From Spin Glasses to Branching Brownian Motion—and Back?

Abstract

I review recent work on the construction of the extremal process of branching Brownian motion. I place this in the context of spin glass theory and in particular the Generalised Random Energy models of Derrida and Gardner. The main emphasis is on a review of the construction of the extremal process of branching Brownian motion, done in collaboration with Louis-Pierre Arguin and Nicola Kistler (Commun Pure Appl Math 64(12):1647–1676, 2011; Ann Appl Probab 22(4):1693–1711, 2012; Probab Theory Relat Fields 157:535–574, 2013). I also present some more recent results on the variable speed Brownian motion, obtained with Lisa Hartung (Electron J Probab 19(18):1–28, 2014; Lat Am J Probab Math Stat 12:261–291, 2015).

Appendix: Point Processes

Here we provide some basic background on point processes and in particular Poisson point processes. For more details, see [24, 37, 48].

1.1 Definition and Basic Properties

Point processes are designed to describe the probabilistic structure of point sets in some metric space. The key idea is to associate to a collection of points a point measure. Let us first consider a single point x. We consider the usual Borel-sigma algebra, $\mathfrak{B} \equiv \mathfrak{B}(\mathbb{R}^{d})$, of $\mathbb{R}^{d}$, that is generated by the open sets in the Euclidean topology of $\mathbb{R}^{d}$. Given $x \in \mathbb{R}^{d}$, we define the Dirac measure, δ _x , such that, for any Borel set $A \in \mathfrak{B}$,

$$\displaystyle{ \delta _{x}(A) = \left \{\begin{array}{@{}l@{\quad }l@{}} 1,\quad &\,\,\text{if}\,x \in A\\ 0,\quad &\,\,\text{if} \,x\not\in A. \end{array} \right. }$$

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A point measure is now a measure, μ, on $\mathbb{R}^{d}$, such that there exists a countable collection of points, $\{x_{i} \in \mathbb{R}^{d},i \in \mathbb{N}\}$, such that

$$\displaystyle{ \mu =\sum _{ i=1}^{\infty }\delta _{ x_{i}} }$$

(261)

and, if K is compact, then $\mu (K) < \infty$.

Note that the points x _i need not be all distinct. The set $S_{\mu } \equiv \{ x \in \mathbb{R}^{d}:\mu (x)\neq 0\}$ is called the support of μ. A point measure such that for all $x \in \mathbb{R}^{d}$, μ(x) ≤ 1 is called simple.

We denote by $M_{p}(\mathbb{R}^{d})$ the set of all point measures on $\mathbb{R}^{d}$. We equip this set with the sigma-algebra $\mathcal{M}_{p}(\mathbb{R}^{d})$, the smallest sigma algebra that contains all subsets of $M_{p}(\mathbb{R}^{d})$ of the form $\{\mu \in M_{m}(\mathbb{R}^{d}):\mu (F) \in B\}$, where $F \in \mathfrak{B}(\mathbb{R}^{d})$ and $B \in \mathfrak{B}([0,\infty ))$. $\mathcal{M}_{p}(\mathbb{R}^{d})$ is also characterised by saying that it is the smallest sigma-algebra that makes the evaluation maps, $\mu \rightarrow \mu (F)$, measurable for all Borel sets $F \in \mathfrak{B}(\mathbb{R}^{d})$.

A point process, N, is a random variable taking values in $M_{p}(\mathbb{R}^{d})$, i.e. a measurable map, $N: (\varOmega,\mathfrak{F}; \mathbb{P}) \rightarrow M_{p}(\mathbb{R}^{d})$, from a probability space to the space of point measures.

This looks very fancy, but in reality things are quite down-to-earth:

Proposition 6.1

N is a point process, if and only if the map $N(\cdot,F):\omega \rightarrow N(\omega,F)$ , is measurable from $(\varOmega,\mathfrak{F}) \rightarrow \mathfrak{B}([0,\infty ))$ , for any Borel set F, i.e. if N(F) is a real random variable.

Proof

Let us first prove necessity, which should be obvious. In fact, since $\omega \rightarrow N(\omega,\cdot )$ is measurable into $(M_{p}(\mathbb{R}^{d}),\mathcal{M}_{p}(\mathbb{R}^{p}))$, and $\mu \rightarrow \mu (F)$ is measurable from this space into $(\mathbb{R}_{+},\mathfrak{B}(\mathbb{R}_{+}))$, the composition of these maps is also measurable.

