hidden gems: aldous's brownian excursions, critical random...
TRANSCRIPT
![Page 1: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/1.jpg)
Hidden Gems: Aldous’s Brownian excursions,critical random graphs and the multiplicative
coalescent
David Clancy
Oct. 5, 2020
![Page 2: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/2.jpg)
Erdos-Renyi random graphs
The Erdos-Renyi random graph G (n, p) is the graph on the vertices
V = [n] := {1, 2, · · · , n}
and the edge connecting i and j is independently added with probability p:
P({i , j} is an edge) = p
![Page 3: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/3.jpg)
Erdos-Renyi random graphs
G (n, p) gives a simple model of a random graph.
Probabilistic method: G (n, p) can be used to prove the existence ofcertain combinatorial objects without a direct construction.Almost equivalent to the study of certain model in epidemiology. Theedges represent disease transmission in the Reed-Frost model.
![Page 4: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/4.jpg)
Erdos-Renyi random graphs
G (n, p) gives a simple model of a random graph.Probabilistic method: G (n, p) can be used to prove the existence ofcertain combinatorial objects without a direct construction.
Almost equivalent to the study of certain model in epidemiology. Theedges represent disease transmission in the Reed-Frost model.
![Page 5: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/5.jpg)
Erdos-Renyi random graphs
G (n, p) gives a simple model of a random graph.Probabilistic method: G (n, p) can be used to prove the existence ofcertain combinatorial objects without a direct construction.Almost equivalent to the study of certain model in epidemiology. Theedges represent disease transmission in the Reed-Frost model.
![Page 6: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/6.jpg)
Criticality of G (n, p)
Basic properties of G (n, p):
Total number of edges is Bin((n
2
), p), E [#total edges] ≈ n2p
2
Expect (n − 1)p other vertices to share an edge with each vertex j .
![Page 7: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/7.jpg)
Criticality of G (n, p)
Basic properties of G (n, p):
Total number of edges is Bin((n
2
), p), E [#total edges] ≈ n2p
2
Expect (n − 1)p other vertices to share an edge with each vertex j .
![Page 8: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/8.jpg)
Criticality of G (n, p)
We expect each vertex to have ≈ np neighbors.
What happens when p = p(n) = cn?
Theorem 1 (Erdos & Renyi [ER60]).
Gn = G (n, c/n) for some constant c ∈ (0,∞). As n→∞:
c > 1: the largest component of Gn is of order n, the second largestcomponent of Gn is of order log n;
c < 1: the largest component of Gn is of order log n;
c = 1: the largest two components of Gn are both order n2/3.
Something interesting happens at p = 1/n.
![Page 9: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/9.jpg)
Criticality of G (n, p)
We expect each vertex to have ≈ np neighbors.What happens when p = p(n) = c
n?
Theorem 1 (Erdos & Renyi [ER60]).
Gn = G (n, c/n) for some constant c ∈ (0,∞). As n→∞:
c > 1: the largest component of Gn is of order n, the second largestcomponent of Gn is of order log n;
c < 1: the largest component of Gn is of order log n;
c = 1: the largest two components of Gn are both order n2/3.
Something interesting happens at p = 1/n.
![Page 10: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/10.jpg)
Criticality of G (n, p)
We expect each vertex to have ≈ np neighbors.What happens when p = p(n) = c
n?
Theorem 1 (Erdos & Renyi [ER60]).
Gn = G (n, c/n) for some constant c ∈ (0,∞). As n→∞:
c > 1: the largest component of Gn is of order n, the second largestcomponent of Gn is of order log n;
c < 1: the largest component of Gn is of order log n;
c = 1: the largest two components of Gn are both order n2/3.
Something interesting happens at p = 1/n.
![Page 11: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/11.jpg)
Criticality of G (n, p)
We expect each vertex to have ≈ np neighbors.What happens when p = p(n) = c
n?
Theorem 1 (Erdos & Renyi [ER60]).
Gn = G (n, c/n) for some constant c ∈ (0,∞). As n→∞:
c > 1: the largest component of Gn is of order n, the second largestcomponent of Gn is of order log n;
c < 1: the largest component of Gn is of order log n;
c = 1: the largest two components of Gn are both order n2/3.
