Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Fluctuation results for Hastings–Levitov planar growth

Abstract

We study the fluctuations of the outer domain of Hastings–Levitov clusters in the small particle limit. These are shown to be given by a continuous Gaussian process \(\mathcal {F}\) taking values in the space of holomorphic functions on \(\{ |z|>1 \}\), of which we provide an explicit construction. The boundary values \(\mathcal {W}\) of \(\mathcal {F}\) are shown to perform an Ornstein–Uhlenbeck process on the space of distributions on the unit circle \(\mathbb {T}\), which can be described as the solution to the stochastic fractional heat equation

$$\begin{aligned} \frac{\partial }{\partial t} \mathcal {W} (t,\vartheta ) = - (-\Delta )^{1/2} \mathcal {W} (t,\vartheta ) + \sqrt{2}\, \xi (t, \vartheta ), \end{aligned}$$

where \(\Delta \) denotes the Laplace operator acting on the spatial component, and \(\xi (t,\vartheta )\) is a space-time white noise. As a consequence we find that, when the cluster is left to grow indefinitely, the boundary process \(\mathcal {W}\) converges to a log-correlated fractional Gaussian field, which can be realised as \((-\Delta )^{-1/4}W\), for W complex white noise on \(\mathbb {T}\).

This is a preview of subscription content, log in to check access.

Fig. 1

Notes

  1. 1.

    Throughout this paper, by conformal map we mean a conformal isomorphism.

  2. 2.

    If \(a,b \rightarrow 0\) (respectively \(a,b\rightarrow \infty \)) write \(a \gg b\) to mean \(\lim _{a,b\rightarrow 0} \frac{b}{a} = 0 \) (respectively \(\lim _{a,b\rightarrow \infty } \frac{b}{a} =0\)).

  3. 3.

    Write \(a_n \approx b_n\) if the sequences \((a_n) , (b_n)\) converge to the same limit as \(n\rightarrow \infty \).

  4. 4.

    Note that, unless \(\dim H <\infty \), a standard Gaussian random variable X on H does not take values in H, but only in the larger Banach space B.

  5. 5.

    Existence and uniqueness follow trivially by existence and uniqueness of \(\mu \).

  6. 6.

    Note that the dynamics of backwards coalescing Brownian Motions is different from the one of forward branching Brownian Motions, since for example in the latter paths can intersect after branching.

References

  1. 1.

    Asselah, A., Gaudillière, A.: From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models. Ann. Probab. 41(3A), 1115–1159 (2013)

  2. 2.

    Ball, R.C., Brady, R.M., Rossi, G., Thompson, B.R.: Anisotropy and cluster growth by diffusion-limited aggregation. Phys. Rev. Lett. 55(13), 1406 (1985)

  3. 3.

    Billingsley, P.: Probability and measure. In: Wiley Series in Probability and Mathematical Statistics. A Wiley-Interscience Publication, 3rd edn. Wiley, New York (1995)

  4. 4.

    Billingsley, P.: Convergence of probability measures. In: Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication, 2nd edn. Wiley, New York (1999)

  5. 5.

    Carleson, L., Makarov, N.: Aggregation in the plane and Loewner’s equation. Commun. Math. Phys. 216(3), 583–607 (2001)

  6. 6.

    Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. In: Encyclopedia of Mathematics and its Applications, vol. 152, 2nd edn. Cambridge University Press, Cambridge (2014)

  7. 7.

    Dharmadhikari, S.W., Fabian, V., Jogdeo, K.: Bounds on the moments of martingales. Ann. Math. Stat. 39(5), 1719–1723 (1968)

  8. 8.

    Duplantier, B., Rhodes, R., Sheffield, S., Vargas, V.: Log-correlated gaussian fields: an overview (2014). arXiv:1407.5605

  9. 9.

    Durrett, R.: Stochastic calculus: a practical introduction. In: Probability and Stochastics Series. CRC Press, Boca Raton (1996)

  10. 10.

    Durrett, R.: Probability: theory and examples. In: Cambridge Series in Statistical and Probabilistic Mathematics, 4th edn. Cambridge University Press, Cambridge (2010)

  11. 11.

    Eden, M.: A two-dimensional growth process. In: Proc. 4th Berkeley Sympos. Math. Statist. and Prob., vol. IV, pp. 223-239. Univ. California Press, Berkeley (1961)

  12. 12.

    Ethier, S.N., Kurtz, T.G.: Markov processes: characterization and convergence, vol. 282. Wiley, New York (2009)

  13. 13.

