of a species F is defined as the formal power series

Z F ω (z 1 , z 2 , . . .) = k≥0 (F,σ)∈Sym(F )[k] ω(F ) k! z σ 1 1 • • • z σ k k in
Fω (z) = Z F ω (z, z 2 , z 3 , . . .).

Constructions on species

There are many ways to form species of structures by combining other species. Most prominently, composite structures are formed by partitioning a set and endowing both the partition classes and the collection of all classes with additional weighted structures. Derived structures are regular structures over a set of labels together with a distinguished * -placeholder that does not count as regular atom. We recall the details following classical literature by Joyal [START_REF]Une théorie combinatoire des séries formelles[END_REF] and Bergeron, Labelle and Leroux [START_REF] Bergeron | Combinatorial species and tree-like structures[END_REF].

Composite structures

Let F ω and G ν be combinatorial species with non-negative weights. We assume that

G ν [∅] = ∅. The composition F ω • G ν = (F • G)
µ is a weighted species that describes partitions of finite sets, where each partition class is endowed with a G-structure, and the collection of partition classes carries an F-structure. That is, for each finite set U

(F • G)[U ] = π F[π] × Q∈π G[Q]
with the index π ranging over all unordered partitions of U with non-empty partition classes. In other words, π is a set of non-empty subsets of

U such that U = Q∈π Q and Q ∩ Q = ∅ for all Q, Q ∈ π with Q = Q . The weight of a composite structure (F, (G Q ) Q∈π ) is given by µ(F, (G Q ) Q∈π ) = ω(F ) Q∈π ν(G Q ).
For any bijection σ : U → V , the corresponding transport function

(F • G)[σ] : (F • G)[U ] → (F • G)[V ]
is given as follows. For each element (F, (G

Q ) Q∈π ) ∈ (F • G)[U ] we let π = {σ(Q) | Q ∈ π} denote the corresponding partition of V and set σ : π → π, Q → π(Q).
For each Q ∈ π we let

σ| Q : Q → σ(Q), x → σ(x)
denote the restriction of σ to the class Q. We set

(F • G)[σ](F, (G Q ) Q∈π ) = (F[σ](F ), (G[σ| Q ](G σ -1 (P ) )) P ∈π ).
The cycle index sum of the composition is given by

Z F ω •G ν (z 1 , z 2 , . . .) = Z F ω (Z G ν (z 1 , z 2 , . . .), Z G ν 2 (z 2 , z 4 , . . .), Z G ν 3 (z 3 , z 6 , . . .), . . .).
Here we let ν i denote the weighting that assigns to each G-object G the weight ν(G) i .

For example, the species SET given by SET[U ] = {U } for all U has cycle index sum given by

Z SET (z 1 , z 2 , . . .) = exp ∞ i=1 z i i .
So, given a weighted species G ν , the generating series for multisets of unlabelled G-objects is given by exp

∞ i=1 Gν i (z i ) i .

Derived structures

Let F ω be a weighted species. The derived species (F ) ω is defined as follows.

For each set U we let * U denote a placeholder object not contained in U . For example, we could set * U := U , as no set is allowed to be an element of itself. By abuse of notation, we will usually drop the index and just refer to it as the * -placeholder atom. We set

F [U ] = F[U ∪ { * U }].
The weight of an element F ∈ F [U ] is its ω-weight as an F-structure. Any bijection σ : U → V may canonically be extended to a bijection

σ : U ∪ { * U } → V ∪ { * V },
and we set

F [σ] = F[σ ].
Thus, an F -object with size n is an F-object with size n + 1, since we do not count the * -placeholder. The cycle index sum of (F ) ω is given by the formal derivative

Z (F ) ω (z 1 , z 2 , . . .) = d dz 1 Z F ω (z 1 , z 2 , . . .).

Pólya-Boltzmann distributions for composite structures

Given a weighted species F ω and a parameter y > 0 with 0 < Fω (y) < ∞, we may consider the corresponding Boltzmann probability measure

P F ω ,y ( F ) = Fω (y) -1 y | F | ω( F ), F ∈ U (F).
Likewise, given parameters y 1 , y 2 , . . . ≥ 0 with

0 < Z F ω (y 1 , y 2 , . . .) < ∞,
we may consider the Pólya-Boltzmann distribution

P Z F ω ,(y j ) j (F, σ) = Z F ω (y 1 , y 2 , . . .) -1 ω(F ) k! y σ 1 1 • • • y σ k k for (F, σ) ∈ k≥0 Sym(F)[k].
Note that if we condition a P F ω ,y -distributed random variable on having a fixed size n, then the result gets drawn from U n (F) with probability proportional to its ω-weight. In a way, this is analogous to the fact that simply generated trees (with analytic weights) may be viewed as Galton-Watson trees conditioned on having a fixed number of vertices, and the viewpoint is equally useful in this context. Lemma 2.1 implies the useful fact, that the orbit of the F-object of a P Z F ω ,(y,y 2 ,...) -distributed symmetry follows a P F ω ,y -distribution. This provides a systematic way for sampling Boltzmann distributed structures, as the cycle index sums for constructions on species admit explicit expressions with concrete combinatorial interpretations. In particular for composite structures, the following result is given in Bodirsky, Fusy, Kang and Vigerske [5, Prop. 25] for species without weights, and the generalization to the weighted setting is straight-forward. Lemma 2.2. Let F ω and G ν be weighted species with G[∅] = ∅. Let y > 0 be a parameter with

F ω • G ν (y) = Z F ω (G ν (y), G ν 2 (y 2 ), G ν 3 (y 3 ), . . .) ∈]0, ∞[.
Then the following procedure terminates with an unlabelled (F ω • G ν )-object that follows a P F ω •G ν ,y -distribution.

1. Let (F, σ) be a P Z F ω ,( Gν (y), Gν 2 (y 2 ),...) -distributed F-symmetry. 3. Construct an F • G-object by assigning for each cycle τ and each atom v of τ an identical copy of G τ to v.

Subexponential sequences

Subexponential sequences correspond up to tilting and rescaling to subexponential densities of random variables with values in a lattice, and may be put into the more general context of subexponential distributions [START_REF] Chover | Functions of probability measures[END_REF][START_REF] Embrechts | The asymptotic behaviour of series and power series with positive coefficients[END_REF][START_REF] Foss | An introduction to heavytailed and subexponential distributions[END_REF].

Definition 2.3. Let d ≥ 1 be an integer. A power series g(z) = ∞ n=0 g n z n with non-negative coefficients and radius of convergence ρ > 0 belongs to the class S d , if g n = 0 whenever n is not divisible by d, and

g n g n+d ∼ ρ d , 1 g n i+j=n g i g j ∼ 2g(ρ) < ∞ (2.1)
as n ≡ 0 mod d becomes large.

We are going to make use of the following basic properties of subexponential sequences.

Lemma 2.4 ([13, Theorems 4.8, 4.11, 4.30], [START_REF] Embrechts | Functions of power series[END_REF]). Let g(z) belong to S d with radius of convergence ρ.

1. For each > 0 there is an n 0 > 0 such that for all n ≥ n 0 with n ≡ 0 mod d and each k ≥ 0

[z n ]g(z) k ≤ c( )(g(ρ) + ) k [z n ]g(z).
2. If f (z) is a non-constant power series with non-negative coefficients that is analytic at ρ, then f (g(z)) belongs to S d and

[z n ]f (g(z)) ∼ f (g(ρ))[z n ]g(z), n → ∞, n ≡ 0 mod d.
3. If a n = h(n)n -β ρ -n for some constants ρ > 0, β > 1 and a slowly varying function h, then the series n∈dN a n z n belongs to the class S d .

The following criterion will prove to be useful as well.

Lemma 2.5 ([13, Thm. 4.9]). Let f (z) belong to S 1 with radius of convergence ρ, and g 1 (z), g 2 (z) be power-series with non-negative coefficients.

If [z n ]g 1 (z) [z n ]f (z) → c 1 and [z n ]g 2 (z) [z n ]f (z) → c 2 as n → ∞ with c 1 , c 2 ≥ 0, then [z n ]g 1 (z)g 2 (z) [z n ]f (z) → c 1 g 2 (ρ) + c 2 g 1 (ρ).
If additionally c 1 g 2 (ρ) + c 2 g 1 (ρ) > 0, then g 1 (z)g 2 (z) belongs to S 1 .

Unlabelled Gibbs partitions

Let F ω and G ν be weighted combinatorial species with G[∅] = ∅, so that the weighted composition

(F • G) µ = F ω • G ν
is well-defined. Throughout we assume that [z k ] Fω (z) > 0 for at least one k ≥ 1 and that F ω • G ν (z) is not a polynomial. For each integer n ≥ 0 with

[z n ] F ω • G ν (z) > 0
we may sample a random composite structure

S n = (F n , (G Q ) Q∈πn )
from the set U n (F • G) with probability proportional to its µ-weight.

We are going to study the asymptotic behaviour of the remainder R n when deleting "the" largest component from S n . More specifically, we pick an arbitrary representative of S n and construct R n as follows. We make a uniform choice of a component Q 0 ∈ π n having maximal size, and let F n denote the F -object obtained from the F-object F n by relabeling the Q 0 atom of F n to a * -placeholder.