Something interesting happens at p = 1/n.
![Page 12: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/12.jpg)
Critical window
What happens near p(n) = 1n?
More formally: What happens when
p(n) =1± ε(n)
nwhere ε(n)→ 0 as n→∞?
Too many results to summarize for general εn windows.We’ll focus on is
p =1 + λn−1/3
n= n−1 + λn−4/3, λ ∈ R.
![Page 13: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/13.jpg)
Critical window
What happens near p(n) = 1n?
More formally: What happens when
p(n) =1± ε(n)
nwhere ε(n)→ 0 as n→∞?
Too many results to summarize for general εn windows.We’ll focus on is
p =1 + λn−1/3
n= n−1 + λn−4/3, λ ∈ R.
![Page 14: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/14.jpg)
Critical window
What happens near p(n) = 1n?
More formally: What happens when
p(n) =1± ε(n)
nwhere ε(n)→ 0 as n→∞?
Too many results to summarize for general εn windows.We’ll focus on is
p =1 + λn−1/3
n= n−1 + λn−4/3, λ ∈ R.
![Page 15: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/15.jpg)
Why n−1 + λn−4/3?
Theorem 2 (Bollobas ’84, Luczak, Pittel, Wierman ’94).
1 Bollobas [Bol84]: If p = n−1 + n−(1+γ) for γ ∈ (0, 1/3) then o(n2/3)vertices appear in components that aren’t trees or uni-cycles (graphswith 1 cycle).
2 L-P-W [ LPW94]: If p = n−1 + λn−4/3 then all components in G (n, p)have at most ξn surplus edges added, and ξn is bounded in probabilityas n→∞.
Surplus edges: #edges−#vertices + 1, the number of edges you have toremove from a graph in order to form a tree.
![Page 16: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/16.jpg)
Why n−1 + λn−4/3?
Theorem 2 (Bollobas ’84, Luczak, Pittel, Wierman ’94).
1 Bollobas [Bol84]: If p = n−1 + n−(1+γ) for γ ∈ (0, 1/3) then o(n2/3)vertices appear in components that aren’t trees or uni-cycles (graphswith 1 cycle).
2 L-P-W [ LPW94]: If p = n−1 + λn−4/3 then all components in G (n, p)have at most ξn surplus edges added, and ξn is bounded in probabilityas n→∞.
Surplus edges: #edges−#vertices + 1, the number of edges you have toremove from a graph in order to form a tree.
![Page 17: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/17.jpg)
Breadth-first tree in a component
Aldous explores the graph G (n, n−1 + λn−4/3) via a breadth-first walk.
![Page 18: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/18.jpg)
Breadth-first tree in a component
Aldous explores the graph G (n, n−1 + λn−4/3) via a breadth-first walk.
![Page 19: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/19.jpg)
Breadth-first tree in a component
![Page 20: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/20.jpg)
Breadth-first tree in a component
![Page 21: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/21.jpg)
Breadth-first tree in a component
![Page 22: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/22.jpg)
Breadth-first tree in a component
![Page 23: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/23.jpg)
Breadth-first tree in a component
![Page 24: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/24.jpg)
Breadth-first tree in a component
![Page 25: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/25.jpg)
Breadth-first tree in a component
![Page 26: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/26.jpg)
Aldous gets breadth-first walk, (Xn(k); k = 0, 1, · · · ).
The increments satisfy:
Xn(k)− Xn(k − 1) = # (vertices discovered by vertex k)− 1.
![Page 27: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/27.jpg)
Aldous gets breadth-first walk, (Xn(k); k = 0, 1, · · · ).The increments satisfy:
Xn(k)− Xn(k − 1) = # (vertices discovered by vertex k)− 1.
![Page 28: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/28.jpg)
Vertex Discovered Vertices1 2, 3, 42 5, 63 None4 7, 85 96 None7 108 None9 None10 None
![Page 29: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/29.jpg)
Vertex Discovered Vertices1 2, 3, 42 5, 63 None4 7, 85 96 None7 108 None9 None10 None
The excursion starts at zero, and ends at -1.