    Fontes, L.R.G., Isopi, M., Newman, C.M., Ravishankar, K.: The Brownian web: characterization and convergence. Ann. Probab. 32(4), 2857–2883 (2004)

  14. 14.

    Gross, L.: Abstract Wiener spaces. In: Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66). Contributions to Probability Theory, Part 1, vol. II, pp. 31-42. Univ. California Press, Berkeley (1967)

  15. 15.

    Hastings, M.B., Levitov, L.S.: Laplacian growth as one-dimensional turbulence. Phys. D Nonlinear Phenom. 116(1), 244–252 (1998)

  16. 16.

    Iglehart, D.L.: Weak convergence of probability measures on product spaces with applications to sums of random vectors. Technical Report No. 120 (1968)

  17. 17.

    Jerison, D., Levine, L., Sheffield, S.: Logarithmic fluctuations for internal DLA. J. Am. Math. Soc. 25(1), 271–301 (2012)

  18. 18.

    Jerison, D., Levine, L., Sheffield, S.: Internal DLA and the Gaussian free field. Duke Math. J. 163(2), 267–308 (2014)

  19. 19.

    Jerison, D., Levine, L., Sheffield, S.: Internal DLA for cylinders. In: Advances in Analysis: The Legacy of Elias M. Stein, p. 189 (2014)

  20. 20.

    Lawler, G.F., Bramson, M., Griffeath, D.: Internal diffusion limited aggregation. Ann. Probab. 20(4), 2117–2140 (1992)

  21. 21.

    Lodhia, A., Sheffield, S., Sun, X., Watson, S.S.: Fractional Gaussian fields: a survey (2014). arXiv:1407.5598

  22. 22.

    McLeish, D.L.: Dependent central limit theorems and invariance principles. Ann. Probab. 2, 620–628 (1974)

  23. 23.

    Meakin, P., Ball, R.C., Ramanlal, P., Sander, L.M.: Structure of large two-dimensional square-lattice diffusion-limited aggregates: approach to asymptotic behavior. Phys. Rev. A 35(12), 5233 (1987)

  24. 24.

    Meakin, Paul, Deutch, John M.: The formation of surfaces by diffusion limited annihilation. J. Chem. Phys. 85(4), 2320–2325 (1986)

  25. 25.

    Niemeyer, L., Pietronero, L., Wiesmann, H.J.: Fractal dimension of dielectric breakdown. Phys. Rev. Lett. 52(12), 1033–1036 (1984)

  26. 26.

    Norris, J., Turner, A.: Hastings–Levitov aggregation in the small-particle limit. Commun. Math. Phys. 316(3), 809–841 (2012)

  27. 27.

    Norris, J., Turner, A.: Weak convergence of the localized disturbance flow to the coalescing Brownian flow. Ann. Probab. 43(3), 935–970 (2015)

  28. 28.

    Rohde, S.: Oded Schramm: from circle packing to SLE. In: Selected Works of Oded Schramm, pp. 3-45. Springer, New York (2011)

  29. 29.

    Rohde, S., Zinsmeister, M.: Some remarks on Laplacian growth. Topol. Appl. 152(1–2), 26–43 (2005)

  30. 30.

    Sheffield, S.: Gaussian free fields for mathematicians. Probab. Theory Relat. Fields 139(3–4), 521–541 (2007)

  31. 31.

    Viklund, F.J., Sola, A., Turner, A.: Scaling limits of anisotropic Hastings–Levitov clusters. Ann. Inst. Henri Poincaré Probab. Stat. 48(1), 235–257 (2012)

  32. 32.

    Viklund, F.J., Sola, A., Turner, A.: Small-particle limits in a regularized Laplacian random growth model. Commun. Math. Phys. 334(1), 331–366 (2015)

  33. 33.

    Whitt, W.: Weak convergence of probability measures on the function space C[0, \(\infty \)). Ann. Math. Stat. 41(3), 939–944 (1970)

  34. 34.

    Witten Jr, T.A., Sander, L.M.: Diffusion-limited aggregation, a kinetic critical phenomenon. Phys. Rev. Lett. 47(19), 1400 (1981)

Download references

Acknowledgments

I am extremely grateful to James Norris for his guidance, support, and help with many of the arguments in the paper. I am also very thankful to Alan Sola for several interesting discussions, to Steffen Rohde for comments about the history of the model, and to Henry Jackson for the simulations in Fig. 1. Finally, I would like to thank the anonymous referee for useful comments and suggestions.