Thus

F n = F[γ](F n ) ∈ F [π n \ {Q 0 }]
for the bijection γ :

π n → (π n \ {Q 0 }) ∪ { * } with γ(Q 0 ) = * and γ(Q) = Q for Q = Q 0 . This yields an unlabelled F • G-object R n := (F n , (G Q ) Q∈πn\{Q 0 } ) ∈ U (F • G).
We let ρ denote the radius of convergence of the ordinary generating series Gν (z) and suppose throughout that

Z F ω ( Gν (ρ) + , Gν 2 ((ρ + ) 2 ), Gν 3 ((ρ + ) 3 ), . . .) < ∞ (3.1)
for some > 0. Let R be a random F • G-element that follows a Boltzmann distribution

P(R = R) = µ(R)ρ |R| (F ) ω • G ν (ρ) , R ∈ U (F • G).
Theorem 3.1. If the series Gν (z) belongs to the class S d , then

d TV (R n , R) → 0, n → ∞, n ≡ 0 mod d.
The main challenge for verifying Theorem 3.1 is that we consider objects up to symmetry. Lemma 2.2 provides a way of sampling S n as a conditioned Boltzmann-distributed composite structure consisting of an Fsymmetry with identical G-objects dangling from each cycle. The key idea will be that the largest G-object is likely to correspond to a fixpoint of the symmetry. A similar congelation phenomenon was observed for random labelled composite structures sampled from (F ω • G ν )[n] with probability proportional to their weight [START_REF] Stufler | Gibbs partitions: the convergent case[END_REF]Thm.3.1]. Our overall strategy is similar, but treating unlabelled structures is more involved. We require the following enumerative result for the proof of our main theorem. Lemma 3.2. If the series Gν (z) belongs to the class S d , then

[z n ] F ω • G ν (z) ∼ (F ) ω • G ν (ρ)[z n ] Gν (z), n → ∞, n ≡ 0 mod d with (F ) ω • G ν (ρ) = d dz 1 Z F ω ( Gν (ρ), Gν 2 (ρ 2 ), . . .).
If Gν (z) is amendable to singularity analysis, then Lemma 3.2 may also be verified using analytic methods [START_REF] Flajolet | Analytic combinatorics[END_REF]. But we make no assumptions at all about the singularities of Gν (z) on the circle |z| = ρ. We only require that this series belongs to the class S d , which is much more general.

Clearly Theorem 3.1 also implies total variational convergence of the number of components, which has been studied in [START_REF] Bell | Asymptotics for the probability of connectedness and the distribution of number of components[END_REF] for the case of weighted multisets where F ω = SET and each F-object receives weight 1.

Corollary 3.3. Suppose that the series Gν (z) belongs to the class S d . Let c(•) denote the number of components in a composite structure. Then c(S n ) converges towards 1 + c(R) in total variation.

If we condition R n on having a fixed size k < n/2, then the G-object of the largest component gets drawn with probability proportional to its weight from U n-k . And clearly R has with probability tending to 1 size less than n/2. Hence we may rephrase Theorem 3.1 as follows.

Proofs

Before starting with the proofs of our main results, we make an elementary observation.

Lemma 4.1. Let F ω and G ν be weighted species with G ν [∅] = ∅, and let (S, σ) be a random symmetry that follows a P Z F ω •G ν ,(ρ j ) j -distribution for some ρ > 0.

The composite structure of S is of the form (F, (G Q ) Q∈π ) with π a partition of a finite set, F an F-structure on π, and

G Q a G-structure on Q for each Q ∈ π. As σ is an automorphism, it follows that σ : π → π, Q → σ(Q)
is well-defined permutation of the collection π of partition classes. For each i ≥ 1 let X i denote the number of cycles of length i in in the induced permutation σ, Y i = i X i the total number of atoms contained in cycles of length i, and Z i the sum of sizes of all G-objects corresponding to atoms of σ that are contained in cycles of length i. Then

E i≥1 x X i i y Y i i z Z i i = Z F ω (x 1 y 1 Gν (z 1 ρ), x 2 y 2 2 Gν 2 ((z 2 ρ) 2 ), x 3 y 3 3
Gν 3 ((z 3 ρ) 3 ), . . .)

F ω • G ν (ρ)
.

Lemma 4.1 is a minor extension of the proof of the well-known enumerative formula

F ω • G ν (z) = Z F ω ( Gν (z), Gν 2 (z 2 ), . . .)
given for example in [14, Theorem 3 and Section 6] or [4, Proposition 11 of Section 2.3]. Instead of using a single formal variable z in the proof for counting the total size, all involved counting series may be replaced by versions with additional formal variables (x i , y i , z i ) i≥1 , that keep track of the required fine-grained statistics. We do not aim to go through the details.

Roughly speaking, the idea behind this is that symmetries of composite F • G-structures correspond, up to a certain relabelling and cycle composition process, to an F-symmetry, where each cycle τ with length |τ | gets endowed with |τ | identical copies of a G-symmetry. Thus, in the sum

Z F ω (x 1 y 1 Gν (z 1 z), x 2 y 2 2
Gν 2 ((z 2 z) 2 ), x 3 y 3 3 Gν 3 ((z 3 z) 3 ), . . .), the variable z keeps track of the total size, the x i of the number of cycles of length i in the symmetry and consequently the y i of the total mass of theses cycles. The powers (z i z) i are due to the fact each G-object assigned to a cycle with length i gets counted i-times due to the identical copies corresponding to each atom of the cycle.

Proof of Lemma 3.2. Throughout, we let n denote an integer that is divisible by d. We assumed that F ω and G ν are weighted species such that the ordinary generating function Gν (z) belongs to S d . We further assumed by Inequality (3.1) that

Z F ω ( Gν (ρ) + , Gν 2 ((ρ + ) 2 ), Gν 3 ((ρ + ) 3 ), . . .) < ∞ (4.1)
for some > 0, with ρ denoting the radius of convergence of the series Gν (z). We start by constructing a P F ω •G ν ,ρ -distributed composite structure according to Lemma 2.2. Let (F, σ) follow a P Z F ω ,( Gν (ρ), Gν 2 (ρ 2 ),...) -distribution. For each cycle τ of σ let |τ | denote its length and draw a G-object G τ according to a P Gν |τ | ,ρ |τ | -distribution. We construct the F • G-object S by assigning for each cycle τ and each atom v of τ an identical copy of G τ to v. Thus S corresponds to (F, (G v ) v ).

Let f denote the number of fixpoints of the permutation σ, and G 1 , . . . G f the corresponding G-structures. We set g i = |G i | for all i. Let H denote the structure obtained from S by deleting all G-objects that correspond to fixpoints of σ, and let h denote the total size of its remaining G-objects.

Thus |S| = f i=1 g i + h. (4.2)
The (g i ) i are independent, but f and h may very well depend on each other. By Lemma 4.1, their joint probability generating function is given by

E[y f w h ] = Z F ω (y Gν (ρ), Gν 2 (w 2 ρ 2 ), Gν 3 (w 3 ρ 3 ), . . .) F ω • G ν (ρ) . (4.3)
Hence the assumption (4.1) states precisely that the vector (f, h) has finite exponential moments. We are going to show that

P(|S| = n) ∼ E[f ]P(g = n) (4.4)
g denoting the size of a P Gν ,ρ -distributed random G-object. Since Equation(4.3) implies that

E[f ] = d dz 1 Z F ω ( Gν (ρ), Gν 2 (ρ 2 ), Gν 3 (ρ 3 ), . . .) Gν (ρ) F ω • G ν (ρ) , it is clear that Equation (4.4) is equivalent to [z n ] F ω • G ν (z) ∼ (F ) ω • G ν (ρ)[z n ] Gν (z), n → ∞, n ≡ 0 mod d.
We have thus successfully reduced the task of determining asymptotically the coefficients of F ω • G ν (z) to the probabilistic task of verifying (4.4), and may apply available results for subexponential probability distributions. Equation (4.2) implies that

P(|S| = n) = P f i=1 g i + h = n = k≥0 P(f = k)P k i=1 g i + h = n | f = k . (4.5)
Let g denote a random variable that is distributed like the size of a P Gν ,ydistributed random G-object. Given f = k, the (g i ) 1≤i≤k are independent and identically distributed copies of g. Lemma 2.4 implies that for each fixed k it holds that

P k i=1 g i = n | f = k = P k i=1 g i = n ∼ kP(g = n).
As the vector (f, h) has finite exponential moments, it also holds that the conditioned version (h | f = k) has finite exponential moments. It follows from Lemma 2.5 that

P k i=1 g i + h = n | f = k ∼ kP(g = n)
and hence

P k i=1 g i + h = n, f = k ∼ P(f = k)kP(g = n).
Consequently, if we can find a summable sequence (C k ) k≥0 such that

P k i=1 g i + h = n, f = k ≤ C k P(g = n), (4.6)
for all k, then it follows by dominated convergence that

P(|S| = n) = k≥0 P(f = k)P k i=1 g i + h = n | f = k ∼ E[f ]P(g = n).
Thus, in order to show (4.4) it remains to establish Inequality (4.6). By Lemma 2.4 for each > 0 there is an integer x 0 = x 0 ( ) > 0 and a constant c( ) > 0 such that for all integers x ≥ x 0 and each k ≥ 0 it holds that

P k i=1 g i = x ≤ c( )(1 + ) k P(g = x). (4.7)
Clearly we have that

P k i=1 g i + h = n, f = k = P k i=1 g i + h = n, h > n -x 0 , f = k + P k i=1 g i + h = n, h ≤ n -x 0 , f = k . (4.8)
Since h has finite exponential moments, there are constants C, c > 0 such that for all n

P k i=1 g i + h = n, h > n -x 0 , f = k ≤ P(h > n -x 0 ) ≤ C exp(-cn).
We know that g is heavy-tailed because it belongs to S d . Hence it follows that

P k i=1 g i + h = n, h > n -x 0 , f = k = o(P(g = n)) (4.9)
uniformly for all k ≥ 0 as n becomes large. As for the other summand in (4.8), it holds that

P k i=1 g i + h = n, h ≤ n -x 0 , f = k = n-x 0 =0 P(h = , f = k)P k i=1 g i = n - .
Since n -k ≥ x 0 , it follows from Inequality (4.7) that for all > 0

n-x 0 =0 P(h = , f = k)P k i=1 g i = n - ≤ c( ) n =0 P(h = , f = k)(1 + ) k P(g = n -). (4.10)
As the vector (f, h) has finite exponential moments, there is a δ > 0 such that

E[(1 + δ) f (1 + δ) w ] < ∞. (4.11)
Since > 0 was arbitrary, we may choose it small enough such that 0 < < δ. Thus 1 + 1 + δ < 1 and

P k i=1 g i + h = n, h ≤ n -x 0 , f = k ≤ c( ) 1 + 1 + δ k n =0 P(h = , f = k)(1 + δ) k P(g = n -) ≤ c( ) 1 + 1 + δ k n =0 p P(g = n -) (4.12) with p = k≥0 P(h = , f = k)(1 + δ) k satisfying ≥0 p (1 + δ) < ∞
by Inequality (4.11). Hence we may apply Lemma 2.5 to obtain n =0 p P(g = n -) ∼ P(g = n).