![Page 30: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/30.jpg)
Vertex Discovered Vertices1 2, 3, 42 5, 63 None4 7, 85 96 None7 108 None9 None10 None
The excursion starts at zero, and ends at -1.
![Page 31: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/31.jpg)
![Page 32: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/32.jpg)
The edges {4, 6} (in red) and {4, 5} (not drawn) are allowable surplusedges, but {4, 9} (in blue) is not.
![Page 33: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/33.jpg)
The edges {4, 6} (in red) and {4, 5} (not drawn) are allowable surplusedges, but {4, 9} (in blue) is not.
![Page 34: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/34.jpg)
The edges {4, 6} (in red) and {4, 5} (not drawn) are allowable surplusedges, but {4, 9} (in blue) is not.The total number of allowable surplus edges is roughly the area under theexcursion on the left.
![Page 35: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/35.jpg)
Properties of Xn(k):
1 Component sizes are Tn(j)− Tn(j − 1) whereTn(j) = min{k : Xn(k) = −j}
2 The number of surplus edges (red edges before) in a component isapproximately
≈Bin (An(j), p)
An(j) = area under the j th excursion
Spencer [Spe97] has a wonderful (and short!) paper on why youshould expect to see the area term An(j).
![Page 36: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/36.jpg)
Properties of Xn(k):
1 Component sizes are Tn(j)− Tn(j − 1) whereTn(j) = min{k : Xn(k) = −j}
2 The number of surplus edges (red edges before) in a component isapproximately
≈Bin (An(j), p)
An(j) = area under the j th excursion
Spencer [Spe97] has a wonderful (and short!) paper on why youshould expect to see the area term An(j).
![Page 37: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/37.jpg)
Properties of Xn(k):
1 Component sizes are Tn(j)− Tn(j − 1) whereTn(j) = min{k : Xn(k) = −j}
2 The number of surplus edges (red edges before) in a component isapproximately
≈Bin (An(j), p)
An(j) = area under the j th excursion
Spencer [Spe97] has a wonderful (and short!) paper on why youshould expect to see the area term An(j).
![Page 38: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/38.jpg)
Properties of Xn(k):
![Page 39: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/39.jpg)
Scaling Limits
Recall p = 1/n then the two largest components are order n2/3.
Size of components are Tn(j + 1)− Tn(j) are order n2/3.
Figure: Brownian scaling: Simulations of n−1/3Xn(bn2/3tc) forλ ∈ {0, 1, 2, 3, 4, 5} and n = 700.
![Page 40: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/40.jpg)
Scaling Limits
Recall p = 1/n then the two largest components are order n2/3.Size of components are Tn(j + 1)− Tn(j) are order n2/3.
Figure: Brownian scaling: Simulations of n−1/3Xn(bn2/3tc) forλ ∈ {0, 1, 2, 3, 4, 5} and n = 700.
![Page 41: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/41.jpg)
Scaling Limits
Recall p = 1/n then the two largest components are order n2/3.Size of components are Tn(j + 1)− Tn(j) are order n2/3.
Figure: Brownian scaling: Simulations of n−1/3Xn(bn2/3tc) forλ ∈ {0, 1, 2, 3, 4, 5} and n = 700.
![Page 42: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/42.jpg)
Scaling Limits
Recall p = 1/n then the two largest components are order n2/3.Size of components are Tn(j + 1)− Tn(j) are order n2/3.Brownian scaling is c−1/2W (ct).
Theorem 3 (Aldous ’97 [Ald97]).
As n→∞n−1/3Xn(bn2/3tc) d−→W (t) + λt − 1
2t2
as processes, and W a standard Brownian motion.
![Page 43: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/43.jpg)
Scaling Limits: Components
Cn(1),Cn(2), · · · components of G (n, p) with
#Cn(1) ≥ #Cn(2) ≥ · · ·
Each of these components is encoded by an excursion like process:(Xn,i (k); k ≥ 0) encodes Cn(i).