Author information

Correspondence to Vittoria Silvestri.

Additional information

Research supported by EPSRC Grant EP/H023348/1 for the Cambridge Centre for Analysis.

Appendices

Appendix 1: Local fluctuations

Our main result on local fluctuations is the following.

Theorem 11

Pick any \(t>0\), and let \(z = e^{ia+\sigma }\), \(w=e^\sigma \) for some \(a \in [-\pi , \pi )\), \(\sigma >0\). Define \(\varepsilon = \delta ^{2/3} \log (1/\delta )\) as in Theorem 3, and assume that \(\sigma \rightarrow 0\) as \(c\rightarrow 0\), with \(\lim _{\varepsilon \rightarrow 0} \frac{\sqrt{\varepsilon }}{\sigma } =0\). Then, as \(n \rightarrow \infty \), \(nc \rightarrow t\) and \(\sigma \rightarrow 0\), it holds:

$$\begin{aligned} \left( \frac{\log \frac{\Phi _n (z)}{z} -nc }{\sqrt{c \log (\frac{1}{2\sigma })}} , \, \frac{\log \frac{\Phi _n (w)}{w} -nc }{\sqrt{c \log (\frac{1}{2\sigma })}} \right) \longrightarrow \, (\mathcal {N}_1 , \mathcal {N}_2 ) \end{aligned}$$

in distribution, where \((\mathcal {N}_1 ,\mathcal {N}_2)\) is a random vector with centred complex Gaussian entries, and covariance structure given by

$$\begin{aligned} {\mathbb {E}}(\mathcal {N}_1 \mathcal {N}_2 ) = \left( \begin{array}{cc} \frac{1}{1+\alpha ^2} &{} -\frac{\alpha }{1+\alpha ^2} \\ \frac{\alpha }{1+\alpha ^2} &{} \frac{1}{1+\alpha ^2} \end{array}\right) , \end{aligned}$$

for \( \alpha = \lim _{\sigma \rightarrow 0} \frac{a}{2\sigma } \in [0, \infty ]\), with the convention that \(\frac{1}{1+\alpha ^2} = \frac{\alpha }{1+\alpha ^2} =0\) when \(\alpha =\infty \).

Theorem 11 follows by the same arguments that lead to Theorem 4, considering now \(\frac{\mathcal {X}_{k,n}^\sigma ( \cdot )}{\sqrt{ \log ( \frac{1}{2\sigma } ) } }\) in place of \(\mathcal {X}_{k,n}^\sigma (\cdot )\). Under the assumption \(\sigma \gg \sqrt{\varepsilon }\) Lemmas 3 and 4 still apply, so that Theorem 2 holds. Together with Proposition 2, this shows that in the limit as \(\sigma \rightarrow 0\), \(n \rightarrow \infty \) and \(nc \rightarrow t\), real and imaginary part of \(\frac{\log \frac{\Phi _n (e^{ia+\sigma })}{e^{ia+\sigma }} -nc }{\sqrt{c \log (\frac{1}{2\sigma })}} \) are asymptotically i.i.d. centred Gaussians, with limiting variance given by

$$\begin{aligned}&\lim _{\sigma \rightarrow 0} \frac{1}{\log \big ( \frac{1}{2\sigma } \big )} \bigg [ \frac{1}{2\pi } \int _{\sigma }^{\sigma +t} \int _{-\pi }^{\pi } \bigg [ \mathrm {Re} \Bigg ( \frac{e^{-i\vartheta + x} +1}{e^{-i\vartheta + x} -1}\Bigg ) \bigg ]^2 \mathrm {d}\vartheta \mathrm {d}x - t \bigg ]\\&\quad = \lim _{\sigma \rightarrow 0} \frac{1}{\log \big ( \frac{1}{2\sigma } \big )} \log \bigg | \frac{1-e^{-2(\sigma +t)}}{1-e^{-2\sigma }} \bigg | = 1 . \end{aligned}$$

