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 Advanced Statistics Advanced Probability and Statistics Math Forum

 May 5th, 2017, 09:17 AM #1 Newbie   Joined: Jun 2016 From: France Posts: 4 Thanks: 0 Diffusion limited aggregation Hello! Firstly, sorry for my bad English. We consider a sequence $A_n$ of subsets of $\mathbb{Z}$, which increase according to the following property: at the step n, a particle is thrown at the origin and is moving according to a random walk - the probabilities to go at the left or at the right are 1/2. The particle is moving until it comes out of $A_n$ in a point $X_{n+1}$. We define : $A_{n+1} = A_n \cup X_n$. The model is initialized with $A_1=\{0\}$. $A_n$ is called aggregate at step n. I can understand that: If $A_1 = \{0\}$, then $A_2 = \{0,1\}$ with probability $1/2$, or $A_2 = \{-1,0\}$ with probability $1/2$. Indeed, the particle comes out of $A_1$ at the point $1$ (by the right) or $-1$ (by the left). If $A_2 = \{0,1\}$, $A_3 = \{0,1,2\}$ or $A_3 = \{-1,0,1\}$. If $A_2 = \{-1,0\}$, $A_3 = \{-1,0,2\}$ or $A_3 = \{-2,-1,0\}$ (always with probability $1/2$). I've done this, until the step 5: I have two questions: Could you explain why the "set" of occupied sites is an interval of integers and what its length? I really can't answer to this question... And we note $A_n = \{G_n, ..., D_n\}$ with $G_n$ and $D_n$ are respectively the leftmost and the rightmost points of the aggregate. We note $Z_n = D_n + G_n$. How we find 'again' $G_n$ and $D_n$ from $Z_n$ ? I've done this, until the step 5, but I don't see how we found $G_n$ and $D_n$ from $Z_n$: If someone could help me, just for one question, I will be very grateful to him! Have a nice day! Last edited by skipjack; August 7th, 2017 at 10:30 PM. August 7th, 2017, 10:32 PM #2 Global Moderator   Joined: Dec 2006 Posts: 20,372 Thanks: 2009 The English is okay. Tags aggregation, diffusion, limited Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post WWRtelescoping Differential Equations 2 September 15th, 2014 01:46 AM juliousceasor Applied Math 0 March 14th, 2011 08:10 AM Smalley Calculus 1 July 15th, 2008 09:59 PM germanaries Applied Math 0 November 15th, 2007 01:53 PM

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