<p>The Cluster-cluster model was introduced by Meakin et al. in 1984. Each <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$x\in \mathbb {Z}^d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> starts with a cluster of size 1 with probability <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p \in (0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> independently. Each cluster <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="MediaObjects/10955_2026_3615_IEq3_HTML.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="120" Type="Linedraw" Width="16" /> </InlineMediaObject> </InlineEquation> performs a continuous-time SRW with rate <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="MediaObjects/10955_2026_3615_IEq4_HTML.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="120" Type="Linedraw" Width="42" /> </InlineMediaObject> </InlineEquation>. If it attempts to move to a vertex occupied by another cluster, it does not move, and instead the two clusters connect via a new edge. Focusing on dimension <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$d=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, we show that for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\alpha >-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, at time <i>t</i>, the cluster size is of order <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$t^{\frac{1}{\alpha + 2}}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>t</mi> <mfrac> <mn>1</mn> <mrow> <mi>α</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </msup> </math></EquationSource> </InlineEquation>, and for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\alpha < -2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo><</mo> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> we get an infinite cluster in finite time a.s. Additionally, for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\alpha = 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> we show convergence in distribution of the scaling limit.</p>

On One-Dimensional Cluster-cluster Model

期刊论文
发表日期：

2026年4月28日

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Noam Berger,
Eviatar B. Procaccia,
Daniel Sharon

摘要

The Cluster-cluster model was introduced by Meakin et al. in 1984. Each $x\in \mathbb {Z}^d$ $x \in Z^{d}$ starts with a cluster of size 1 with probability $p \in (0,1]$ $p \in (0, 1]$ independently. Each cluster performs a continuous-time SRW with rate . If it attempts to move to a vertex occupied by another cluster, it does not move, and instead the two clusters connect via a new edge. Focusing on dimension $d=1$ $d = 1$ , we show that for $\alpha >-2$ $α > - 2$ , at time t, the cluster size is of order $t^{\frac{1}{\alpha + 2}}$ $t^{\frac{1}{α + 2}}$ , and for $\alpha < -2$ $α < - 2$ we get an infinite cluster in finite time a.s. Additionally, for $\alpha = 0$ $α = 0$ we show convergence in distribution of the scaling limit.