The conformally invariant measure on self-avoiding loops
The conformally invariant measure on self-avoiding loops
复制标题
自回避环的共形不变测度
DOI：
10.1090/s0894-0347-07-00557-7
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发表时间：
2005
影响因子：
3.9
通讯作者：
W. Werner
中科院分区：
文献类型：
--
作者：
W. Werner
本文的目的是构造和描述平面上和任意黎曼曲面上的自避环集上的自然测度。所谓曲面S上的自避免环，我们指的是从单位圆到S模单调重参数化的连续内射映射（即我们只看环的迹而忽略它的参数化）。我们将构造一个具有某些强共形不变性性质的测度，并且我们将看到这个测度是唯一具有这些性质的测度。让我们首先描述这个强共形不变性，用平面中自避免环集上的测度来表述：我们说这样的测度fi满足共形限制，如果对任意两个共形等价的域D和Df，（即，使得存在从D到D '的保形映射），测度fi的像被限制到停留在D中的环的集合，通过从D到D '的任何共形映射，x被限制到留在D '中的循环的集合。注意，这个条件特别意味着测度fi是平移不变的，尺度不变的（因此，它有无限大的总质量，只要fi t？0），并且如果fi满足共形约束，则对于任意正常数c，cfi也满足共形约束。我们将说自避环集上的测度是非平凡的，如果对于某个0 < S < A < oo，直径至少为S且停留在某个半径为A的圆盘中的环集的质量既不是0也不是无穷大。我们将证明以下结果：
The aim of the present paper is to construct and describe a natural measure on the set of self-avoiding loops in the plane and on any Riemann surface. By a self-avoiding loop on a surface S, we mean a continuous injective map from the unit circle into S modulo monotone reparametrizations (i.e. we look only at the trace of the loop and forget about its parametrization). We will construct a measure that possess some strong conformai invariance properties, and we shall see that this measure is the only one with these properties. Let us first describe this strong conformai invariance property, phrased in terms of a measure on the set of self-avoiding loops in the plane: We say that such a measure fi satisfies conformai restriction if for any two conformally equivalent domains D and Df (i.e. such that there exists a conformai map from D onto D') in the plane, the image of the measure fi restricted to the set of loops that stay in D, via any conformai map $ from D onto D', is exactly the measure ?x restricted to the set of loops that stay in D'. Note that this condition implies in particular that the measure fi is translation invariant, scale-invariant (and therefore that it has infinite total mass provided that fi t? 0), and that if fi satisfies conformai restriction, then so does cfi for any positive constant c. We are going to say that a measure on the set of self-avoiding loops is non-trivial if for some 0 < S < A < oo, the mass of the set of loops of diameter at least S and that stay in some disc of radius A is neither 0 nor infinite. We shall prove the following result: