<p>Diffusion limited aggregation and its generalization, dielectric-breakdown model play an important role in physics, approximating a range of natural phenomena. Yet little is known about them, with the famous Kesten’s estimate on the DLAs growth being perhaps the most important result. Using a different approach we prove a generalisation of this result for the DBM in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5276_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Z}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5276_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Z}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation>. The obtained estimate depends on the DBM parameter, and matches with the best known results for DLA. In particular, since our methods are different from Kesten’s, our argument provides a new proof for Kesten’s result both in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5276_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Z}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5276_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Z}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation>.</p>

How Long are the Arms in DBM?

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2025年4月2日

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Ilya Losev,
Stanislav Smirnov

摘要

Diffusion limited aggregation and its generalization, dielectric-breakdown model play an important role in physics, approximating a range of natural phenomena. Yet little is known about them, with the famous Kesten’s estimate on the DLAs growth being perhaps the most important result. Using a different approach we prove a generalisation of this result for the DBM in $\mathbb {Z}^2$ $Z^{2}$ and $\mathbb {Z}^3$ $Z^{3}$ . The obtained estimate depends on the DBM parameter, and matches with the best known results for DLA. In particular, since our methods are different from Kesten’s, our argument provides a new proof for Kesten’s result both in $\mathbb {Z}^2$ $Z^{2}$ and $\mathbb {Z}^3$ $Z^{3}$ .