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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Connective constant ： ウィキペディア英語版 Connective constant In mathematics, the connective constant is a numerical quantity associated with self-avoiding walks on a lattice. It is studied in connection with the notion of universality in two-dimensional statistical physics models. While the connective constant depends on the choice of lattice so itself is not universal (similarly to other lattice-dependent quantities such as the critical probability threshold for percolation), it is nonetheless an important quantity that appears in conjectures for universal laws. Furthermore, the mathematical techniques used to understand the connective constant, for example in the recent rigorous proof by Duminil-Copin and Smirnov that the connective constant of the hexagonal lattice has the precise value $\backslash sqrt\}〕\; to\; a\; possible\; approach\; for\; attacking\; other\; important\; open\; problems\; in\; the\; study\; of\; self-avoiding\; walks,\; notably\; the\; conjecture\; that\; self-avoiding\; walks\; converge\; in\; the\; scaling\; limit\; to\; theSchramm–Loewner\; evolution.$ ==Definition== The connective constant is defined as follows. Let $c\_n$ denote the number of ''n''-step self-avoiding walks starting from a fixed origin point in the lattice. Since every ''n'' + ''m'' step self avoiding walk can be decomposed into an ''n''-step self-avoiding walk and an m-step self-avoiding walk, it follows that $c\_\; \backslash leq\; c\_n\; c\_m$. Then by applying Fekete's lemma to the logarithm of the above relation, the limit $\backslash mu\; =\; \backslash lim\_\; c\_n^$ can be shown to exist. This number $\backslash mu$ is called the connective constant, and clearly depends on the particular lattice chosen for the walk since $c\_n$ does. The value of $\backslash mu$ is precisely known only for two lattices, see below. For other lattices, $\backslash mu$ has only been approximated numerically. It is conjectured that $c\_n\; \backslash approx\; \backslash mu^n\; n^$ as n goes to infinity, where $\backslash mu$ depends on the lattice, but the critical exponent $\backslash gamma$ is universal (it depends on dimension, but not the specific lattice). In 2-dimensions it is conjectured that $\backslash gamma\; =\; 43/32$ 〔 〕〔 〕 ==Known values〔 〕== These values are taken from the 1998 Jensen–Guttmann paper. The connective constant of the $(3.12^2)$ lattice, since each step on the hexagonal lattice corresponds to either two or three steps in it, can be expressed exactly as a root of the polynomial : $x^\; -\; 4x^8\; -\; 8x^7\; -\; 4x^6\; +\; 2x^4\; +\; 8x^3\; +\; 12x^2\; +\; 8x\; +\; 2$ given the exact expression for the hexagonal lattice connective constant. More information about these lattices can be found in the percolation threshold article. ==Duminil-Copin–Smirnov proof== In 2010, Hugo Duminil-Copin and Stanislav Smirnov published the first rigorous proof of the fact that $\backslash mu=\backslash sqrt(a,b)$ as the total rotation of the direction in radians when $\backslash gamma$ is traversed from $a$ to $b$. The aim of the proof is to show that the partition function : $Z(x)=\backslash sum\_x^\; =\; \backslash sum\_^c\_n\; x^n$ converges for Given a domain $\backslash Omega$ in the hexagonal lattice, a starting mid-edge $a$, and two parameters $x$ and $\backslash sigma$, we define the parafermionic observable $F(z)=\backslash sum\_\; e^x^.$ If $x\; =\; x\_c=\; 1/\backslash sqrt$ with 2L cells forming the left hand side, T cells across, and upper and lower sides at an angle of $\backslash pm\; \backslash pi/3$. (Picture needed.) We embed the hexagonal lattice in the complex plane so that the edge lengths are 1 and the mid-edge in the center of the left hand side is positioned at −1/2. Then the vertices in $S\_$ are given by : $V(S\_)=\backslash ,\; \backslash ;\; |\backslash sqrtIm(z)-Re(z)|\; \backslash leq\; 3L\backslash \}.$ We now define partition functions for self-avoiding walks starting at $a$ and ending on different parts of the boundary. Let $\backslash alpha$ denote the left hand boundary, $\backslash beta$ the right hand boundary, $\backslash epsilon$ the upper boundary, and $\backslash bar$ the lower boundary. Let : $$ A_^x:=\sum_} x^,\quad B_^x:=\sum_ x^, \quad E_^x:=\sum_} x^. By summing the identity : $(p-v)F(p)\; +\; (q-v)F(q)\; +\; (r-v)F(r)\; =\; 0$ over all vertices in $V(S\_)$ and noting that the winding is fixed depending on which part of the boundary the path terminates at, we can arrive at the relation : $1=\; \backslash cos(3\backslash pi/8)\; A\_^\; +\; B\_^\; +\; \backslash cos(\backslash pi/4)\; E\_^$ after another clever computation. Letting $L\backslash to\backslash infty$, we get a strip domain $S\_T$ and partition functions : $$ A_^x:=\sum_} x^,\quad B_^x:=\sum_ x^, \quad E_^x:=\sum_} x^. It was later shown that $E\_^=0$, but we do not need this for the proof.〔 〕 We are left with the relation : $1=\; \backslash cos(3\backslash pi/8)\; A\_^\; +\; B\_^$. From here, we can derive the inequality : $A\_^\; -\; A\_^\; \backslash leq\; x\_c\; (B\_^)^2$ And arrive by induction at a strictly positive lower bound for $B\_^$. Since $Z(x\_c)\backslash geq\backslash sum\_B\_T^\; =\backslash infty$, we have established that and $T\_0>\backslash cdots\; >\; T\_j$. Note that we can bound : $B\_T^x\backslash leq\; (x/x\_c)^T\; B\_T^\backslash leq\; (x/x\_c)^T$ which implies . Finally, it is possible to bound the partition function by the bridge partition functions : And so, we have that $\backslash mu\; =\; \backslash sqrt\{2+\backslash sqrt\{2\}\}$ as desired. 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「Connective constant」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.