A proof of Theorem 1.2.4 and Lemma 1.2.5

implies that H(Xn) ≥x or |B| ≥ x. Since Xn conditioned on the event |Xn|= ` is distributed like the uniform P´olya treeA`, it follows by Lemma 1.2.6 below that there are constantsC₁, c₁ >0 such that for all y≥0 andn the probabilityP(H(Xn)≥y) is bounded by

n

X

`=1

P(|Xn|=`)P(H(A`)≥y)≤

n

X

`=1

P(|Xn|=`)C1exp(−c1y<sup>2</sup>/`)

≤C1exp(−c1y²/n).

Moreover, by Propositions 3.1.5 and 3.1.6 we know that there are constantsC₃>0 and 0< γ <1 such that for ally ≥0 andn we have that

P(|B| ≥y| |V|=n)≤C₃n^3/2γ^y.

It follows that there are constants C4, c2 > 0 such that we have uniformly for all x≥√

n

P(Hn≥x)≤C1exp(−c1y²/n) +C3n^3/2γ^x ≤C4exp(−c2x²/n).

This concludes the proof.

3.1.2 A proof of Theorem 1.2.4 and Lemma 1.2.5

with respect to the Gromov-Hausdorff metric. Moreover, there are constantsC, c >0 such that for all n and x≥0 we it holds that

P(D(Tn)≥x)≤Cexp(−cx²/n).

Proof. We have that

S ' X ·(SETΩ◦ AΩ^∗)

, hence we do not require cycle pointing techniques in this case. Let (Sn, σ_n) be drawn uniformly at random from the set Sym(S)[n]. Letπndenote the corresponding partition. By the discussion in Section 2.2.4,σ_ninduces an automorphism

¯

σn:πn→πn

of the SETΩ-object. Moreover, let F_n ⊂π_n denote the fixpoints of ¯σ_n, f_n =|F_n| their number and for each fixpoint Q ∈ Fn let (AQ, σQ) denote the corresponding symmetry from Sym(A^Ω∗)(Q). Let Hn denote the total size of the trees dangling from cycles with length at least 2. We are going to show the following claims.

1) There are constantsC₁>0 and 0< γ <1 such that for allnand x≥0 we have that

P(Hn≥x)≤C1n^3/2γ^x and

P(fn≥x)≤C1n^3/2γ^x.

2) For anyδ >0 the maximum size maxQ∈Fn|AQ|of the trees corresponding to the fixpoints of ¯σ_n satisfies

P( max

Q∈Fn|AQ| ≤n−n^δ) =o(1).

3) There is a constantC2 >0 such that

E[fn]≤C2

for all n.

We may deduce the tail bound for the diameter as follows. First, it suffices to show such a bound for all √

n ≤ x ≤ n. If D(Sn) ≥ x, then we have H_n ≥ x/2 or maxQ∈F_nH(AQ)≥x/2−1. By 1), we have

P(Hn≥x/2)≤C1n^3/2γ^x/2 and there are constants C4, c4>0 such that

C₁n^3/2γ^x/2≤C₄exp(−c₄x²/n)

for allnand√

n≤x≤n. Let E_n denote the event maxQH(AQ)≥x/2−1. It holds that

P(E_n)≤X

F

P(F_n=F)P(E_n|F_n=F).

with F ranging over all subsets of partitions of [n] with P(F_n = F) > 0. By the discussion of symmetries in Section 2.2.4 we have that givenFn=F, the symmetries (AQ, σQ)Q∈F are independent and for eachQ∈F we have that (AQ, σQ) gets drawn uniformly at random from the set Sym(AΩ^∗)[Q]. That is,AQ gets drawn uniformly at random from all unlabelled P´olya trees with outdegrees in the set Ω^∗. By Lemma 1.2.6 it follows that there are positive constantsC5, c5 such that uniformly for alln andx

P(E_n|F_n=F)≤C₅ X

Q∈F

exp(−c₄x²/|Q|)≤ |F|C₄exp(−c₅x²/n).

It follows that

P(En)≤C₅exp(−c₅x²/n)X

F

P(F_n=F)|F| ≤E[f_n]C₅exp(−c₅x²/n).

By 3) we have that

E[f_n]≤C₂ for alln. Thus, for some C₆, c₆ >0, it holds that

P(D(Sn)≥x)≤C₄exp(−c₄x²/n) +C₂C₅exp(−c₅x²/n)≤C₆exp(−c₆x²/n) uniformly for allnand√

n≤x≤n. Thus the claims 1) and 3) imply the tail bound for the diameter.