So Equation (4.9) and Inequality (4.12) imply that for all k ≥ 0

P k i=1 g i + h = n, f = k ≤ C k P(g = n)
for a summable sequence (C k ) k≥0 . This verifies Inequality (4.6) and hence (4.4) follows by dominated convergence.

Proof of Theorem 3.1. We use the same notation as in the proof of Lemma 3.2, that is, we let S denote a random P F ω •G ν ,ρ -distributed composite structure assembled according to Lemma 2.2 as follows: We sample an F-symmetry (F, σ) following a P Z F ω ,( Gν (ρ), Gν 2 (ρ 2 ),...) -distribution and let f denote the number of fixpoints of σ. We let (G i ) i≥1 denote an independent family of P Gν ,ρdistributed G-objects, of which we match the first f to the fixpoints of σ in any canonical order. For example, we may order the fixpoints according to their labels in {1, . . . , |F|}, but any canonical order will do by exchangability of the G i . Likewise, for each cycle τ of σ with length |τ | ≥ 2 we draw a G-object G τ according to a P Gν |τ | ,ρ |τ | -distribution, and assign to each atom of τ an identical copy of G τ . We let H denote structure obtained from (F, σ) by only attaching the G-objects to atoms of cycles with length at least 2. Then S is fully described by the vector

(H, G 1 , . . . , G f ).
It holds that

|S| = f i=1 g i + h.
with h denoting the number of atoms of H and g i = |G i | for all i. As discussed in Subsection 2.4, the result of conditioning a Boltzmann-distributed object on having a fixed size gets sampled with probability proportional to its weight among all objects of this size. Hence

S n d = (S | |S| = n).
Similarly, the P (F ) ω •G ν ,ρ -distributed F • G-object R may, by virtue of Lemma 2.2, be sampled analogously as follows: We draw an F-symmetry (F , σ) following a P Z (F ) ω ,( Gν (ρ), Gν 2 (ρ 2 ),...) -distribution and let f denote the number of fixpoints of σ . Note that σ is a permutation of the non- * -atoms of F, hence we do not count the place-holder atom. We let (G i ) i≥1 denote a list of independent copies of a P Gν ,ρ -distributed G-object, and match the first f to the fixpoints of σ in a canonical way. For each cycle τ of σ with length |τ | ≥ 2 we draw a G-object G τ according to a P Gν |τ | ,ρ |τ | -distribution, and assign to each atom of τ an identical copy of G τ . We let H denote the pruned structure where only atoms of non-fixpoints of σ receive a G-object. Thus R is fully determined by the vector (H , G 1 , . . . , G f ), and we set g i = |G i | for all i and let h denote the number of atoms in H . If R has size less than n, we let Ŝn denote the result of assigning to the * -placeholder atom a random unlabelled G-structure G * sampled from U n-|R| (G) with probability proportional to its weight. If R ≥ n, we let Ŝn assume some placeholder value Ŝn = . We are going to show that

d TV (S n , Ŝn ) → 0, n → ∞, n ≡ 0 mod d. (4.13)
If R < n/2, then G * is the largest G-object of Ŝn . Since R is almost surely finite, this event takes place with probability tending to 1 as n becomes large. Hence (4.13) implies that

d TV (R n , R) → 0.
Thus verifying (4.13) is sufficient to conclude the proof. If we interpret F as an F-object F * (rather than an F -object), then the permutation σ extends to an F-automorphism σ * of F * such that the *vertex is a fixpoint. The distributions of (F, σ) and (F * , σ * ) differ in the fact that σ always has at least one fixpoint, and that the probability to assume a fixed size is different. However, given integers m, k ≥ 1 it holds that up to relabelling

((F, σ) | f = k, |F| = m) d = ((F * , σ * ) | f = k -1, |F | = m -1). ( 4.14)
This may be verified as follows. The left-hand side gets drawn with probability proportional to its weight from the subset A k ⊂ Sym(F)[m] of all symmetries with k ≥ 1 fixpoints. Likewise, the right-hand side gets drawn with probability proportional to its weight from the subset B k ⊂ Sym(F )[m -1] of symmetries with k fixpoints in total (counting the * -atom). There is a weight-preserving bijection between Sym(F )[m -1] and the symmetries in Sym(F) [m] where the atom m is a fixpoint. It follows that there is a weightpreserving 1 to m correspondence between Sym(F )[m -1] and the set of symmetries in Sym(F)[m] with a distinguished fixpoint. Now, for each symmetry in A k there are precisely k ways to distinguish a fixpoint, hence there is a weight-preserving 1 to km relation between A k and B k . Thus (4.14) holds. Let x 1 , . . . , x k ≥ 1 and r ≥ 0 be given with

x 1 + . . . + x k + r = n.
It follows from (4.14) and the construction of H and H , that

(H | f = k, h = r) d = (H | f = k -1, h = r).
If we condition the left-hand side additionally on g i = x i for all 1 ≤ i ≤ k, then the distribution of H does not change and G i gets drawn from G[x i ] with probability proportional to its ν-weight. Likewise, if we condition the right-hand side additionally on g i = x i for all 1 ≤ i ≤ k -1, then for each i it holds that G i gets drawn from G[x i ] with probability proportional to its weight, and G * gets drawn with probability proportional to its weight among all unlabelled G-objects with n -r - We let g denote a random variable that is distributed like the size of a random G-object with a P Gν ,ρ distribution. Since Gν (z) belongs to S d , it holds that

P(g = n + d) ∼ P(g = n), n → ∞.
This implies that there is a sequence t n of non-negative integers such that t n → ∞ and For each integer m with m > D(y) we also set σ m (y) := {(y 1 , . . . , y j-1 , m -D(y), y j , . . . , y k ) | 1 ≤ j ≤ k}.

Finally, we set

M n := {(k, r, y) | k ≥ 1, r ≥ 0, y ∈ N k-1 , r + D(y) ≤ t n }.
We will show that as n becomes large, it holds uniformly for all (k, r, y) ∈ M n that P(f = k, h = r, (g 1 , . . . , g k ) ∈ σ n-r (y) | g 1 + . . . + g f + h = n) ∼ P(f = k -1, h = r, (g 1 , . . . , g k-1 ) = y). (4.17)

For D(y) + r ≤ t n < n/2, the (g 1 , . . . , g k ) ∈ σ n (y) corresponds to k distinct outcomes, depending on the unique location for the maximum of the g i . Thus the left-hand side in (4.17 Note that kP(f = k, h = r)

P(f = k -1, h = r) = Gν (ρ) (F ) ω • G ν (ρ) F ω • G ν (ρ) = E[f ].
By Lemma 3.2 it holds that P(g 1 + . . .

+ g f + h = n) ∼ E[f ]P(g = n).
Equation (4.16) and D(y) + r ≤ t n yield that P(g = n -D(y) -r) ∼ P(g = n)

uniformly for (k, r, y) ∈ M n . This verifies the asymptotic equality in (4.17). As t n → ∞, it clearly holds that (f + 1, r , (g 1 , . . . , g f )) ∈ M n with probability tending to 1 as n becomes large. Hence it follows from (4.17

Lemma 2 . 1 .

 21 countably infinitely many indeterminates z 1 , z 2 , . . .. The following standard result is given for example by Bergeron, Labelle and Leroux [4, Ch. 2.3] and shows how the ordinary generating series and the cycle index sum of a species are related. For any finite set U and any unlabelled F-object F ∈ F[U ] there are precisely |U |! many symmetries (F, σ) ∈ Sym(F)[U ] such that F belongs to the orbit F . Consequently:

2 .

 2 For each cycle τ of σ let |τ | denote its length and draw a G-object G τ according to a P Gν |τ | ,y |τ | -distribution.

Corollary 3 . 4 .

 34 Suppose that the series Gν (z) belongs to the class S d . If R has size less than n, let Ŝn denote the random unlabelled F • G-object constructed by drawing a G-object G n-|R| from U n-|R| with probability proportional to its weight. If R ≥ n, set Ŝn to some placeholder value . Then d TV (S n , Ŝn ) → 0, n → ∞, n ≡ 0 mod d.

x 1 -

 1 . . . -x k-1 = x k atoms. Thus (S | f = k, h = r, g 1 = x 1 , . . . , g k = x k ) d = ( Ŝn | f = k -1, h = r, g 1 = x 1 , . . . , g k-1 = x k-1).(4.15)

  |P(g = n + y)/P(g = n) -1| = 0. (4.16)Without loss of generality we may assume that t n < n/2 for all n. For any sequence y = (y 1 , . . . , y k-1 ) of positive integers we set D(y) := y 1 + . . . , y k-1 .

  ) divided by the right-hand side equals kP(f = k, h = r)P(g = n -D(y) -r)P(f = k -1, h = r)P(g 1 + . . . + g f + h = n) .