Theorem 4 (It’s complicated, [Ald97, ABBG12, CG20]).
As n→∞ (n−1/3Xn,i (bn2/3tc); t ≥ 0
)i≥1
d−→ (ei (t); t ≥ 0)i≥1
where ei are excursions of W (t) + λt − 12 t2 above its running minimum
re-ordered by length.
![Page 44: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/44.jpg)
Scaling Limits: Components
Cn(1),Cn(2), · · · components of G (n, p) with
#Cn(1) ≥ #Cn(2) ≥ · · ·
Each of these components is encoded by an excursion like process:(Xn,i (k); k ≥ 0) encodes Cn(i).
Theorem 4 (It’s complicated, [Ald97, ABBG12, CG20]).
As n→∞ (n−1/3Xn,i (bn2/3tc); t ≥ 0
)i≥1
d−→ (ei (t); t ≥ 0)i≥1
where ei are excursions of W (t) + λt − 12 t2 above its running minimum
re-ordered by length.
![Page 45: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/45.jpg)
Scaling Limits: Components
Cn(1),Cn(2), · · · components of G (n, p) with
#Cn(1) ≥ #Cn(2) ≥ · · ·
Each of these components is encoded by an excursion like process:(Xn,i (k); k ≥ 0) encodes Cn(i).
Theorem 4 (It’s complicated, [Ald97, ABBG12, CG20]).
As n→∞ (n−1/3Xn,i (bn2/3tc); t ≥ 0
)i≥1
d−→ (ei (t); t ≥ 0)i≥1
where ei are excursions of W (t) + λt − 12 t2 above its running minimum
re-ordered by length.
![Page 46: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/46.jpg)
Scaling Limits: Surplus
For the i th largest component, the surplus edges are
Bin(
An(i), n−1 + λn−4/3)
≈ Bin
(n
∫ei (s) ds, n−1 + λn−4/3
)≈ Poisson
(∫ei (s) ds
)
Theorem 5 (Aldous [Ald97]).
As n→∞, the largest componets rescale as:
#Cn(j)
n2/3
d−→ ζj := life-time of ej .
The number of surplus edges of the corresponding component
surplus(Cn(j))d−→ Poisson
(∫ ζj
0ej(s) ds
)
![Page 47: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/47.jpg)
Scaling Limits: Surplus
For the i th largest component, the surplus edges are
Bin(
An(i), n−1 + λn−4/3)
≈ Bin
(n
∫ei (s) ds, n−1 + λn−4/3
)
≈ Poisson
(∫ei (s) ds
)
Theorem 5 (Aldous [Ald97]).
As n→∞, the largest componets rescale as:
#Cn(j)
n2/3
d−→ ζj := life-time of ej .
The number of surplus edges of the corresponding component
surplus(Cn(j))d−→ Poisson
(∫ ζj
0ej(s) ds
)
![Page 48: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/48.jpg)
Scaling Limits: Surplus
For the i th largest component, the surplus edges are
Bin(
An(i), n−1 + λn−4/3)
≈ Bin
(n
∫ei (s) ds, n−1 + λn−4/3
)≈ Poisson
(∫ei (s) ds
)
Theorem 5 (Aldous [Ald97]).
As n→∞, the largest componets rescale as:
#Cn(j)
n2/3
d−→ ζj := life-time of ej .
The number of surplus edges of the corresponding component
surplus(Cn(j))d−→ Poisson
(∫ ζj
0ej(s) ds
)
![Page 49: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/49.jpg)
Scaling Limits: Surplus
For the i th largest component, the surplus edges are
Bin(
An(i), n−1 + λn−4/3)
≈ Bin
(n
∫ei (s) ds, n−1 + λn−4/3
)≈ Poisson
(∫ei (s) ds
)
Theorem 5 (Aldous [Ald97]).
As n→∞, the largest componets rescale as:
#Cn(j)
n2/3
d−→ ζj := life-time of ej .
The number of surplus edges of the corresponding component
surplus(Cn(j))d−→ Poisson
(∫ ζj
0ej(s) ds
)
![Page 50: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/50.jpg)
Subsequent Work
Aldous gave a process-level scaling limit for a function which encodes theinformation of the random graph.