We may deduce the convergence towards the CRT as follows. Select one of the partition classes fromFnwith maximal size uniformly at random and letXn denote the corresponding tree. By claim 2) we have

P(|Xn| ≤n−n^1/4) =o(1) and thus

P(d_GH(Xn,Sn)≥n^1/4) =o(1).

It follows that

dGH(cΩ^∗Sn/√

n, cΩ^∗Xn/√

n)−→^p 0.

Hence it suffices to show

cΩ^∗Xn/√

n−→ T^(d) e.

Letf :K→R denote a bounded Lipschitz-continuos function defined on the space (K, d_GH) of isometry classes of compact metric spaces equipped with the Gromov-Hausdorff metric. By claim 2) it follows that

E[f(cΩ^∗

√nXn)] =o(1) +X

`

P(|Xn|=`)E[f(cΩ^∗

√nXn)| |Xn|=`].

with the index of the sum ranging over all integers n−n^1/4 ≤`≤n satisfying P(|Xn| = `) > 0, in particular ` ≡ 1 mod gcd(Ω<sup>∗</sup>). Since ` > n/2 we have by the discussion of the structure of symmetries in Section 2.2.4 that Xn conditioned

|Xn|=`is distributed like a uniformly drawn P´olya treeA` of size`with outdegrees in Ω^∗. Hence

E[f(cΩ^∗

√nXn)| |Xn|=`] =E[f(cΩ^∗

√nA`)] =E[f(cΩ^∗

√`A`)] +R_` with

|R_`| ≤C| 1

√n− 1

√`|E[D(A`)]

for a fixed constant C > 0 that does not depend on `. We have by Lemma 1.2.6 that

E[D(A`)] =O(√

`), hence

X

`

R_`=o(1).

By assumption,

E[f(cΩ^∗

√`A`)]→E[f(T^e)]

and hence it follows that

E[c_Ω^∗Xn/√

n]→E[f(Te)].

Thus claim 2) implies that

cΩ^∗Sn/√

n−→ T^(d) e.

It remains to verify claims 1) - 3). The probability generating function of Hn is given by

E[w^Hn] = [zⁿ⁻¹]Z_SETΩ( ˜AΩ^∗(ρz),A˜Ω^∗((ρwz)²),A˜Ω^∗((ρwz)³), . . .) [zⁿ⁻¹]Z_SETΩ( ˜A^Ω∗(ρz),A˜^Ω∗((ρz)²), . . .)

Since 1∈ Ω we may bound the denominator from below by [zⁿ⁻¹] ˜AΩ^∗(ρz) and by Proposition 3.1.5 we have that

[zⁿ⁻¹] ˜A^Ω∗(ρz)∼Cn^−3/2

for some constant C >0 asn≡2 mod gcd(Ω^∗) tends to infinity. Moreover, for all n the polynomial in the indeterminatew in the numerator is dominated coefficient wise by the series

Z_SETΩ( ˜AΩ^∗(ρ),A˜Ω^∗((ρw)²), . . .)

which by Proposition 3.1.5 has radius of convergence strictly greater than 1. In particular we have that

X

k≥x

[w^k]Z_SETΩ( ˜AΩ^∗(ρ),A˜Ω^∗((ρw)²), . . .) =O(γ^x)

for some constant 0 < γ < 1. Hence there is a constant C⁰ such that P(Hn ≥ x)≤C⁰n^3/2γ^x for all n and x. The probability generating function for the random numberf_n is given by

E[w^fn] = [zⁿ⁻¹]Z_SETΩ(wA˜Ω^∗(ρz),A˜Ω^∗((ρz)²), . . .) [zⁿ⁻¹]Z_SETΩ( ˜AΩ^∗(ρz),A˜Ω^∗((ρz)²), . . .)

and the corresponding bound for the eventfn ≥x follows by the same arguments.

This proves claim 1).

We proceed with showing claim 2). Let xn be a given sequence of positive numbers. The event

Q∈Fmaxn|AQ| ≤xn

would imply that

n−1 =Hn+ X

Q∈Fn

|AQ| ≤Hn+xnfn.

In particular it holds thatHn≥(n−1)/2 or fn≥(n−1)/(2xn). Thus, for xn=cn/log(n)

withc >0 a sufficiently small number, it follows by the tail bounds of claim 1) that P( max

Q∈Fn|AQ| ≤xn) =o(1).

Thus, setting

y_n=n−n^2/3+

for any small >0, we have that P( max

Q∈Fn|AQ| ≤yn) =o(1) + X

xn≤k≤y_n

P( max

Q∈Fn|AQ|=k).