  ) thatP((f, h, (g 1 , . . . , g k )) ∈ {{(k, r)} × σ n-r (y) | (k, r, y) ∈ M n }) → 1as n becomes large. Thus, we have that uniformly for all sets E of n-sized unlabelled F • G-objectsP(S n ∈ E) = P(S ∈ E | g 1 + . . . g f + h = n) = o(1) + (k,r,y)∈Mn P(S ∈ E, (f, h) = (k, r), (g 1 , . . . , g k ) ∈ σ n (y)) P(g 1 + . . . g f + h = n) .The summand for (k, r, y) may be expressed by the productP(S ∈ E | (f, h) = (k, r), (g 1 , . . . , g k ) ∈ σ n (y)) P((f, h) = (k, r), (g 1 , . . . , g k ) ∈ σ n (y) | g 1 + . . . g f + h = n).

The author is supported by the German Research Foundation DFG, STU 679/1-

Equation (4.15) yields that the first factor is equal to

By Equation (4.17), the second factor is asymptotically equivalent to

uniformly for all (k, r, y) ∈ M n as n becomes large. Thus

This completes the proof.


Benedikt Stufler
email: benedikt.stufler@ens-lyon.fr

Gibbs partitions: the convergent case

Keywords: Gibbs partitions, random partitions of sets, random graphs, graph classes, graph limits 1

come

Introduction

The motivation for the present work stems from various areas, starting with enumerative combinatorics. It was conjectured by Bernardi, Noy and Welsh [START_REF] Bernardi | On the growth rate of minor-closed classes of graphs[END_REF] that proper minor-closed addable classes have smooth growth. Such a condition crops up in related contexts, for example, in McDiarmid, Steger and Welsh [START_REF] Mcdiarmid | Random planar graphs[END_REF][START_REF]Random graphs from planar and other addable classes[END_REF]. The conjecture was confirmed by McDiarmid [START_REF]Random graphs from a minor-closed class[END_REF]. One of the methods used is an approach used in Bender, Canfield and Richmond [START_REF] Bender | Coefficients of functional compositions often grow smoothly[END_REF], who proved smoothness for classes of graphs embeddable on any fixed surface. It was established in [START_REF]Random graphs from a minor-closed class[END_REF] furthermore that random graphs from proper minor-closed addable classes typically admit a giant component, and that the remaining fragments converge in total variation toward a limit called the Boltzmann Poisson random graph of the class. Such a behaviour had previously been observed for random planar graphs by McDiarmid [START_REF] Mcdiarmid | Random graphs on surfaces[END_REF]. The enumerative study of minor-closed classes and related classes of graphs has since then received growing attention in the literature, see Noy [START_REF] Noy | Random planar graphs and beyond[END_REF] for a comprehensive survey.

A link from this topic to general models of random partitions can be found in the work by Barbour and Granovsky [START_REF] Barbour | Random combinatorial structures: the convergent case[END_REF]. The authors use a perturbed Stein recursion approach to study the asymptotic behaviour of random partitions satisfying a conditioning relation. Various regimes with differing behaviour are known for this general model of partitions [START_REF] Arratia | Logarithmic combinatorial structures: a probabilistic approach[END_REF], and the "convergent case" setting of [START_REF] Barbour | Random combinatorial structures: the convergent case[END_REF] is characterized by exhibiting a giant component, whose remainder converges toward an almost surely finite limit. It is an interesting observation, that the distribution of the remainder in [START_REF] Barbour | Random combinatorial structures: the convergent case[END_REF] belongs to a 1 INTRODUCTION 2 family encompassing the components of the Boltzmann Poisson random graph constructed by Mc-Diarmid [START_REF]Random graphs from a minor-closed class[END_REF], and only natural, to check whether these results on random partitions can be applied to random graphs. Clearly great care was taken in [START_REF] Barbour | Random combinatorial structures: the convergent case[END_REF] to use only a minimum set of requirements, but too little is known apart from smoothness about the asymptotic number of graphs in an arbitrary proper addable minor-closed classes.

As it is not clear whether these results apply, other options have to be considered. Apart from random partitions that are characterized by a conditioning relation, there is another well-known model that encompasses the component distribution of random graphs from proper addable minor-closed classes: Gibbs partitions. Both families of random structures are quite general and have a non-trivial intersection, but neither contains the other. The term was coined by Pitman [START_REF] Pitman | Combinatorial stochastic processes[END_REF] in his comprehensive survey on combinatorial stochastic processes, and since then further important additions to the theory were made [START_REF] Erlihson | Limit shapes of Gibbs distributions on the set of integer partitions: the expansive case[END_REF]. Gourdon [15,Thm. 1] gave results in a specific setting, where a giant component emerges, and the size of the remainder converges in distribution. He required the exponential generating function of the structures on the components to be amendable to singularity analysis [START_REF] Flajolet | Analytic combinatorics[END_REF]Thm. 1], such that its coefficients are asymptotically close to c(log n) β n -α for some constants c, β and 1 < α < 2. Whenever methods from analytic combinatorics apply, they yield results of great precision, which is impressively demonstrated in the tail-bounds [START_REF] Flajolet | Analytic combinatorics[END_REF]Thm. 2] for the size of the remainder. However, these requirements are much more specific than in the mentioned work [START_REF] Barbour | Random combinatorial structures: the convergent case[END_REF] for partitions with a conditioning relation, and there are known examples of minor-closed addable classes such as random planar graphs [START_REF] Giménez | Asymptotic enumeration and limit laws of planar graphs[END_REF], for which α = 7/2 lies outside of the considered interval.

For these reasons, it is desirable to establish a "convergent case" regime for Gibbs partitions, that is as general as possible, and in which a similar behaviour as in Barbour and Granovsky's setting [START_REF] Barbour | Random combinatorial structures: the convergent case[END_REF] may be observed. In the present work, we consider Gibbs partitions with a subcritical composition scheme, such that the generating series of the structures on the components belongs to the family of subexponential sequences studied in [START_REF] Chover | Functions of probability measures[END_REF][START_REF] Embrechts | The asymptotic behaviour of series and power series with positive coefficients[END_REF][START_REF] Embrechts | Functions of power series[END_REF]. The elements of this family correspond up to tilting and normalizing to subexponential densities of lattice distributed random variables, and hence may be put in the general context of heavy-tailed and subexponential distributions [START_REF] Foss | An introduction to heavy-tailed and subexponential distributions[END_REF]. Our first main result establishes that Gibbs partitions in this setting exhibit a giant component, and the small rest converges in total variation toward a limit structure following a weighted Boltzmann distribution. In order to demonstrate its broad scope and relevance for combinatorial questions, we use the strong ratio property and a number of results related to simply generated trees [START_REF] Janson | Random cutting and records in deterministic and random trees[END_REF], to show how analytic assemblies of arbitrary tree-like combinatorial structures belong to this regime.

We apply our results to small block-stable classes of graphs. For any such class A, we partition the integers into a finite set of shifted lattices of the form a + dZ for 0 a < d, along which the class A grows smoothly. This allows us to characterize precisely when A belongs to the family of smooth graph classes. The uniform n-sized random graph from A is shown to form a giant component with a stochastically bounded remainder. The fragments not contained in the giant component converge to different Boltzmann Poisson random graphs, depending along which lattice we let n tend to infinity. Any proper addable minor-closed class of graphs is small and block-stable, but the converse does not hold. Hence this recovers and generalizes corresponding results by McDiarmid [23, Theorems 1.2, 1.7], who established smooth growth and convergence of the small fragments for proper addable minor-closed classes. Our approach also works in various other settings, for which we provide some examples, including random graphs drawn with probability proportional to weights assigned to their blocks.

Plan of the paper

In Section 2 we fix notations and recall necessary background related to Gibbs partitions, graph classes and subexponential sequences. Section 3 presents our results on Gibbs partitions in the convergent case, and Section 4 discusses their applications to small block-stable graph classes. Section 5 discusses extensions to similar settings. In Section 6 we collect all proofs.

Preliminaries

Notation

Throughout, we set

N = {1, 2, . . .}, N 0 = {0} ∪ N, [n] = {1, 2, . . . , n}, n ∈ N 0 .
We usually assume that all considered random variables are defined on a common probability space (Ω, F , P). All unspecified limits are taken as n becomes large, possibly along an infinite subset of N.

The total variation distance between two random variables X and Y with values in a countable state space S is defined by

d TV (X, Y) = sup E⊂S |P(X ∈ E) -P(Y ∈ E)|. A sequence of R-valued random variables (X n ) n 1 is stochastically bounded, if for each > 0 there is a constant M > 0 with lim sup n→∞ P(|X n | M) .
We let R >0 and R 0 denote the sets of positive and non-negative real numbers, respectively. A function h : R >0 → R >0 is called slowly varying, if

lim x→∞ h(tx) h(x) = 1
for all fixed t > 0. For any power series f(z), we let [z n ]f(z) denote the coefficient of z n .

Weighted combinatorial species and generating functions

Let F ω denote a species of combinatorial structures with non-negative weights in the sense of Joyal [START_REF] Joyal | Une théorie combinatoire des séries formelles[END_REF]. That is, for each finite set U we are given a finite set F[U] of F-structures and a map

ω U : F[U] → R 0 .
Moreover, for each bijection σ : U → V the species F produces a corresponding bijection

F[σ] : F[U] → F[V]
that preserves the ω-weights. This may be expressed by requiring that the diagram

F[U] F[σ] / / ω U # # F[V] ω V R 0
commutes. Species are also subject to the usual functoriality requirements: the identity map id U on U gets mapped to the identity map

F[id U ] = id F[U] on the set F[U]. For any bijections σ : U → V and τ : V → W the diagram F[U] F[σ] / / F[τσ] # # F[V] F[τ] F[W]
commutes. As a last requirement, we also assume that F[U] ∩ F[V] = ∅ whenever U = V. This is not much of a restriction, as we may always replace F[U] by {U} × F[U] for all sets U, to make sure that it is satisfied. Two weighted species F ω and H γ are structurally equivalent or isomorphic, denoted by F ω H γ , if there is a family of weight-preserving bijections (α U : F[U] → H[U]) U with U ranging over all finite sets, such the following diagram commutes for each bijection σ : U → V of finite sets.

