Historical Lattice Trees

Abstract

We prove that the rescaled historical processes associated to critical spread-out lattice trees in dimensions $d>8$ converge to historical Brownian motion. This is a functional limit theorem for measure-valued processes that encodes the genealogical structure of the underlying random trees. Our results are applied elsewhere to prove that random walks on lattice trees, appropriately rescaled, converge to Brownian motion on super-Brownian motion.

Fig. 1

A (nearest neighbour) lattice tree in 2 dimensions

Fig. 2

The MVP $X_5^{(1)}$ assigns masses to points in the tree at distance 5 from the root, while $H_5^{(1)}$ assigns the same masses to paths in the tree leading to these points

Fig. 3

On the left is the index set I drawn (with labels as edges) up to and including generation 3. On the right is an example of a Galton–Watson tree (with edge labels $\alpha$), where $e^{\alpha }=0$ for all $\alpha \in \{000,0010,0011,01\}$, while $e^{\alpha }=2$ for $\alpha \in \{0,00,001\}$. Note that we have dropped the parentheses and commas in the notation for elements of I to declutter the pictures

Fig. 4

A (binary) branching Brownian motion in 1-dimension, with time on the x axis, drawn up to the third branch time, 3/n. In the corresponding G–W tree, the root 0 has two children, exactly one of which has 2 children

Fig. 5

A depiction of a shape $ϝ\in {\mathrm{\Sigma }}_4$ with vertex labels above vertices and edge labels in brackets. The set of edges in the path from vertex 0 to vertex 1 is ${\mathcal{E}}_1(ϝ)=\{1,5,6\}$. Variables $u_i$ are associated to each of the vertices i, describing a ‘length’ from 0 to i, to form $\mathcal{T}(ϝ,u)$. Differences in these $u_i$ are then the “edge lengths”

Fig. 6

The tree $\mathbb{T}(ϝ,u)$ together with times $\mathit{s}$. The ($m^{(1)}=7$) $▵$ symbols represent times $s_1^{(1)},\dots ,s_7^{(1)}$. Similarly $\square$ symbols represent times $s_j^{(2)}$ (with $m^{(2)}=6$) and $\otimes$ symbols represent times $s_j^{(3)}$ (with $m^{(3)}=3$) respectively. In this example there is one point (on edge 5) that is both square and triangle simultaneously. The ‘subinterval’ lengths ${\check{s}}_{5,i}$ are indicated for edge 5

Fig. 7

On the left is (part of) a GW tree with ${\beta }^1,{\beta }^2,{\beta }^3$ indicated. Here $|{\beta }^1|=|{\beta }^5|=2$, $|{\beta }^2|=|{\beta }^3|=3$, and $|{\beta }^4|=1$, and this contributes to (2.13) when $t_2,t_3\in [3/n,4/n)$ and $t_1\in [2/n,3/n)$ as depicted. On the right is the corresponding tree shape. The edge lengths associated to the latter are determined by taking differences of the $u_e$, where $u_4=1/n$, $u_5=2/n$, $u_1=t_1$, $u_2=t_2$, $u_3=t_3$

Fig. 8

A depiction of the event in the detailed 1-particle function with $n=1$, $t=6$, $s_1=1$ and $s_2=4$, with the path $s\mapsto w_s(6,x_3)$ in bold (recall the notation from (1.1))

Fig. 9

A depiction of the event in the 3-particle function ${\rho }_{(3,3,6)}(y_1,y_2,y_3)$

Fig. 10

An example of a marked skeleton network from a shape $ϝ\in {\mathrm{\Sigma }}_4$. Branch points and leaves are $\blacksquare$, marked points are $\star$

Fig. 11

A graph $\mathrm{\Gamma }$ on a marked skeleton network ${\mathcal{N}}^{+}$, with b denoting the branch point nearest to the root. The rightmost bond is in $\mathcal{R}$ since it covers two special points. Also, $\mathrm{\Gamma }\in {\mathcal{E}}_{{\mathcal{N}}^{+}}^b$ since ${\mathcal{A}}_b(\mathrm{\Gamma })$ (highlighted) contains a neighbour of a marked point

Fig. 12

A skeleton network ${\mathcal{N}}^{+}$ with a bond in $\mathcal{R}$. This bond has endpoints in the marked edge $\check{e}\in {\mathcal{N}}^{+}$ and the “phantom” marked edge ${\check{e}}^{\prime }\in {\mathcal{N}}^{++}$ of lengths ${\check{n}}_{\check{e}}=6$ and ${\check{n}}_{{\check{e}}^{\prime }}^{\prime }=0$ respectively. We write $st(\check{e},{\check{e}}^{\prime },{\check{m}}_{\check{e}},{\check{m}}_{{\check{e}}^{\prime }}^{\prime })$ for this bond. Here, ${\check{m}}_{\check{e}}=2$ is indicated, while ${\check{m}}_{{\check{e}}^{\prime }}^{\prime }=0$. The set of marked edges ${\mathit{E}}_{\check{e},{\check{e}}^{\prime }}^{+}$ on the path from $\check{e}$ to ${\check{e}}^{\prime }$ is $\{f_1,f_2\}$ from to and to as indicated