We can form any unlabelled S-object by taking an ordered pair of unlabelledAΩ^∗ -objects, connecting their roots by an edge, and declaring the root of the first object as the new root of the resulting tree. It follows that the number of unlabelled S -objects with size n having the property that at least one of the subtrees dangling from the root has sizekis bounded bya_kan−k witha_i= [zⁱ] ˜AΩ^∗(z) for alli. Hence

P( max

Q∈Fn|AQ|=k)≤a_kan−k/[zⁿ] ˜S(z).

By Proposition 3.1.5 we know that ai ∼Ci^−3/2ρ⁻ⁱ asi≡1 mod gcd(Ω^∗) tends to infinity. Thus

P( max

Q∈Fn|AQ| ≤yn)≤o(1) +C⁰ X

xn≤k≤yn

(k(n−k)/n)^−3/2

for someC⁰ >0. Writingk=n/2+twe obtaink(n−k)/n= ((n/2)²−t²)/nand this quantity strictly decreases as|t|grows. Hence we have (k(n−k)/n)^−3/2≤n^2/3+(1+

o(1)) uniformly for allxn≤k≤yn, and thusP(maxQ∈Fn|AQ| ≤yn) =o(1). Setting z_n=n−n^23(23+)+0 for a small⁰>0 we may repeat the same arguments to obtain

P( max

Q∈F_n|AQ| ≤z_n)≤o(1) +C⁰ X

yn≤k≤z_n

(k(n−k)/n)^−3/2

≤o(1) +O(1)(z_n−y_n)(n^23(23+)+0)^−3/2

and this quantity tends to zero. We may repeat the same argument arbitrarily many times and hence obtain that for anyδ >0 we have that

P( max

Q∈Fn|AQ| ≤n−n^δ) =o(1).

This proves claim 2).

It remains to prove claim 3), i.e. we have to show that E[fn] =O(1). If Ω⊂N is bounded, then this is trivial. Otherwise it seems to require some work. We have that

E[fn] = [zⁿ⁻¹](s₁^∂Z_∂s^SETΩ

1 )( ˜AΩ^∗(z),A˜Ω^∗(z²), . . .) [zⁿ⁻¹]Z_SETΩ( ˜AΩ^∗(z),A˜Ω^∗(z²), . . .) .

Since 1∈Ω we have that the denominator is bounded from below by [zⁿ⁻¹] ˜AΩ^∗(z).

By Proposition 3.1.5 it follows that

([zⁿ⁻¹] ˜AΩ^∗(z))⁻¹=O(n^3/2ρⁿ).

The power series inz in the numerator is bounded coefficient wise by (s₁∂Z_SET

∂s1

)( ˜AΩ^∗(z),A˜Ω^∗(z²), . . .) = ˜AΩ^∗(z) exp(

∞

X

i=1

A˜Ω^∗(zⁱ)/i) =h( ˜AΩ^∗(z))g(z) with

h(w) =wexp(w) analytic onCand

g(w) = exp(X

i≥2

A˜Ω^∗(zⁱ)/i)

having radius of convergence strictly larger than ρ since ρ < 1. By a singularity analysis using results from [BBY06] and [FS09, Thm. VI.5] it follows that

[zⁿ⁻¹]h( ˜AΩ^∗(z))g(z) =O(n^−3/2ρ⁻ⁿ).

The detailed arguments are identical as in the proof of Proposition 3.1.6 below. This concludes the proof.

Lemma 3.1.3. Let Vn be drawn uniformly from the unlabelled V = (SET^~_Ω}AΩ^∗)?X

objects of size n. Then we have c_Ω^∗

√nVn

−→ T(d) e.

Moreover, there are constantsC, c >0such that for all x≥0and nwe have the tail bound

P(D(Vn)≥x)≤Cexp(−cx²/n).

Proof. The proof is analogous to the proof of Lemma 3.1.2, only with pointed cycle index sums replacing the role of cycle index sums. Let (Vn, τn, σn, vn) be a rooted c-symmetry drawn uniformly at random from the set RSym(S)[n]. In particular,Vn

is distributed like the uniformly at random chosen unlabelledV-object with sizen.

Let πn denote the corresponding partition. By the discussion in Section 2.2.4, σn

induces an automorphism

¯

σn:πn→πn

of the SETΩ-object. Moreover, let Fn ⊂ πn denote the fixpoints of ¯σn, fn = |Fn| their number and for each fixpoint Q ∈ Fn let (AQ, σQ) denote the corresponding symmetry from Sym(AΩ^∗)(Q). Let H_n denote the total size of the trees dangling from cycles with length at least 2. We are going to show the following claims.

1) There are constantsC₁ >0 and 0< γ <1 such that for all nandx≥0 we have that

P(H_n≥x)≤C₁n^3/2γ^x and

P(fn≥x)≤C1n^3/2γ^x.