F[U] α U F[σ] / / F[V] α V H[U] G[σ] / / H[V]
We will often write ω(F) instead of ω U (F) for the weight of a structure

F ∈ F[U]. For any F-object F ∈ F[U] we let |F| := |U| ∈ N 0
denote its size. It will be convenient to use the notation

U (F) = n 0 F[n].
Here we write F[n] = F[{1, . . . , n}] for all non-negative integers n. This allows us to define the exponential generating series

F ω (z) = F∈U (F) ω(F) z |F| |F|!
as a formal power series with non-negative coefficients. A simple example is the species SET where SET[U] = {U} for each finite set U and each object receives weight 1. Hence SET(z) = exp(z).

Two F-objects F 1 ∈ F[U] and F 2 ∈ F[V] are termed isomorphic, denoted by F 1 F 2 , if there is a bijection σ : U → V such that F[σ](F 1 ) = F 2 .
An unlabelled F-object is formally defined as a maximal class of pairwise isomorphic objects. The unlabelled object corresponding to a given F-object F is also termed its isomorphism type and denoted by F.

We are going to consider probability measures on the collection of all unlabelled F-objects. From a formal viewpoint, this may be slightly problematic, because infinite collections of proper classes are not well-defined objects. But this more of a notational issue that could easily be resolved by working with a fixed set of representatives instead.

Composite and derived structures

Let F ω and G ν be combinatorial species with non-negative weights, such that G ν [∅] = ∅. The composition F ω • G ν = (F • G) µ of the two species describes partitions of finite sets where each partition class is endowed with a G-structure, and the collection of partition classes carries an F-structure. Formally, it is defined by setting for each finite set U

(F • G)[U] = π F[π] × Q∈π G[Q]
with the index π ranging over all unordered partitions of U with non-empty partition classes. That is, π is a set of non-empty subsets of

U such that U = Q∈π Q and Q ∩ Q = ∅ for all Q, Q ∈ π with Q = Q . The weight of a composite structure (F, (G Q ) Q∈π ) is defined by µ(F, (G Q ) Q∈π ) = ω(F) Q∈π ν(G Q ).
For any bijection σ : U → V, the corresponding function

(F • G)[σ] : (F • G)[U] → (F • G)[V] is defined as follows. For each element (F, (G Q ) Q∈π ) ∈ (F • G)[U] we let π = {σ(Q) | Q ∈ π} denote a partition of V and set σ : π → π, Q → π(Q).
For each

Q ∈ π we let σ| Q : Q → σ(Q), x → σ(x)
denote the restriction of σ to the class Q. We set

(F • G)[σ](F, (G Q ) Q∈π ) = (F[σ](F), (G[σ| Q ](G σ -1 (P) )) P∈ π).
The generating series of the composition satisfies [START_REF] Joyal | Une théorie combinatoire des séries formelles[END_REF]Prop. 24]

(F ω • G ν )(z) = F ω (G ν (z)).
A further construction that we are going to use is the derived species (F ) ω defined as follows. For each set U we let * U denote a placeholder object not contained in U. For example, we could define * U = U, as no set is allowed to be an element of itself. We set

F [U] = F[U ∪ { * U }].
The weight of an element F ∈ F [U] is its ω-weight as F-structure. Any bijection σ : U → V may canonically be extended to a bijection

σ : U ∪ { * U } → V ∪ { * V },
and we set

F [σ] = F[σ ].
Thus, an F -object with size n is an F-object with size n + 1, since we do not count the * -placeholder.

The exponential generating series of (F ) ω is given by the formal derivative

(F ) ω (z) = d dz F ω (z).

Kolchin's representation theorem and Boltzmann distributions

Given a weighted species F ω and a parameter y > 0 with 0 < F ω (y) < ∞, we may consider the corresponding Boltzmann probability measure

P F ω ,y (F) = F ω (y) -1 y |F| ω(F)/|F|!, F ∈ U (F).
In a certain sense, Boltzmann measures are invariant under relabelling: The Boltzmann distribution for composite structures admits a useful canonical coupling, which is a combinatorial interpretation of Kolchin's representation theorem [START_REF] Pitman | Combinatorial stochastic processes[END_REF]Thm. 1.2], and also known as the substitution rule for Boltzmann samplers. It is constructed in [START_REF] Bodirsky | Boltzmann samplers, Pólya theory, and cycle pointing[END_REF] for species without weights, and the generalization to the weighted setting is straight-forward.

Lemma 2.2 ([6]

). Let F ω and G ν be weighted species with

G[∅] = ∅. Let x > 0 be a parameter with 0 < F ω (G ν (x)) < ∞ and y := G ν (x) < ∞. If we sample a P F ω ,y -distributed F-object F, and for each 1 i |F| and independent P G ν ,x -distributed G-object G i , then the tupel (F, G 1 , . . . , G |F| ) may be interpreted as an F • G- object S on the disjoint union V = 1 i |F i | [|G i |].
Let σ : V → [|V|] be a uniformly at random sampled bijection. Then the relabelled object

(F • G)[σ](S) ∈ U (F • G) follows a P F ω •G ν ,x -distribution.

Graph classes

A simple finite graphs is a pair

G = (V, E) of a finite set V = V(G) of vertices or labels together with a set E = E(G) of edges E ⊂ {{x, y} | x, y ∈ V, x = y}.
To avoid notational ambiguities we assume additionally that

V ∩ E = ∅. Two vertices x, y ∈ V are adjacent, if {x, y} ∈ E. We also say that y is a neighbour of x. A subgraph of G is a graph H with V(H) ⊂ V(G). We say H is a proper subgraph, if additionally H = G.
We term G connected, if its vertex set is non-empty, and for all x, y ∈ V we can reach y by starting at x and traversing edges. A connected component of G is a connected subgraph H that is maximal with this property. That is, no other connected subgraph of G exists that contains H as a proper subgraph. We say G is 2-connected, if G is connected, has at least 3 vertices, and deleting an arbitrary single vertex does not disconnect the graph. A subgraph H of G is termed a block, if it is either an isolated vertex with no neighbours, or two vertices joined by a single edges whose deletion would increase the number of connected components, or a 2-connected subgraph that is maximal with this property.

For each bijection σ : V → U between the vertex set of G and an arbitrary finite set U we may form the relabelled graph σ.G := (U, {{σ(x), σ(y)} | {x, y} ∈ E(G)}.

Two graphs are termed isomorphic if one is a relabelled version of the other. For any edge e = {x, y} ∈ E(G) we may form a new graph G/e by contracting e. The graph G/e is formed by replacing x and y with a single vertex v x,y that is adjacent to all former neighbours of x and y. A graph H is a minor of a graph G, if there are graphs G 0 , . . . , G t such that G 0 G and G t H and for each G i+1 arises from G i by deleting an edge, contracting an edge, or deleting a vertex. A collection A of graphs that is closed under relabelling is termed a graph class. For notational convenience, we will always assume that A contains the trivial graph whose vertex-set is the empty set. We may interpret A as a combinatorial species by letting, for each finite set V, A[V] ⊂ A denote the finite subset of all graphs in A with vertex set V, and defining

A[σ] : A[V] → A[U], G → σ.G
for each finite set U and bijection σ : V → U. Moreover, we assign weight 1 to each A-object.

We say a graph class A is 1. proper, if there exists a graph that is not contained in A.

2. small, if the radius of convergence of the exponential generating series A(z) is positive.

(z) = zφ(zC (z)) with φ(z) = exp(B (z)). (2.2)
It is well-known that all minor-closed classes are block-stable. To see this, suppose that A is a minor-closed graph class. An excluded minor of A is a graph that does not belong to A, but all its proper minors do. If M denotes the collection of excluded minors of A, then a graph lies in A if and only if none of its minors belongs to M. A moment's though verifies that A is decomposable, if and only if all excluded minors are connected, and addable, if and only if all excluded minors are 2-connected. Graph classes defined by excluding 2-connected minors must be block-stable, because every 2-connected subgraph of a graph G is a subgraph of one of its blocks.

It was shown by Norine, Seymour, Thomas and Wollan [START_REF] Norine | Proper minor-closed families are small[END_REF] that proper minor-closed classes are small. Thus proper addable minor-closed classes are examples of small block-stable classes containing all trees.

Subexponential sequences

We consider power series whose coefficients belong to the family of subexponential sequences studied by Chover, Ney and Wainger [START_REF] Chover | Functions of probability measures[END_REF] and Embrechts [START_REF] Embrechts | The asymptotic behaviour of series and power series with positive coefficients[END_REF]. Up to tilting and rescaling, these sequences correspond to subexponential densities of random variables with values in a lattice. Hence they may be put into the more general context of heavy-tailed and subexponential distributions, for which a comprehensive treatment is given in the book by Foss, Korshunov, and Zachary [START_REF] Foss | An introduction to heavy-tailed and subexponential distributions[END_REF].

Definition 2.3. Let d 1 be an integer. A power series g(z) = ∞ n=0 g n z n with non-negative coefficients and radius of convergence ρ > 0 belongs to the class S d , if g n = 0 whenever n is not divisible by d, and

g n g n+d ∼ ρ d , 1 g n i+j=n g i g j ∼ 2g(ρ) < ∞ (2.3)
as n ≡ 0 mod d becomes large.

The following theorem describes the behaviour of randomly stopped sums.

Theorem 2.4 ([13, Thm. 4.8, 4.30]). If g(z) belongs to S d with radius of convergence ρ, and f(z) is a non-constant power series with non-negative coefficients that is analytic at ρ, then f(g(z)) belongs to S d and

[z n ]f(g(z)) ∼ f (g(ρ))[z n ]g(z), n → ∞, n ≡ 0 mod d.
The broad scope of this setting is illustrated by the following easy observation, which has been noted in various places, see for example [START_REF] Embrechts | Functions of power series[END_REF].