Next we prove sufficiency. Define the set

$$\displaystyle{ \mathfrak{G} \equiv \{ A \in \mathcal{M}_{p}(\mathbb{R}^{d}): N^{-1}A \in \mathfrak{F}\} }$$

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This set is a sigma-algebra and N is measurable from $(\varOmega,\mathfrak{F}) \rightarrow (M_{p}(\mathbb{R}^{d}),\mathfrak{G})$ by definition. But $\mathfrak{G}$ contains all sets of the form $\{\mu \in M_{p}(\mathbb{R}^{d}):\mu (F) \in B\}$, since

$$\displaystyle{ N^{-1}\{\mu \in M_{ p}(\mathbb{R}^{d}):\mu (F) \in B\} =\{\omega \in \varOmega: N(\omega,F) \in B\} \in \mathfrak{F}, }$$

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since N(⋅ , F) is measurable. Thus $\mathfrak{G} \supset \mathcal{M}_{p}(\mathbb{R}^{d})$, and N is measurable a fortiori as a map from the smaller sigma-algebra. □ 

We will have need to find criteria for convergence of point processes. For this we recall some notions of measure theory. If $\mathfrak{B}$ is a Borel-sigma algebra, of a metric space E, then $\mathcal{T} \subset \mathfrak{B}$ is called a Π-system, if $\mathcal{T}$ is closed under finite intersections; $\mathfrak{G} \subset \mathfrak{B}$ is called a $\lambda$-system, or a sigma-additive class, if

• (i) $E \in \mathfrak{G}$,

• (ii) If $A,B \in \mathfrak{G}$, and $A \supset B$, then $A\setminus B \in \mathfrak{G}$,

• (iii) If $A_{n} \in \mathfrak{G}$ and $A_{n} \subset A_{n+1}$, then $\lim _{n\uparrow \infty }A_{n} \in \mathfrak{G}$.

The following useful observation is called Dynkin’s theorem.

Theorem 6.2

If $\mathcal{T}$ is a Π-system and $\mathfrak{G}$ is a $\lambda$ -system, then $\mathfrak{G} \supset \mathcal{T}$ implies that $\mathfrak{G}$ contains the smallest sigma-algebra containing $\mathcal{T}$ .

The most useful application of Dynkin’s theorem is the observation that, if two probability measures are equal on a Π-system that generates the sigma-algebra, then they are equal on the sigma-algebra (since the set on which the two measures coincide forms a $\lambda$-system containing $\mathcal{T}$).

As a consequence we can further restrict the criteria to be verified for N to be a Point process. In particular, we can restrict the class of F’s for which N(⋅ , F) need to be measurable to bounded rectangles.

Proposition 6.3

Suppose that $\mathcal{T}$ are relatively compact sets in $\mathfrak{B}$ satisfying

• (i) $\mathcal{T}$ is a Π-system,

• (ii) The smallest sigma-algebra containing $\mathcal{T}$ is $\mathfrak{B}$ ,

• (iii) Either, there exists $E_{n} \in \mathcal{T}$ , such that E _n ↑ E, or there exists a partition, {E _n }, of E with $\cup _{n}E_{n} = E$ , with $E_{n} \subset \mathcal{T}$ .

Then N is a point process on $(\varOmega,\mathfrak{F})$ in $(E,\mathfrak{B})$ , if and only if the map $N(\cdot,I):\omega \rightarrow N(\omega,I)$ is measurable for any $I \in \mathcal{T}$ .

Corollary 6.4

Let $\mathcal{T}$ satisfy the hypothesis of Proposition 6.3 and set

$$\displaystyle{ \mathfrak{G} \equiv \left \{\{\mu:\mu (I_{j}) = n_{j},1 \leq j \leq k\},k \in \mathbb{N},I_{j} \in \mathcal{T},n_{j} \geq 0\right \}. }$$

(264)

Then the smallest sigma-algebra containing $\mathfrak{G}$ is $\mathcal{M}_{p}(\mathbb{R}^{d})$ and $\mathfrak{G}$ is a Π-system.

Next we show that the law, P _N , of a point process is determined by the law of the collections of random variables N(F _n ), $F_{n} \in \mathfrak{B}(\mathbb{R}^{d})$.

