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math105-s22:s:sk [2022/04/18 04:27]
griffinke [Brownian sphere] 		math105-s22:s:sk [2026/02/21 14:41] (current)
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 Definition[Topological path]: Fix a topology space $(X,\theta_X)$. A topological path in $X$ is a continuous mapping $\gamma:[a,b]\subset\mathbb{R}\rightarrow X$. Definition[Topological path]: Fix a topology space $(X,\theta_X)$. A topological path in $X$ is a continuous mapping $\gamma:[a,b]\subset\mathbb{R}\rightarrow X$.
  
-Definition[Real tree]: a compact metric space $(X,d)$ is a real tree if (1) $\forall x,y\in X$, there exists a shortest path $ [ [ x , y ] ] $ from $x$ to $y$ with length $d(x,y)$ ;  (2) $\forall x,y\in X$, the only non-self-intersecting path from $x$ to $y$ is $ [ [ x , y ] ] $.+Definition[Real/continuum tree]: a compact metric space $(X,d)$ is a real tree if (1) $\forall x,y\in X$, there exists a shortest path $ [ [ x , y ] ] $ from $x$ to $y$ with length $d(x,y)$ ;  (2) $\forall x,y\in X$, the only non-self-intersecting path from $x$ to $y$ is $ [ [ x , y ] ] $.
  
 +Example: we can construct a real tree with an undergraph of a function. Please refer to Christina Goldschmidt's lecture https://www.stats.ox.ac.uk/~goldschm/WarwickLecture2.pdf
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 +How can we generate a random tree? There are several methods. For example, we can generate the tree via a branching process (Galton–Watson process).
 ==== References ==== ==== References ====
  
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 3. Lecture 2: The continuum random tree (continued). Christina Goldschmidt. Retrieved from https://www.stats.ox.ac.uk/~goldschm/WarwickLecture2.pdf 3. Lecture 2: The continuum random tree (continued). Christina Goldschmidt. Retrieved from https://www.stats.ox.ac.uk/~goldschm/WarwickLecture2.pdf
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 +4. The Continuum Random Tree I. David Aldous. Retrieved from https://projecteuclid.org/journals/annals-of-probability/volume-19/issue-1/The-Continuum-Random-Tree-I/10.1214/aop/1176990534.full
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 +5. The Continuum Random Tree II: an overview. David Aldous. Retrieved from https://www.stat.berkeley.edu/~aldous/Papers/me55.pdf
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 +6. The Continuum Random Tree III. David Aldous. Retrieved from https://projecteuclid.org/journals/annals-of-probability/volume-21/issue-1/The-Continuum-Random-Tree-III/10.1214/aop/1176989404.full
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 +7. Gwynne, E., Miller, J.P., Sheffield, S. The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity. https://arxiv.org/pdf/1705.11161.pdf
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 +8. Gwynne, Ewain, Jason Miller and Scott Sheffield. “Harmonic functions on mated-CRT maps.” Electronic journal of probability, 24, 58 (May 2019) https://dspace.mit.edu/bitstream/handle/1721.1/126714/1807.07511.pdf?sequence=2&isAllowed=y
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 +9. Jason Miller - 1/4 Equivalence of Liouville quantum gravity and the Brownian map. 
 +Institut des Hautes Études Scientifiques (IHÉS). https://www.youtube.com/watch?v=NB9iZ8ZX4dQ
 +
 ===== Additional material ===== ===== Additional material =====
  
math105-s22/s/sk.1650256055.txt.gz · Last modified: 2026/02/21 14:43 (external edit)

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