Proposition 2.5. If g n = h(n)n -β ρ -n for some constants ρ > 0, β > 1 and a slowly varying function h, then the series n∈dN g n z n belongs to the class S d .

We will make use of the following criterion related to sums of random variables. Lemma 2.6 ([13, Thm. 4.9]). Let f(z) belong to S 1 with radius of convergence ρ, and g 1 (z), g 2 (z) be powerseries with non-negative coefficients. If

[z n ]g 1 (z) [z n ]f(z) → c 1 , and
[z n ]g 2 (z) [z n ]f(z) → c 2 as n → ∞ with c 1 , c 2 0, then [z n ]g 1 (z)g 2 (z) [z n ]f(z) → c 1 g 2 (ρ) + c 2 g 1 (ρ).
If additionally c 1 g 2 (ρ) + c 2 g 1 (ρ) > 0, then g 1 (z)g 2 (z) belongs to S 1 .

Convergent Gibbs partitions

Suppose that we are given combinatorial species F ω and G ν with G

[∅] = ∅ and [z k ]F ω (z) > 0 for at least one k 1. For each integer n 0 with [z n ](F ω •G ν )(z) > 0 we may sample a random composite structure S n = (F n , (G Q ) Q∈π n ) from the set (F • G)[n]
with probability proportional to its weight. The corresponding random partition π n of the set [n] is termed a Gibbs partition. We assume throughout that (F ω • G ν )(z) is not a polynomial, so that we may study S n as n tends to infinity. We are interested in the behaviour of the remainder R n when deleting "the" largest component from S n . More specifically, we construct R n as follows. We make a uniform choice of a component Q 0 ∈ π n having maximal size, and let F n denote the F -object obtained from the F-object F n by relabeling the Q 0 atom of F n to a * -placeholder. In more formal words, we set

F n = F[γ](F n ) ∈ F [π n \ {Q 0 }] for the bijection γ : π n → (π n \ {Q 0 }) ∪ { * } with γ(Q 0 ) = * and γ(Q) = Q for Q = Q 0 . This yields an F • G-object (F n , (G Q ) Q∈π n \{Q 0 } ) ∈ (F • G)[[n] \ Q 0 ].
We are not interested in the precise content of the underlying set

[n] \ Q 0 . Any (n -|Q 0 |)-sized subset of [n]
is equally likely. Hence we define the unique order-preserving map

σ : [n] \ {Q 0 } → [n -|Q 0 |]
and set

R n = (F • G)[σ](F n , (G Q ) Q∈π n \{Q 0 } ) ∈ U (F • G).
Alternatively, we could have chosen σ uniformly at random, it wouldn't have changed the distribution of the outcome. We let µ denote the weighting on

(F ) ω • G ν , that is, µ(F, (G Q ) Q ) = ω(F) Q ν(G Q ), (F, (G Q ) Q ) ∈ U (F • G).
In a very general setting the remainder R n converges in total variation toward a limit object following a Boltzmann distribution. Theorem 3.1. Suppose that the power series G ν (z) belongs to the class S d with radius of convergence ρ, and that F ω (z) has radius of convergence strictly larger than G ν (ρ). Let R be a random element of the set U (F • G) that follows a Boltzmann distribution

P(R = R) = µ(R) ρ |R| |R|! ((F ) ω • G ν )(ρ)) -1 , R ∈ U (F • G).
Then

d TV (R n , R) → 0, n → ∞, n ≡ 0 mod d. (3.1)
This implies convergence in total variation of the number of components, which has also been studied in [START_REF] Mutafchiev | Local limit theorems for sums of power series distributed random variables and for the number of components in labelled relational structures[END_REF][START_REF] Bell | Asymptotics for the probability of connectedness and the distribution of number of components[END_REF] for the case F ω = SET. We may also verify convergence of moments. Proposition 3.2. Suppose that the assumptions of Theorem 3.1 hold. Let c(•) denote the number of components in a composite structure. Then c(S n ) converges towards 1 + c(R) in total variation and arbitrarily high moments.

Roughly speaking, the following lemma shows that Theorem 3.1 applies, whenever the species G ν is related to structures admitting a tree-like decomposition. This encompasses important families of enumerative series in combinatorics, for which it is not known whether all members fall into the setting of Proposition 2.5. We demonstrate its usefulness in Section 4 with a novel application to small block-stable graph classes. The proof of Lemma 3.3 uses the strong ratio property and a variety of results related to simply generated trees. Lemma 3.3. Let Z(z) = n 1 Z n z n and φ(z) = k 0 ω k z k be power series with non-negative coefficients that are related by the equation

Z(z) = zφ(Z(z)).
Suppose that ω 0 > 0, ω k > 0 for at least one k 2, and let d denote the greatest common divisor of all k with ω k > 0. By [18, Lem. 13.3], we know that Z n = 0 if n -1 is not a multiple of d, and Z n > 0 if n ≡ 1 mod d is large enough. Suppose that Z(z) has non-zero radius of convergence ρ Z . Then

Z -1 n i+j=n+1 Z i Z j ∼ 2Z(ρ Z )/ρ Z , n → ∞, n ≡ 1 mod d.
This implies that the shifted series Z(z)/z belongs to S d , since it was shown in [18, Rem. 7.5] that Z(ρ Z ) < ∞, and in [18, Thm. 18.6, 18.10], that

Z n /Z n+d → ρ d Z and Z 1/n n → 1/ρ Z as n ≡ 1 mod d becomes large.
If the series G ν (z) is periodic with a shift, then different behaviour may occur depending along which lattice we let n tend to infinity.

Applications to random graphs

In the following, we let A denote a small block-stable class of graphs, C ⊂ A its subclass of connected graphs, and B ⊂ A the subclass of all graphs that are 2-connected or consist of two vertices joined by a single edge. To exclude the case where A is the trivial class of all graphs consisting of isolated points, we assume that B is non-empty. Hence we may let d denote the greatest common divisor of all integers k with [z k ] exp(B (z)) > 0. We let ρ > 0 denote the radius of convergence of the exponential generating series A(z) = exp(C(z)). Clearly ρ is also the radius of convergence C(z). Using Lemma 3.3 and the robustness of subexponential sequences against perturbation, we deduce the following enumerative result.

c n c n+d ∼ ρ d , 1 c n i+j=n+1 c i c j ∼ 2C(ρ)/ρ
as n ≡ 1 mod d becomes large.

For each integer i we let SET i denote the restriction of the species SET to objects whose size lies in the lattice i + dZ. For each 0 a < d we let G a denote a random graph from the class A following a P SET a •C,ρ Boltzmann distribution. That is, R a is a random graph from the class A whose number of components lie in the shifted lattice a -1 + dZ, and its distribution is given by

P(G a = G) = ρ |G| |G|!    k 0 k∈a-1+Z C(ρ) k k!    -1 , G ∈ U (SET a-1 • C).
For each integer n ∈ N 0 let A n denote the random graph sampled uniformly from the set A[n] of graphs in A with vertex set [n]. We let frag(A n ) denote the graph obtained by deleting a uniformly at random drawn largest component of A n , and relabelling the rest in a canonical order-preserving way to a set of the form [k] for some k 0. Alternatively, we may relabel by choosing a bijection uniformly at random, it makes no difference for the resulting distribution. Theorem 3.4 yields our main application.

[z n ]A(z) ∼ C a-1 [z n+1-a ]C(z) with C a-1 = k 0 k≡a-1 mod d C(ρ) k /k!.
This allows us to precisely describe under which conditions the graph class A is smooth. Indeed, for d = 1 the coefficients of A(z) behave asymptotically up to a constant factor like those of C(z), and hence grow smoothly. For d 2, the only way for A to be smooth is when

C 0 = . . . = C d-1 .
There is a beautiful reason, why this may never happen. If C i = C i+1 would hold for all i, then we could select a d-th root of unity ζ = 1 and deduce the contradiction

exp(ζC(ρ)) = C 0 + C 1 ζ + . . . + C d-1 ζ d-1 = C 0 (1 + ζ + . . . + ζ d-1 ) = 0.
Hence A cannot be smooth for d 2.

Extensions

Many classes of weighted combinatorial composite structures may be expressed by a subcritical substitution scheme F ω • G ν , such that Lemma 3.3 may be applied either directly, or similarly as in the proof of Theorem 4.1, to show that G ν belongs up to a constant shift to the class S d for some d 1.

We illustrate this with some examples. There are of course many more, but we do not aim to provide an exhaustive list.

Random graphs with block-weights. Random graphs from block-stable classes have a natural generalization to the weighted setting. Suppose that we are given a weighting γ on the species B of all graphs that are two-connected or consist of two vertices joined by a single edge. This yields a weighting on the species of connected graphs C, given by

ν(C) = B γ(B),
with the index B ranging over all blocks of the connected graph C. Here we set ν(•) = 1 for the graph "•" consisting of a single vertex. Likewise we may define a weighting µ on the species G of all graphs in the same way, such that

G µ SET • C ν .
We may consider a random n-vertex graph G µ n drawn from G µ [n] with probability proportional to its weight. This encompasses uniform random graphs from block-stable classes, which correspond precisely to the case where γ(B) ∈ {0, 1} for all blocks B.