Proposition 6.5

Let N be a point process in $(\mathbb{R}^{d},\mathfrak{B}(\mathbb{R}^{d})$ and suppose that $\mathcal{T}$ is as in Proposition 6.3 . Define the mass functions

$$\displaystyle{ P_{I_{1},\ldots,I_{k}}(n_{1},\ldots,n_{k}) \equiv \mathbb{P}\left [N(I_{j}) = n_{j},\forall 1 \leq j \leq k\right ] }$$

(265)

for $I_{j} \in \mathcal{T}$ , n _j ≥ 0. Then P _N is uniquely determined by the collection

$$\displaystyle{ \{P_{I_{1},\ldots,I_{k}},k \in \mathbb{N},I_{j} \in \mathcal{T}\} }$$

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We need some further notions. First, if N ₁, N ₂ are point processes, we say that they are independent, if and only if, for any collection $F_{j} \in \mathfrak{B}$, $G_{j} \in \mathfrak{B}$, the vectors

$$\displaystyle{ (N_{1}(F_{j}),1 \leq j \leq k)\quad \text{and}\quad (N_{2}(G_{j}),1 \leq j \leq \ell) }$$

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are independent random vectors.

The intensity measure, $\lambda$, of a point process N is defined as

$$\displaystyle{ \lambda (F) \equiv \mathbb{E}N(F) =\int _{M_{p}(\mathbb{R}^{d})}\mu (F)P_{N}(\mathrm{d}\mu ) }$$

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for $F \in \mathfrak{B}$.

For measurable functions $f: \mathbb{R}^{d} \rightarrow \mathbb{R}_{+}$, we define

$$\displaystyle{ N(\omega,f) \equiv \int _{\mathbb{R}^{d}}f(x)N(\omega,\mathrm{d}x) }$$

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Then N(⋅ , f) is a random variable. We have that

$$\displaystyle{ \mathbb{E}N(f) =\lambda (f) =\int _{\mathbb{R}^{d}}f(x)\lambda (\mathrm{d}x). }$$

(270)

1.2 Laplace Functionals

If Q is a probability measure on $(M_{p},\mathcal{M}_{p})$, the Laplace transform of Q is a map, ψ from non-negative Borel functions on $\mathbb{R}^{d}$ to $\mathbb{R}_{+}$, defined as

$$\displaystyle{ \psi (f) \equiv \int _{M_{p}}\exp \left (-\int _{R^{d}}f(x)\mu (\mathrm{d}x)\right )Q(\mathrm{d}\mu ). }$$

(271)

If N is a point process, the Laplace functional of N is

$$\displaystyle\begin{array}{rcl} \psi _{N}(f)& \equiv & \mathbb{E}\mathrm{e}^{-N(f)} =\int \mathrm{ e}^{-N(\omega,f)}\mathbb{P}(\mathrm{d}\omega ) \\ & =& \int _{M_{p}}\exp \left (-\int _{R^{d}}f(x)\mu (\mathrm{d}x)\right )P_{N}(\mathrm{d}\mu ){}\end{array}$$

(272)

Proposition 6.6

The Laplace functional, ψ _N , of a point process, N, determines N uniquely.

Proof

For k ≥ 1, and $F_{1},\ldots,F_{k} \in \mathfrak{B}$, $c_{1},\ldots,c_{k} \geq 0$, let $f =\sum _{ i=1}^{k}c_{i}\mathbb{1}_{F_{i}}(x)$. Then

$$\displaystyle{ N(\omega,f) =\sum _{ i=1}^{k}c_{ i}N(\omega,F_{i}) }$$

(273)

and

$$\displaystyle{ \psi _{N}(f) = \mathbb{E}\exp \left (-\sum _{i=1}^{k}c_{ i}N(F_{i})\right ). }$$

(274)

This is the Laplace transform of the vector (N(F _i ), 1 ≤ i ≤ k), that determines uniquely its law. Hence the proposition follows from Proposition 6.5 □ 

1.3 Poisson Point Processes

The most important class of point processes for our purposes will be Poisson point processes.

Definition 6.7

Let $\lambda$ be a $\sigma$-finite, positive measure on $\mathbb{R}^{d}$. Then a point process, N, is called a Poisson point process with intensity measure $\lambda$ ($PPP(\lambda )$), if

• (i) For any $F \in \mathfrak{B}(\mathbb{R}^{d})$, and $k \in \mathbb{N}$,

  $$\displaystyle{ \mathbb{P}\left [N(F) = k\right ] = \left \{\begin{array}{@{}l@{\quad }l@{}} \mathrm{e}^{-\lambda (F)}\frac{(\lambda (F))^{k}} {k!},\quad &\quad \text{if}\,\,\lambda (F) < \infty \\ 0, \quad &\quad \text{if}\,\,\lambda (F) = \infty,\end{array} \right. }$$

  (275)

• (ii) If $F,G \in \mathfrak{B}$ are disjoint sets, then N(F) and N(G) are independent random variables.