Theorem 6 (Addario-Berry, Broutin, Goldschmidt[ABBG12]).
There exists sequence of random metric spaces (Mi ; i ≥ 1) such that(n−1/3Cn(i); i ≥ 1
)d−→ (Mi ; i ≥ 1)
![Page 51: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/51.jpg)
Subsequent Work
Aldous gave a process-level scaling limit for a function which encodes theinformation of the random graph.
Theorem 6 (Addario-Berry, Broutin, Goldschmidt[ABBG12]).
There exists sequence of random metric spaces (Mi ; i ≥ 1) such that(n−1/3Cn(i); i ≥ 1
)d−→ (Mi ; i ≥ 1)
![Page 52: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/52.jpg)
What are the metric spaces?
![Page 53: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/53.jpg)
What are the metric spaces?
![Page 54: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/54.jpg)
What are the metric spaces?
![Page 55: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/55.jpg)
What are the metric spaces?
![Page 56: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/56.jpg)
What are the metric spaces?
![Page 57: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/57.jpg)
What are the metric spaces?
![Page 58: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/58.jpg)
What are the metric spaces?
![Page 59: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/59.jpg)
An epidemic model
Population of m + k people.
On day zero k people are infected with a disease, and m people are healthy.Infected person infects healthy person with probability p on that day, andare forever cured after that day.Then
Bin (m, q) are infected by the next day where
q = (1− p)k = probability of not being infected by the k infected people.
![Page 60: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/60.jpg)
An epidemic model
Population of m + k people.On day zero k people are infected with a disease, and m people are healthy.
Infected person infects healthy person with probability p on that day, andare forever cured after that day.Then
Bin (m, q) are infected by the next day where
q = (1− p)k = probability of not being infected by the k infected people.
![Page 61: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/61.jpg)
An epidemic model
Population of m + k people.On day zero k people are infected with a disease, and m people are healthy.Infected person infects healthy person with probability p on that day, andare forever cured after that day.
ThenBin (m, q) are infected by the next day where
q = (1− p)k = probability of not being infected by the k infected people.
![Page 62: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/62.jpg)
An epidemic model
Population of m + k people.On day zero k people are infected with a disease, and m people are healthy.Infected person infects healthy person with probability p on that day, andare forever cured after that day.Then
Bin (m, q) are infected by the next day where
q = (1− p)k = probability of not being infected by the k infected people.
![Page 63: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/63.jpg)
This leads to the following Markov chain:
Zn(0) = k , Cn(0) = k
Zn(h + 1) =#infected people the next day with Zn(h)infected people and n − Cn(h) healthy people
Cn(h) =h∑
j=0
Zn(j)
Theorem 7 (C. [Cla20]).
When p = n−1 + λn−4/3 and kn−1/3 → x as n→∞ then(n−1/3Zn(bn1/3tc), n−2/3Cn(bn1/3tc)
)d−→ (Z (t),C (t))
where (Z ,C ) solves
Z (t) = x + Xλ ◦C (t), C (t) =
∫ t
0Z (s) ds, Xλ(t) = W (t) + λt − 1
2t2.
![Page 64: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/64.jpg)
This leads to the following Markov chain:
Zn(0) = k , Cn(0) = k
Zn(h + 1) =#infected people the next day with Zn(h)infected people and n − Cn(h) healthy people
Cn(h) =h∑
j=0
Zn(j)
Theorem 7 (C. [Cla20]).
When p = n−1 + λn−4/3 and kn−1/3 → x as n→∞ then(n−1/3Zn(bn1/3tc), n−2/3Cn(bn1/3tc)
)d−→ (Z (t),C (t))
where (Z ,C ) solves
Z (t) = x + Xλ ◦C (t), C (t) =
∫ t
0Z (s) ds, Xλ(t) = W (t) + λt − 1
2t2.
![Page 65: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/65.jpg)
This leads to the following Markov chain:
Zn(0) = k , Cn(0) = k
Zn(h + 1) =#infected people the next day with Zn(h)infected people and n − Cn(h) healthy people
Cn(h) =h∑
j=0
Zn(j)
Theorem 7 (C. [Cla20]).