The block-decomposition of connected graphs into 2-connected components described for example in Labelle [21, 2.10] can easily be seen to preserve the weights, yielding a weighted version of Equation (2.2):

z(C ) ν (z) = z exp((B ) γ (z(C ) ν (z)).
Hence a straight-forward analogon of Theorem 4.2 also holds for the random graph G µ n . The corresponding proof requires no modification at all. Forests of Galton-Watson trees with a random number of trees. Let (T i ) i 1 be a family of independent copies of a subcritical or critical Galton-Watson tree T with offspring distribution ξ. Let K denote an independent random non-negative integer having finite exponential moments. We may consider a Galton-Watson forest F with a random number of trees

F = (T 1 , . . . , T K ),
f(z) = E[z |F| ], ψ(z) = E[z K ], φ(z) = E[z ξ ], and
Z(z) = E[z |T| ] are related by f(z) = ψ(Z(z)) and Z(z) = zφ(Z(z)).
Obviously Lemma 3.3 applies to Z(z), so we obtain an analogon of Theorem 4.2 that describes the asymptotic behaviour if we condition the forest F to be large and cut down the largest tree.

Proofs

We list the proofs of our results in order of their appearance.

Proofs from Section 3

Proof of Theorem 3.1. In the following, we assume tacitly that n is divisible by d, and large enough such that

[z n ](F ω • G ν )(z) > 0.
The distributions of R n and R are both invariant under relabelling uniformly at random. This implies that conditioned on having a fixed isomorphism type T of an F • G-object with positive µ-weight, each possible labelling of T is equally likely. In particular, it follows that

(R n | R = T ) = (R | R = T ).
Hence it suffices to establish total variational convergence of the isomorphism type of R n to the isomorphism type of R, that is

lim n→∞ d TV ( Rn , R) = 0. (6.1)
Let F denote a random F-object following a P F ω ,G ν (ρ) -distribution, and for each 1 i |F| let G i be an independent P G ν ,ρ distributed G-object. Lemma 2.2 states that, up to relabelling uniformly at random, the tupel

S := (F, G 1 , . . . , G |F| )
follows a P F ω •G ν ,ρ distribution. Consequently, if we condition S on having size n, then it corresponds to an element from (F • G)[n] that is sampled with probability proportional to its weight. That is, setting f = |F| and g i = |G i | for all i, it follows that as unlabelled objects

S n d = (S | g 1 + . . . + g f = n).
The random F • G-object R follows a P (F) ω •G ν ,ρ distribution. Hence we may apply Lemma 2.2 again to identify it up to relabelling with a tuple

R = (F , G 1 , . . . , G |F | ),
where F follows a P (F ) ω ,G ν (ρ) distribution, and the G i are independent and P G ν ,ρ distributed. To simplify notation, we set f := |F | and g i = |G i | for all i.

Let Ŝn denote the composite F • G-structure obtained by sampling a random G-structure G * from G[n -|R|] with probability proportional to its ν-weight, and assigning it to the * -vertex of F . This is only well-defined if

n -|R| > 0 and [z n-|R| ]G ν (z) > 0, (6.2)
otherwise we set Ŝn to some place-holder value. The probability for the event (6.2) tends to 1 as n becomes large, since |R| is almost surely finite and a multiple of d, and [z kd ]G ν (z) > 0 for all sufficiently large k by assumption. We are going to show that as unlabelled F • G-objects

lim n→∞ d TV (S n , Ŝn ) = 0. (6.3)
This implies Equation (6.1), since the probability, that R is the largest component of Ŝn , tends to 1 as n becomes large. Let g denote a random variable that is distributed like the size of a random G-object with a P G ν ,ρ distribution. Since G ν (z) belongs to S d , it holds that

P(g = n + d) ∼ P(g = n), n → ∞.
Consequently, there is a sequence t n of non-negative integers such that t n → ∞ and Let k, x 1 , . . . , x k 1 be integers with x 1 + . . . + x k = n. If we condition on f = k and g i = x i for all 1 i k, then F gets drawn from F[k] with probability proportional to its ω-weight, and likewise G i gets drawn from G[x i ] with probability proportional to its ν-weight for all i. Conditioned on having size k -1, F gets drawn from F[[k -1] ∪ { * }] with probability proportional to its ω-weight. Thus, up to relabeling uniformly at random, In order for Equation (6.4) to hold, we may replace the sequence t n by any other sequence that tends to infinity more slowly. So without loss of generality, we may assume that t n < n/2 for all n, and set

(F | f = k -1) d = (F | f = k). Since x 1 + . . . + x k = n, it follows that as unlabelled F • G-objects (S | f = k, g i = x i , 1 i k) d = ( Ŝn | f = k -1, g i = x i , 1 i k -1). ( 6
M n := {(k, y) | k 1, y ∈ N k-1 , D(y) t n }.
We are going to verify that

P(f = k, (g 1 , . . . , g k ) ∈ σ n (y) | g 1 + . . . + g f = n) ∼ P(f = k -1, (g 1 , . . . , g k-1 ) = y) (6.6)
uniformly for all (k, y) ∈ M n . Since t n < n/2, it holds that given g 1 + . . . g f = n and f = k, the event (g 1 , . . . , g k ) ∈ σ n (y) corresponds to k distinct outcomes, depending on the unique location for the maximum of the g i . Each outcome is equally likely, so the left-hand side in (6.6) divided by the right-hand side equals kP(f = k)P(g = n -D(y))

P(f = k -1)P(g 1 + . . . + g f = n) , with kP(f = k) P(f = k -1) = (F ) ω (G ν (ρ))G ν (ρ) F ω (G ν (ρ)) = E[f].
As D(y) t n , it follows by Equation (6.4) that uniformly for (k, y) ∈ M n P(g = n -D(y)) ∼ P(g = n).

By Theorem 2.4 it holds that

P(g 1 + . . . + g f = n) ∼ E[f]P(g = n),
and (6.6) follows.

To complete the proof, we first note that

lim n→∞ P((f + 1, (g 1 , . . . , g f )) ∈ M n ) = 1. (6.7)
Hence (6.6) yields that with probability tending to 1 as n becomes large

((f, (g 1 , . . . , g f )) | g 1 + . . . g f = n) ∈ {k} × σ n (y) for some (k, y) ∈ M n .
Using Equation (6.5), it follows that uniformly for all sets E of n-sized unlabelled F • G-objects

P(S n ∈ E) = P(S ∈ E | g 1 + . . . g f = n) = o(1) + (k,y)∈M n P(S ∈ E, f = k, (g 1 , . . . , g f ) ∈ σ n (y) | g 1 + . . . g f = n).
For each (k, y) ∈ M n , the corresponding summand may be simplified to

P(S ∈ E, f = k, (g 1 , . . . , g f ) ∈ σ n (y))/P(g 1 + . . . g f = n),
and then expressed as the product

P(S ∈ E | f = k, (g 1 , . . . , g k ) ∈ σ n (y))P(f = k, (g 1 , . . . , g k ) ∈ σ n (y) | g 1 + . . . g f = n).
We treat the two factors separately. For the first, Equation (6.5) yields

P(S ∈ E | f = k, (g 1 , . . . , g k ) ∈ σ n (y)) = P( Ŝn ∈ E | f = k -1, g i = y i , 1 i k -1).
By (6.6) it holds that

P(f = k, (g 1 , . . . , g k ) ∈ σ n (y) | g 1 + . . . g f = n) ∼ P(f = k -1, (g 1 , . . . , g k-1 ) = y)
uniformly for all (k, y) ∈ M n . Using Equation (6.7) it follows that

P(S n ∈ E) = o(1) + (k,y)∈M n (1 + o(1))P( Ŝn ∈ E, f = k -1, (g 1 , . . . , g k-1 ) = y) = o(1) + P( Ŝn ∈ E).
This completes the proof.

Proof of Proposition 3.2. By Theorem 3.1 we need only check the convergence of the moments. With

F ω (z) = ∞ i=0 f i z i , set f(z) = ∞ i=1 i k f i z i . Theorem 2.4 implies that E[c(S n ) k ] = [z n ]f(G ν (z)) [z n ]F ω (G ν (z)) ∼ f (G ν (ρ)) (F ) ω (G ν (ρ)) = E[(c(R) + 1) k ].
Proof of Lemma 3.3. We will tacitly assume that n ≡ 1 mod d. The number Z n is known as the partition function of simply generated trees. That is, the random plane tree T n with distribution given by

P(T n = T ) = Z -1 n v∈T ω d + (v)
for any plane tree T , with d + (v) denoting outdegree of a vertex v, that is, its number of sons. There is a well-known connection between simply generated trees and branching processes [START_REF]Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation[END_REF][START_REF] Drmota | Random trees[END_REF]: Precisely when ρ Z > 0, there is a critical or subcritical Galton-Watson tree T, with

P(|T| = n) = Z n ρ n Z /Z(ρ Z ),
such the simply generated tree T n is distributed like the T conditioned on having |T| = n vertices:

T n d = (T | |T| = n).
Hence in order to verify

Z -1 n i+j=n+1 Z i Z j ∼ 2Z(ρ Z )/ρ Z ,
we need to show that

P(|T| + |T | = n + 1) ∼ 2P(|T| = n), (6.8)
with T denoting an independent copy of T. Let ξ with E[ξ] 1 denote the offspring distribution of the Galton-Watson tree T. Let (ξ i ) i 1 a family of independent copies of ξ, and A classical combinatorial observation, also called the cycle lemma, states that for any sequence x 1 , . . . , x s -1 of integers satisfying s i=1

S n = ξ 1 + . . . +
x i = -r for some r 1, there are precisely r integers 1 u s such that the cyclically shifted sequence

x (u) i = x 1+(i+u) mod s satisfies i=1 x (u) i > r
for all 1 s -1; see for example [START_REF]Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation[END_REF]Lem. 15.3]. Hence

P(|T| = n) = P(ξ 1 + . . . + ξ n = n -1, ξ 1 + . . . + ξ k k for k < n) = 1 n P(S n = n -1).
Likewise, any list d 1 , . . . , d n+1 in N 0 corresponds to the concatenation of the depth-first-search ordered lists of outdegrees of two plane trees with total size n + 1, if and only if

n+1 i=1 d i = n -1 and k i=1 d i k -1 for all k n.
This yields

P(|T| + |T | = n + 1) = P(ξ 1 + . . . + ξ n+1 = n -1, ξ 1 + . . . + ξ k k -1 for k n) = 2 n + 1 P(S n+1 = n -1). Hence P(|T| + |T | = n + 1) P(|T| = n) = 2 n + 1 n P(S n+1 = n -1) P(S n = n -1)
.

