翻訳と辞書 Words near each other ・ Loop splitting ・ Loop start ・ Loop subdivision surface ・ Loop the Loop (Coney Island) ・ Loop the Loop (Olentangy Park) ・ Loop the Loop (Young's Million Dollar Pier) ・ Loop theorem ・ Loop tiling ・ Loop unrolling ・ Loop unswitching ・ Loop v. Litchfield ・ Loop variant ・ Loop, Germany ・ Loop, Texas ・ Loop, West Virginia ・ Loop-erased random walk ・ Loop-invariant code motion ・ Loop-mediated isothermal amplification ・ Loop-O-Plane ・ Loop-switch sequence ・ Loop-the-Loop (disambiguation) ・ Loop-the-Loop (song) ・ Loopallu Festival ・ Loopback ・ Loopback device ・ LoopCo ・ Loope, California ・ Loopealse ・ Loopebach ・ Looped Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Loop-erased random walk ： ウィキペディア英語版 Loop-erased random walk In mathematics, loop-erased random walk is a model for a random simple path with important applications in combinatorics and, in physics, quantum field theory. It is intimately connected to the uniform spanning tree, a model for a random tree. See also ''random walk'' for more general treatment of this topic. ==Definition== Assume ''G'' is some graph and $\backslash gamma$ is some path of length ''n'' on ''G''. In other words, $\backslash gamma(1),\backslash dots,\backslash gamma(n)$ are vertices of ''G'' such that $\backslash gamma(i)$ and $\backslash gamma(i+1)$ are connected by an edge. Then the loop erasure of $\backslash gamma$ is a new simple path created by erasing all the loops of $\backslash gamma$ in chronological order. Formally, we define indices $i\_j$ inductively using :$i\_1\; =\; 1\backslash ,$ :$i\_=\backslash max\backslash +1\backslash ,$ where "max" here means up to the length of the path $\backslash gamma$. The induction stops when for some $i\_j$ we have $\backslash gamma(i\_j)=\backslash gamma(n)$. Assume this happens at ''J'' i.e. $i\_J$ is the last $i\_j$. Then the loop erasure of $\backslash gamma$, denoted by $\backslash mathrm(\backslash gamma)$ is a simple path of length ''J'' defined by :$\backslash mathrm(\backslash gamma)(j)=\backslash gamma(i\_j).\backslash ,$ Now let ''G'' be some graph, let ''v'' be a vertex of ''G'', and let ''R'' be a random walk on ''G'' starting from ''v''. Let ''T'' be some stopping time for ''R''. Then the loop-erased random walk until time ''T'' is LE(''R''(())). In other words, take ''R'' from its beginning until ''T'' — that's a (random) path — erase all the loops in chronological order as above — you get a random simple path. The stopping time ''T'' may be fixed, i.e. one may perform ''n'' steps and then loop-erase. However, it is usually more natural to take ''T'' to be the hitting time in some set. For example, let ''G'' be the graph Z2 and let ''R'' be a random walk starting from the point (0,0). Let ''T'' be the time when ''R'' first hits the circle of radius 100 (we mean here of course a ''discretized'' circle). LE(''R'') is called the loop-erased random walk starting at (0,0) and stopped at the circle. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Loop-erased random walk」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.