In the next theorem we will assert the existence of a Poisson point process with any desired intensity measure. In the proof we will give an explicit construction of such a process.

Proposition 6.8

• (i) $PPP(\lambda )$ exists and its law is uniquely determined by the requirements of the definition.

• (ii) The Laplace functional of $PPP(\lambda )$ is given, for f ≥ 0, by

  $$\displaystyle{ \varPsi _{N}(f) =\exp \left (\int _{\mathbb{R}^{d}}(\mathrm{e}^{-f(x)} - 1)\lambda (\mathrm{d}x)\right ). }$$

  (276)

Proof

Since we know that the Laplace functional determines a point process, in order to prove that the conditions of the definition uniquely determine the $PPP(\lambda )$, we show that they determine the form (276) of the Laplace functional. Thus suppose that N is a $PPP(\lambda )$. Let $f = c\mathbb{1}_{F}$. Then

$$\displaystyle\begin{array}{rcl} \varPsi _{N}(f)& =& \mathbb{E}\exp \left (-N(f)\right ) = \mathbb{E}\exp \left (-cN(F)\right ) \\ & =& \sum _{k=0}^{\infty }\mathrm{e}^{-ck}\mathrm{e}^{-\lambda (F)}\frac{(\lambda (F))^{k}} {k!} =\mathrm{ e}^{(\mathrm{e}^{-c}-1)\lambda (F) } \\ & =& \exp \left (\int (\mathrm{e}^{-f(x)}-)\lambda (\mathrm{d}x)\right ), {}\end{array}$$

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which is the desired form. Next, if F _i are disjoint, and $f =\sum _{ i=1}^{k}c_{i}\mathbb{1}_{F-i}$, it is straightforward to see that

$$\displaystyle{ \varPsi _{N}(f) = \mathbb{E}\exp \left (-\sum _{i=1}^{k}c_{ i}N(F_{i})\right ) =\prod _{ i=^{1}}^{k}\mathbb{E}\exp \left (-c_{ i}N(F_{i})\right ) }$$

(278)

due to the independence assumption (ii); a simple calculations shows that this yields again the desired form. Finally, for general f, we can choose a sequence, f _n , of the form considered, such that f _n ↑ f. By monotone convergence then N(f _n ) ↑ N(f). On the other hand, since e^−N(g) ≤ 1, we get from dominated convergence that

$$\displaystyle{ \varPsi _{N}(f_{n}) = \mathbb{E}\mathrm{e}^{-N(f_{n})} \rightarrow \mathbb{E}\mathrm{e}^{-N(f)} =\varPsi _{ N}(f). }$$

(279)

But, since $1 -\mathrm{ e}^{-f_{n}(x)} \uparrow 1 -\mathrm{ e}^{-f(x)}$, and monotone convergence gives once more

$$\displaystyle{ \varPsi _{N}(f_{n}) =\exp \left (\int (1 -\mathrm{ e}^{-f_{n}(x)})\lambda (\mathrm{d}x)\right ) \uparrow \exp \left (\int (1 -\mathrm{ e}^{-f(x)})\lambda (\mathrm{d}x)\right ) }$$

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On the other hand, given the form of the Laplace functional, it is trivial to verify that the conditions of the definition hold, by choosing suitable functions f.

Finally we turn to the construction of $PPP(\lambda )$. Let us first consider the case $\lambda (\mathbb{R}^{d}) < \infty$. Then construct

• (i) A Poisson random variable, τ, of parameter $\lambda (\mathbb{R}^{d})$.

• (ii) A family, X _i , $i \in \mathbb{N}$, of independent, $\mathbb{R}^{d}$- valued random variables with common distribution $\lambda$. This family is independent of τ.

Then set

$$\displaystyle{ N^{{\ast}}\equiv \sum _{ i=1}^{\tau }\delta _{ X_{i}} }$$

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It is not very hard to verify that N ^∗ is a $PPP(\lambda )$.