By the strong ratio property [START_REF] Kemeny | A probability limit theorem requiring no moments[END_REF], it holds that P(S n+1 = n -1) ∼ P(S n = n -1).

This verifies (6.8) and completes the proof.

Proof of Theorem 3.4. Let F denote a random F-object following a P F ω ,G ν (ρ) -distribution, and for each 1 i |F| let G i be an independent P G ν ,ρ distributed G-object. Then Lemma 2.2 yields that the tupel

S := (F, G 1 , . . . , G |F| )
follows up to relabelling a P F ω •G ν ,ρ distribution. We set f = |F| and g i = |G i | for all i. The Gibbs partition S n is a random structure sampled from (F•G)[n] with probability proportional to its weight. Hence it is distributed like the Boltzmann structure S conditioned on having size n: For n 1, the event f ≡ a mod D has positive probability if and only if the restriction F ω a to objects with size in a + DZ has a non-constant generating function F ω a (z). Let us fix an integer 0 a < D with this property, and suppose that n ≡ am mod d. Then Equations (6.9) and (6.10) imply that Conditioned on having size in a + DZ, the random F-object F follows a P F ω a •G ν ,ρ distribution. Let F a denote a P F ω a ,G ν (ρ) distributed F a -object that is independent from all previously considered random variables. It follows that Thus it remains to check (6.11). Let g be distributed according to the size of a random G-object following a P G ν ,ρ Boltzmann distribution. We may write g = m + dḡ, f a = a + fa D, n = am + nd, with n ∈ N 0 , and ḡ, fa random non-negative integers. We let (ḡ (j) i ) i,j 0 , denote independent copies of ḡ, and set S Since G ν (z)/z m lies in the class S d by assumption, it follows that the probability weight sequence of ḡ lies in S 1 . By Theorem 2.4 it follows that the densities of S This verifies (6.11) and thus completes the proof.

S n d = (S | g 1 + . . . + g f = n). ( 6

Proofs from Section 4

Proof of Theorem 4.1. Equation (2.2) yields that zC (z) = zφ(zC (z))

for the power series φ(z) = exp(B (z)). Hence we may apply Lemma 3.

Proposition 2 . 1 .

 21 Let F follow a P F ω ,y distribution and let N = |F| denote its random size. If we draw a permutation σ : [N] → [N] uniformly at random, then the corresponding relabelled object F[σ](F) is also P F ω ,y distributed.

Theorem 3 . 4 .

 34 Suppose that there is an integer 0 m < d such that G ν (z)/z m belongs to the class S d . Let D = d/ gcd(m, d) and for each 0 a < D, let F ω a denote the restriction of F ω to objects whose size lies in a + DZ. If the exponential generating series F ω a (z) is not constant, thend TV (R n , R(a)) → 0, n → ∞, n ≡ am mod dwith the limit object R(a) following a P (F a ) ω •G ν ,ρ Boltzmann distribution.

Theorem 4 . 1 .

 41 The series C (z) and C(z)/z both lie in the class S d of subexponential sequences with span d. That is C (ρ), C(ρ) < ∞, and the coefficients c n = [z n ]C(z) satisfy

Theorem 4 . 2 .

 42 For each 0 a < d, it holds that d TV (frag(A n ), G a ) → 0 as n ≡ a mod d becomes large. The coefficients of A(z) along the lattice a + dZ belong, after a shift by -a, to the class S d of subexponential sequences with span d. As n ≡ a mod d becomes large, it holds that

Theorem 4 . 3 .

 43 The graph class A is smooth, if and only if d = 1.

  its size. The probability generating functions

  n y≡0 mod d |P(g = n + y)/P(g = n) -1| = 0. (6.4)Indeed, for each > 0 and t > 0 there is a constant N ,t 1 such that for all n N sup0 y t y≡0 mod d |P(g = n + y)/P(g = n) -1|.Setting t n = 1 for n < N 2,1/2 , t n = 2 for N 2,1/2 x < N 2,1/2 + N 3,1/3 ,and in general t n = k for N 2,1/2 + . . . + N k,1/k n N 2,1/2 + . . . + N k+1,1/(k+1) , yields a sequence with the desired properties.

. 5 )

 5 For any sequence y = (y 1 , . . . , y k-1 ) of positive integers with D(y) := y 1 + . . . , y k-1 < n set σ n (y) = {(y 1 , . . . , y j-1 , n -D(y), y j , . . . , y k ) | 1 j k}.

  | g 1 + . . . g f = n, f ≡ a mod D).

  F a , G 1 , . . . , G |F a | ) | g 1 + . . . + g |F a | = n).It follows by Lemma 2.2, that the vector(F a , G 1 , . . . , G |F a | ) has a P F ω a •G ν ,ρBoltzmann distribution, and consequently S n is distributed like the n-sized Gibbs partition for F ω a • G ν . If we can verify thatP(g 1 + . . . + g f a = n) ∼ E[f a ]P(g = n -(a -1)m), n → ∞, n ≡ am mod d,(6.11)then the convergence in total variation of R n toward R(a) follows in an entirely analogous manner as in the proof of Theorem 3.1.

1 +

 1 . . . + g (j) i . Thus P(g 1 + . . . + g f a = n) = P(S (0) a + (S(1)D + Dm/d) + . . . + (S ( fa ) D + Dm/d) = n).

DDD

  +Dm/d belong to S 1 . Applying Theorem 2.4 again yields that the same holds for the randomly stopped sum fa j=1 (S (j) D +Dm/d), with P(S (0) a = n) ∼ aP(ḡ = n) + Dm/d) = x) ∼ DE[ fa ]P(ḡ = n) as n → ∞. Hence Lemma 2+ Dm/d) + . . . + (S ( fa ) D + Dm/d) = n) ∼ (a + DE[ fa ])P(ḡ = n).

c n i+j=n+1 c i c j = 1 →

 1 3 and obtain that the series C (z) belongs to the class S d . That is, the coefficientsx n = [z n ]C (z) = (n + 1)c n+1 satisfy C (ρ) < ∞, x n x n+d ∼ ρ d , 1 x n i+j=n x i x j ∼ 2C (ρ) < ∞,as n ≡ 0 mod d becomes large. It is clear, that this also impliesC(ρ) < ∞ and c n c n+d ∼ ρ d , n → ∞, n ≡ 1 mod d.A characterization of subexponential series given for example in Foss, Korshunov, Zachary[START_REF] Foss | An introduction to heavy-tailed and subexponential distributions[END_REF] Thm. 4.21] states that for any sequence k n → ∞ with k n < n/2 it holds that1 x n i+j=n i,j k n x i x j → 0, n → ∞, n ≡ 0 mod d.As x n /x n+d ∼ ρ d , we may choose a sequence k n that tends to infinity slowly enough such thatlim n→∞ sup 0 y k n y≡0 mod d x n x n+y -ρ y = 0, n ≡ 0 mod d.Without loss of generality we may additionally assume that k n = o(n). Hence 1 2C(ρ)/ρ. as n ≡ 1 mod d becomes large. Thus, the shifted series C(z)/z belongs to the class S d . Proof of Theorem 4.2. The convergence of the small fragments follows directly by Theorem 3.4. The asymptotic expression of [z n ]A(z) follows from the observation that n ≡ a mod d implies [z n ]A(z) = [z n ](SET a • C)(z) ∼ C a-1 [z n-(a-1) ]C(z), similar as in Equation (6.11).

  This expresses the fact the connected components of a graph form a partition of the vertex set of the graph, and any combination of connected graphs from C must lie in A, since A is decomposable.If the graph class A is block-stable, then it is also decomposable. We may consider the subclass B ⊂ C of all graphs in C that are 2-connected or consist of two vertices joined by a single edge. It was noted byHarary and Palmer [16, 1.3.3, 8.7.1], Robinson[START_REF] Robinson | Enumeration of non-separable graphs[END_REF] Thm. 4], and Labelle[21, 2.10] that

		two
	species are related by a canonical isomorphism
	A SET • C.	(2.1)
	zC

3. decomposable, if any graph lies in A if and only if all its connected components do.

4. bridge-addable, if, for each graph G ∈ A and each pair of vertices x, y ∈ V(G) contained in different connected components of G, the graph obtained by adding the edge {x, y} to G also belongs to A.

5. addable, if it is both decomposable and bridge-addable.

6. minor-closed, if for each G ∈ A and each minor H of G it also holds that H ∈ A.

7. block-stable, if it contains the graph consisting of a single vertex, and any graph lies in A if and only if all its blocks do.

8. smooth, if its generating series A(z) has a finite positive radius of convergence and satisfies the ratio test.

If A is decomposable and C ⊂ A denotes the subclass of all connected graphs in A, then the

  ξ n the associated random-walk. Any list d 1 , . . . , d n in N 0 corresponds to the out-degrees of a depthfirst-search ordered list of vertices of a plane tree with size n, if and only if

	n	k
	d i = n -1	and	d i k for all k < n.
	i=1	i=1

  .9)By assumption, there is an integer 0 m < d such that G ν (z)/z m lies in the class S d . In particular, we have g i ≡ m mod d.

for all 1 i f. Recall that D = d/ gcd(m, d). If g 1 + . . . + g f = n, then for all 0 a < D it holds that f ≡ a mod D if and only if n ≡ am mod d. (6.10)

The author is supported by the German Research Foundation DFG, STU 679/1-