Lace Expansion and Mean-Field Behavior for the Random Connection Model

Lace Expansion and Mean-Field Behavior for the Random Connection Model
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随机连接模型的花边扩展和平均场行为

DOI：
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发表时间：
2019
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通讯作者：
Kilian Matzke
Kilian Matzke
中科院分区：
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作者：
M. Heydenreich;R. Hofstad;G. Last;Kilian Matzke

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我们考虑三个版本的连接函数$varphi$的随机连接模型：有限方差版本（包括布尔模型），扩展版本，和远程版本。我们适应花边扩展，以适应基本的连续空间泊松点过程的框架，推导出足够高的维度的三角形条件，并进一步建立红外线的界限。由此，可以推导出模型的平均场行为。作为一个例子，我们表明，临界指数$伽玛$需要其平均场值$伽玛=1$和渗流函数是连续的。
We consider the random connection model for three versions of the connection function $varphi$: A finite-variance version (including the Boolean model), a spread-out version, and a long-range version. We adapt the lace expansion to fit the framework of the underlying continuum-space Poisson point process to derive the triangle condition in sufficiently high dimension and furthermore to establish the infra-red bound. From this, mean-field behavior of the model can be deduced. As an example, we show that the critical exponent $gamma$ takes its mean-field value $gamma=1$ and that the percolation function is continuous.