To deal with the case when $\lambda (\mathbb{R}^{d})$ is infinite, decompose $\lambda$ into a countable sum of finite measures, $\lambda _{k}$, that are just the restriction of $\lambda$ to a finite set F _k , where the F _k form a partition of $\mathbb{R}^{d}$. Then N ^∗ is just the sum of independent $PPP(\lambda _{k})\ N_{k}^{{\ast}}$. □ 

1.4 Convergence of Point Processes

Before we turn to applications to extremal processes, we still have to discuss the notion of convergence of point processes. As point processes are probability distributions on the space of point measures, we will naturally think about weak convergence. This means that we will say that a sequence of point processes, N _n , converges weakly to a point process, N, if for all continuous functions, f, on the space of point measures,

$$\displaystyle{ \mathbb{E}f(N_{n}) \rightarrow \mathbb{E}f(N). }$$

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However, to understand what this means, we must discuss what continuous functions on the space of point measures are, i.e. we must introduce a topology on the set of point measures. The appropriate topology for our purposes will be that of vague convergence.

1.4.1 Vague Convergence

We consider the space $\mathbb{R}^{d}$ equipped with its natural Euclidean metric. Clearly $\mathbb{R}^{d}$ is a complete, separable metric space. We will denote by $C_{0}(\mathbb{R}^{d})$ the set of continuous real-valued functions on $\mathbb{R}^{d}$ that have compact support; $C_{0}^{+}(\mathbb{R}^{n})$ denotes the subset of non-negative functions. We consider $\mathcal{M}_{+}(\mathbb{R}^{d})$ the set of all $\sigma$-finite, positive measures on $(\mathbb{R}^{d},\mathfrak{B}(\mathbb{R}^{d}))$. We denote by $\mathcal{M}_{+}(\mathbb{R}^{d})$ the smallest sigma-algebra of subsets of $M_{+}(\mathbb{R}^{d})$ that makes the maps $m \rightarrow m(f)$ measurable for all $f \in C_{0}^{+}(\mathbb{R}^{d})$.

We will say that a sequence of measures, $\mu _{n} \in M_{+}(\mathbb{R}^{d})$ converges vaguely to a measure $\mu \in M_{+}(\mathbb{R}^{d})$, if, for all $f \in C_{0}^{+}(\mathbb{R}^{d})$,

$$\displaystyle{ \mu _{n}(f) \rightarrow \mu (f) }$$

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Note that for this topology, typical open neighbourhoods are of the form

$$\displaystyle{ B_{f_{1},\ldots,f_{k},\epsilon }(\mu ) \equiv \{\nu \in M_{+}(\mathbb{R}^{d}): \forall _{ i=1}^{k}\,\vert \nu (f_{ i}) -\mu (f_{i})\vert <\epsilon \}, }$$

(284)

i.e. to test the closeness of two measures, we test it on their expectations on finite collections of continuous, positive functions with compact support. Given this topology, on can of course define the corresponding Borel sigma algebra, $\mathfrak{B}(M_{+}(\mathbb{R}^{d}))$, which (fortunately) turns out to coincide with the sigma algebra $\mathcal{M}_{+}(\mathbb{R}^{d})$ introduced before.

The following properties of vague convergence are useful.

Proposition 6.9

Let μ _n , $n \in \mathbb{N}$ be in $M_{+}(\mathbb{R}^{d})$ . Then the following statements are equivalent:

• (i) μ _n converges vaguely to μ, $\mu _{n}\buildrel v \over \rightarrow \mu$ .

• (ii) $\mu _{n}(B) \rightarrow \mu (B)$ for all relatively compact sets, B, such that μ(∂B) = 0.

• (iii) $\limsup _{n\uparrow \infty }\mu _{n}(K) \leq \mu (K)$ and $\limsup _{n\uparrow \infty }\mu _{n}(G) \geq \mu (G)$ , for all compact K, and all open, relatively compact G.

1.4.2 Weak Convergence

Having established the space of $\sigma$-finite measures as a complete, separable metric space, we can think of weak convergence of probability measures on this space just as if we were working on an Euclidean space.

Definition 6.10

A sequence of point processes N _n converges weakly with respect to the vague topology to a point process, N, iff for all functions $F: \mathcal{M}_{+}(\mathbb{R}^{d}) \rightarrow \mathbb{R}$ that are continuous with respect to the vague topology,

$$\displaystyle{ \lim _{n\uparrow \infty }\mathbb{E}F(N_{n}) = \mathbb{E}F(N). }$$

(285)

An important convergence criterion is given by the following theorem (see [24, Chap. 11]).

Theorem 6.11

A sequence of point processes N _n converges weakly to a point process N, if for all positive continuous functions, ϕ, with compact support,

$$\displaystyle{ \lim _{n\uparrow \infty }\varPsi _{N_{n}}(\phi ) =\varPsi _{N}(\phi ). }$$

(286)