A third rigorous result about DLA

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This post highlights recent work which constitutes only the third rigorous result about planar diffusion-limited aggregation (DLA), a paradigmatic model of random, dendritic growth.

The definition of DLA

$$ A_{t+1} = A_t \cup \\{X\\},$$$$ P (A_{t+1} = A_t \cup \\{x\\} \bigm\vert A_t) = H_{\partial A_t} (x), $$

where $H_{\partial A_t}$ is the harmonic measure of $\partial A_t$,3 and where $\partial A_t$ denotes the boundary of $A_t$, i.e., elements $y$ which do not belong to $A_t$, but are at distance $\| y - x\| = 1$ from some element $x$ which does.

State-of-the-art simulations show that large DLA clusters have a cross-shaped backbone with many smaller scales of branching, the confluence of which forms narrow channels, called “fjords” (Figure 1).4 Since the “tips” of the cluster are exposed, their boundaries have greater growth probabilities than those at the bottom of the fjords. The multifractal conjecture concerns the way these growth probabilities scale with the linear size (i.e., the diameter) of the cluster.

Figure 1. A DLA cluster with roughly 145 million particles simulated by Denis Grebenkov and Dmitry Beliaev.

Figure 1. A DLA cluster with roughly 145 million particles simulated by Denis Grebenkov and Dmitry Beliaev.

There are few rigorous results about DLA

$$ \mathrm{diam} (A_t) \leq C t^{\frac{2}{3}}. $$

This result is a consequence of an upper bound on the harmonic measure of a connected set. The second, due to Eberz-Wagner,6 states that, almost surely, the DLA cluster will have infinitely-many holes.

The third rigorous result about DLA comes from a harmonic measure lower bound I recently proved with Shirshendu Ganguly and Alan Hammond. This result is related to the conjectural multifractality of DLA.

The multifractal conjecture

$$ \log p_{min} (t) = E \big[ \log \min \\{ H_{\partial A_t} (x): H_{\partial A_t} (x) > 0 \\} \big]. $$$$ \min \\{ H_{\partial A_t} (x): H_{\partial A_t} (x) > 0 \\}. $$$$ p_{min} (t) \approx L(t)^{-\alpha}, $$$$ c g(t) \leq f(t) \leq C g(t) $$

for all sufficiently large $t$.

A rigorous result concerning the failure of multifractality

$$ p_{min} (t) \approx L(t)^{-\alpha}, $$

then it is natural to wonder: How rapidly can $p_{min} (t)$ decrease, as a function of $L(t)$?

$$ \log \min \\{ H_B (x): H_B (x) > 0 \\} \geq -C_1 n. $$$$ \log p_{min} (t) \geq -C_2 t. $$

Hence, the smallest growth probability can be no smaller than exponential in $t$.

$$ \log \widetilde p \approx - \widetilde L^2, $$$$ \log p_{min} (t) \approx - t^\beta, $$

which our result constrains to $\beta < 1$.

Takeaway: A result about harmonic measure leads to the third rigorous result about DLA, which is a lower bound on the smallest growth probability.

  1. It is also possible to define DLA on other graphs. ↩︎

  2. Thomas A. Witten Jr and Leonard M. Sander. Diffusion-limited aggregation, a kinetic critical phenomenon. Physical Review Letters, 47(19):1400, 1981. ↩︎

  3. This post defines harmonic measure. ↩︎

  4. Denis S. Grebenkov and Dmitry Beliaev. How anisotropy beats fractality in two-dimensional on-lattice diffusion-limited-aggregation growth. Phys. Rev. E, 96:042159, Oct 2017. ↩︎

  5. Harry Kesten. Upper bounds for the growth rate of DLA. Physica A: Statistical Mechanics and its Applications, 168(1):529–535, 1990. ↩︎

  6. Dorothea M. Eberz-Wagner. Discrete growth models. PhD thesis, University of Washington, 1999. ↩︎

  7. Paul Meakin. Scaling properties for the growth probability measure and harmonic measure of fractal structures. Phys. Rev. A, 35:2234–2245, Mar 1987. ↩︎

  8. These are the two papers.

    Benny Davidovitch, Mogens~H. Jensen, Anders Levermann, Joachim Mathiesen, and Itamar Procaccia. Thermodynamic formalism of the harmonic measure of diffusion limited aggregates: Phase transition. Phys. Rev. Lett., 87:164101, Oct 2001.

    Mogens H. Jensen, Anders Levermann, Joachim Mathiesen, and Itamar Procaccia. Multifractal structure of the harmonic measure of diffusion-limited aggregates. Phys. Rev. E, 65:046109, Mar 2002. ↩︎

  9. There are many; here are two.

    C. J. G. Evertsz, P. W. Jones, and B. B. Mandelbrot. Behaviour of the harmonic measure at the bottom of fjords. Journal of Physics A: Mathematical and General, 24(8):1889, 1991.

    H. G. E. Hentschel. Wedge fjord model for dendritic growth. Physical Review A, 46(12):R7379–R7382, 12 1992. ↩︎

  10. Jacob Calvert, Shirshendu Ganguly, and Alan Hammond. Collapse and diffusion in harmonic activation and transport. arXiv preprint arXiv:2110.13895, 2021. ↩︎

  11. Jysoo Lee and H. Eugene Stanley. Phase transition in the multifractal spectrum of diffusion-limited aggregation. Phys. Rev. Lett., 61:2945–2948, Dec 1988. ↩︎

  12. Raphael Blumenfeld and Amnon Aharony. Breakdown of multifractal behavior in diffusion-limited aggregates. Phys. Rev. Lett., 62:2977–2980, Jun 1989. ↩︎

probability harmonic measure paper highlights Laplacian growth diffusion-limited aggregation