2) For anyδ >0 the maximum size maxQ∈F_n|AQ|of the trees corresponding to the fixpoints of ¯σn satisfies

P( max

Q∈Fn|AQ| ≤n−n^δ) =o(1).

3) There is a constantC2 >0 such that

E[f_n]≤C₂ for all n.

From these claims we may deduce the tail bounds for the diameter and the conver-gence towards the CRT in an identical manner as in the proof of Lemma 3.1.2. It remains to verify claims 1)-3). We start with claim 1). The probablity generating function of Hn is given by

E[w^Hn] = [zⁿ⁻¹] ¯Z_SET~

Ω( ˜AΩ^∗(ρz),A˜^◦Ω^∗(ρz); ˜AΩ^∗((ρwz)²),A˜^◦Ω^∗((ρwz)²);. . .) [zⁿ⁻¹] ¯Z_SET^~

Ω( ˜AΩ^∗(ρz),A˜^◦_Ω^∗(ρz); ˜AΩ^∗((ρz)²),A˜^◦_Ω^∗((ρz)²);. . .) . Since 1∈Ω and there is a number k≥3 withk∈Ω it follows that the denominator is bounded from below by

[zⁿ⁻¹]z^k−1A˜Ω^∗(ρz) = [z^n−k] ˜AΩ^∗(ρz).

We have that

n−k≡1 mod gcd(Ω^∗) and thus, by Proposition 3.1.5, we have that

[z^n−k] ˜AΩ^∗(ρz)∼Cn^−3/2

as n ≡ 2 mod gcd(Ω^∗) tends to infinity. The polynomial in the numerator with indeterminate wis bounded coefficient wise by the series

Z¯_SET~

Ω( ˜AΩ^∗(ρ),A˜^◦Ω^∗(ρ); ˜AΩ^∗((ρw)²),A˜^◦Ω^∗((ρw)²);. . .)

which does not depend on n and, by Proposition 3.1.6, has radius of convergence strictly larger than 1. It follows that there is a constant C⁰ such that

P(H_n≥x)≤C⁰n^3/2γ^x

for allnandx. The probability generating function for the random number number f_n is given by

E[w^fn] = [zⁿ⁻¹] ¯Z_SET^~

Ω(wA˜Ω^∗(ρz), wA˜^◦Ω^∗(ρz); ˜AΩ^∗((ρz)²),A˜^◦Ω^∗((ρz)²);. . .) [zⁿ⁻¹] ¯Z

SET^~_Ω( ˜AΩ^∗(ρz),A˜^◦_Ω^∗(ρz); ˜AΩ^∗((ρz)²),A˜^◦_Ω^∗((ρz)²);. . .) . and the corresponding bound for the event fn≥x follows by the same arguments.

This proves claim 1).

We proceed with showing claim 2). Let xn be a given sequence of positive numbers. The event

Q∈Fmax_n|AQ| ≤x_n would imply that

n−1 =H_n+ X

Q∈Fn

|A_Q| ≤H_n+x_nf_n.

In particular it holds thatHn≥(n−1)/2 or fn≥(n−1)/(2xn). Thus, for xn=cn/log(n)

withc >0 a sufficiently small number, it follows by the tail bounds of claim 1) that P( max

Q∈Fn|AQ| ≤xn) =o(1).

Setting

y_n=n−n^2/3+

for any small >0, we have that P( max

Q∈Fn|AQ| ≤y_n) =o(1) + X

xn≤k≤yn

P( max

Q∈Fn|AQ|=k).

Any unlabelled V-object with a tree of size kdangling from the root that does not contain any vertex of the marked cycle can be formed by connecting the roots of an unlabelled AΩ^∗-object of size k and an unlabelled SET^~_Ω^∗}AΩ^∗ object of size n−k. By a singularity analysis similiar to the proof of claim 3) in Lemma 3.1.2 we have that the numberb_i of unlabelledSET^~_Ω^∗}AΩ^∗-objects of sizei is at most O(i^−3/2ρ⁻ⁱ). It follows that

P( max

Q∈Fn|AQ|=k)≤([z^k] ˜AΩ^∗(z))bn−k/([zⁿ] ˜V(z)) =O((k(n−k)/n)^−3/2) uniformly for allxn≤k≤yn and thus

P( max

Q∈Fn|AQ| ≤y_n) =o(1) +O(1) X

xn≤k≤yn

(k(n−k)/n)^−3/2.

In order to finish the proof of claim 2) we may now follow precisely the same argu-ments as in the proof of claim 2) in Lemma 3.1.2.

Claim 3) follows by similar arguments as in the proof of claim 3) in Lemma 3.1.2.

This completes the proof.

Im Dokument Scaling limits of random trees and graphs (Seite 60-68)