When p = n−1 + λn−4/3 and kn−1/3 → x as n→∞ then(n−1/3Zn(bn1/3tc), n−2/3Cn(bn1/3tc)
)d−→ (Z (t),C (t))
where (Z ,C ) solves
Z (t) = x + Xλ ◦C (t), C (t) =
∫ t
0Z (s) ds, Xλ(t) = W (t) + λt − 1
2t2.
![Page 66: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/66.jpg)
Subsequent Works
Generalization of Erdos-Renyi random graph: Rank-1 inhomogeneousmodel.
Graph on n vertices with edges included
P ({i , j} is an edge) = 1− exp (−qwiwj) ,
where w1 ≥ w2 ≥ · · · ≥ wn > 0 and some q ∈ [0,∞).Weights are a propensity to have neighbors.
![Page 67: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/67.jpg)
Subsequent Works
Generalization of Erdos-Renyi random graph: Rank-1 inhomogeneousmodel.Graph on n vertices with edges included
P ({i , j} is an edge) = 1− exp (−qwiwj) ,
where w1 ≥ w2 ≥ · · · ≥ wn > 0 and some q ∈ [0,∞).
Weights are a propensity to have neighbors.
![Page 68: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/68.jpg)
Subsequent Works
Generalization of Erdos-Renyi random graph: Rank-1 inhomogeneousmodel.Graph on n vertices with edges included
P ({i , j} is an edge) = 1− exp (−qwiwj) ,
where w1 ≥ w2 ≥ · · · ≥ wn > 0 and some q ∈ [0,∞).Weights are a propensity to have neighbors.
![Page 69: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/69.jpg)
Subsequent Works
Theorem 8 (Aldous, Limic [AL98], Broutin ,Duquesne,Wang [BDW20]).
Under some assumptions (some technical, some natural) the rank-1inhomogeneous model:
1 A-L [AL98] a breadth-first walk has a rescaling limit:
σW (t) + λt − 1
2σ2t2 +
∑j≥1
(cj1(Ej≤t) − c2
j t),
for a Brownian motion W and some exponential random variables Ej
with E[Ej ] = 1/cj .
2 B-D-W [BDW20]. The components of the model have scaling metricspace limits.
![Page 70: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/70.jpg)
Subsequent Works
A different way to construct random graphs: graphs from degreesequences.
Have n vertices where each vertex has degree dj ≥ 1 with∑n
j=1 dj is even.
Figure: From [vdH17].
![Page 71: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/71.jpg)
Subsequent Works
A different way to construct random graphs: graphs from degreesequences.Have n vertices where each vertex has degree dj ≥ 1 with
∑nj=1 dj is even.
Figure: From [vdH17].
![Page 72: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/72.jpg)
Subsequent Works
A different way to construct random graphs: graphs from degreesequences.Have n vertices where each vertex has degree dj ≥ 1 with
∑nj=1 dj is even.
Figure: From [vdH17].
![Page 73: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/73.jpg)
Subsequent Works
Theorem 9 (Vaguely stated below [MR95, MR98],[Jos14], [CG20]).
1 Molloy, Reed [MR95, MR98]: when the degrees are i.i.d. randomvariables, there is a phase transition (like for Erdos-Renyi randomgraphs) where a giant component emerges.
2 Joseph [Jos14]: For Dj i.i.d. with power law distribution, there is anencoding walk with scaling limit as
α-stable analog of W (t)− 1
2t2.
3 Conchon-Kerjan, Goldschmidt [CG20]: There exists metric spacescaling limits which are the α-stable versions of the “Browniangraphs” in the Erdos-Renyi case.
![Page 74: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/74.jpg)
Subsequent Works
Theorem 9 (Vaguely stated below [MR95, MR98],[Jos14], [CG20]).
1 Molloy, Reed [MR95, MR98]: when the degrees are i.i.d. randomvariables, there is a phase transition (like for Erdos-Renyi randomgraphs) where a giant component emerges.
2 Joseph [Jos14]: For Dj i.i.d. with power law distribution, there is anencoding walk with scaling limit as
α-stable analog of W (t)− 1
2t2.
3 Conchon-Kerjan, Goldschmidt [CG20]: There exists metric spacescaling limits which are the α-stable versions of the “Browniangraphs” in the Erdos-Renyi case.
![Page 75: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/75.jpg)
Subsequent Works
Theorem 9 (Vaguely stated below [MR95, MR98],[Jos14], [CG20]).
1 Molloy, Reed [MR95, MR98]: when the degrees are i.i.d. randomvariables, there is a phase transition (like for Erdos-Renyi randomgraphs) where a giant component emerges.
2 Joseph [Jos14]: For Dj i.i.d. with power law distribution, there is anencoding walk with scaling limit as
α-stable analog of W (t)− 1
2t2.
3 Conchon-Kerjan, Goldschmidt [CG20]: There exists metric spacescaling limits which are the α-stable versions of the “Browniangraphs” in the Erdos-Renyi case.
![Page 76: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/76.jpg)
L. Addario-Berry, N. Broutin, and C. Goldschmidt.The continuum limit of critical random graphs.Probab. Theory Related Fields, 152(3-4):367–406, 2012.
David Aldous and Vlada Limic.The entrance boundary of the multiplicative coalescent.Electron. J. Probab., 3:No. 3, 59 pp. 1998.
David Aldous.Brownian excursions, critical random graphs and the multiplicativecoalescent.Ann. Probab., 25(2):812–854, 1997.
Nicolas Broutin, Thomas Duquesne, and Minmin Wang.Limits of multiplicative inhomogeneous random graphs and Levy trees:Limit theorems.arXiv e-prints, page arXiv:2002.02769, February 2020.
![Page 77: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/77.jpg)
Bela Bollobas.The evolution of random graphs.Trans. Amer. Math. Soc., 286(1):257–274, 1984.
Guillaume Conchon–Kerjan and Christina Goldschmidt.The stable graph: the metric space scaling limit of a critical randomgraph with i.i.d. power-law degrees.arXiv e-prints, page arXiv:2002.04954, February 2020.
David Clancy, Jr.A new relationship between Erdos-Renyi graphs, epidemic models andBrownian motion with parabolic drift.arXiv e-prints, page arXiv:2006.06838, June 2020.
P. Erdos and A. Renyi.On the evolution of random graphs.Magyar Tud. Akad. Mat. Kutato Int. Kozl., 5:17–61, 1960.
![Page 78: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/78.jpg)
Adrien Joseph.The component sizes of a critical random graph with given degreesequence.Ann. Appl. Probab., 24(6):2560–2594, 2014.
Tomasz Luczak, Boris Pittel, and John C. Wierman.The structure of a random graph at the point of the phase transition.Trans. Amer. Math. Soc., 341(2):721–748, 1994.
Michael Molloy and Bruce Reed.A critical point for random graphs with a given degree sequence.In Proceedings of the Sixth International Seminar on Random Graphsand Probabilistic Methods in Combinatorics and Computer Science,“Random Graphs ’93” (Poznan, 1993), volume 6, pages 161–179,1995.
Michael Molloy and Bruce Reed.The size of the giant component of a random graph with a givendegree sequence.Combin. Probab. Comput., 7(3):295–305, 1998.
![Page 79: Hidden Gems: Aldous's Brownian excursions, critical random ...djclancy/files/AldousERHiddenGems.pdfThe edges f4;6g(in red) and f4;5g(not drawn) are allowable surplus edges, but f4;9g(in](https://reader035.vdocument.in/reader035/viewer/2022081621/612191e106fe642e99624e70/html5/thumbnails/79.jpg)
Joel Spencer.Enumerating graphs and Brownian motion.Comm. Pure Appl. Math., 50(3):291–294, 1997.
Remco van der Hofstad.Random graphs and complex networks. Vol. 1.Cambridge Series in Statistical and Probabilistic Mathematics, [43].Cambridge University Press, Cambridge, 2017.