For two point correlation we reason as in Sect. 4.1, to gather that the limiting covariance between \(\mathrm {Re} \Bigg ( \frac{\log \frac{\Phi _n (e^{ia+\sigma })}{e^{ia+\sigma }} -nc }{\sqrt{c \log (\frac{1}{2\sigma })}} \Bigg )\) and \(\mathrm {Re} \Bigg ( \frac{\log \frac{\Phi _n (e^{\sigma })}{e^{\sigma }} -nc }{\sqrt{c \log (\frac{1}{2\sigma })}} \Bigg )\) is given by

$$\begin{aligned} \begin{aligned}&\lim _{\sigma \rightarrow 0} \, \frac{1}{\log \big ( \frac{1}{2\sigma } \big )} \bigg [ \frac{1}{2\pi } \int _{\sigma }^{\sigma +t} \int _{-\pi }^{\pi } \mathrm {Re} \Bigg ( \frac{e^{-i(\vartheta -a) + x} +1}{e^{-i(\vartheta -a) + x} -1} \Bigg ) \mathrm {Re} \Bigg ( \frac{e^{-i\vartheta + x} +1}{e^{-i\vartheta + x} -1}\Bigg ) \mathrm {d}\vartheta \mathrm {d}x - t \bigg ] \\&\quad = \lim _{\sigma \rightarrow 0} \frac{1}{\log \big ( \frac{1}{2\sigma } \big )} \log \bigg | \frac{1-e^{-2(\sigma +t)+ia}}{1-e^{-2\sigma +ia}} \bigg | \\&\quad = \lim _{\sigma \rightarrow 0} \frac{ \log ( 1 + e^{-4\sigma } -2e^{-2\sigma } \cos a )}{2 \log 2\sigma } = \frac{1}{1+\alpha ^2} \end{aligned} \end{aligned}$$

whenever \(a/2\sigma \rightarrow \alpha \in [0,\infty ]\), where the last equality is obtained by Taylor expanding around \(a=0 , \sigma =0\). The same holds for imaginary parts correlation.

Finally, the asymptotic covariance between \(\mathrm {Re} \Bigg ( \frac{\log \frac{\Phi _n (e^{ia+\sigma })}{e^{ia+\sigma }} -nc }{\sqrt{c \log (\frac{1}{2\sigma })}} \Bigg )\) and \(\mathrm {Im} \Bigg ( \frac{\log \frac{\Phi _n (e^{\sigma })}{e^{\sigma }} -nc }{\sqrt{c \log (\frac{1}{2\sigma })}} \Bigg )\) is given by

$$\begin{aligned} \begin{aligned}&\lim _{\sigma \rightarrow 0} \, \frac{1}{\log \big ( \frac{1}{2\sigma } \big )} \bigg [ \frac{1}{2\pi } \int _{\sigma }^{\sigma +t} \int _{-\pi }^{\pi } \mathrm {Re} \Bigg ( \frac{e^{-i(\vartheta -a) + x} +1}{e^{-i(\vartheta -a) + x} -1} \Bigg ) \mathrm {Im} \Bigg ( \frac{e^{-i\vartheta + x} +1}{e^{-i\vartheta + x} -1}\Bigg ) \mathrm {d}\vartheta \mathrm {d}x - t \bigg ]\\&\quad = \lim _{\sigma \rightarrow 0} \frac{1}{\log \big ( \frac{1}{2\sigma } \big )} \mathrm {Arg} \Bigg ( \frac{1-e^{-2(\sigma +t)+ia}}{1-e^{-2\sigma +ia}} \Bigg )\\&\quad = \lim _{\sigma \rightarrow 0} \frac{1}{\log 2\sigma } \arctan \Bigg ( \frac{\sin a }{\cos a - e^{2\sigma }} \Bigg ) = \frac{\alpha }{1+\alpha ^2} \end{aligned} \end{aligned}$$

whenever \(a/2\sigma \rightarrow \alpha \in [0,\infty ]\), where the last equality is obtained by Taylor expanding around \(a=0 , \sigma =0\). This concludes the proof of Theorem 11.

Appendix 2: Proof of Lemma 7

Fix any \(k\le {\lfloor ns \rfloor }\) and set for simplicity \( Z_1 = Z_{k,{\lfloor nt \rfloor }} (a) , \quad Z_2 = Z_{k,{\lfloor ns \rfloor }} (0) , \quad W_1 = e^{ia + \sigma + ({\lfloor nt \rfloor }-k)c} \), \( W_2 = e^{\sigma + ({\lfloor ns \rfloor }-k)c }\). Moreover, introduce the functions \(g_\vartheta ( z) = \mathrm {Re} \big ( \log \frac{F(e^{-i\vartheta } z)}{e^{-i\vartheta } z} \big ) \), \(h_\vartheta ( z) = c \,\mathrm {Re} \big (\frac{e^{-i\vartheta }z +1}{e^{-i\vartheta }z-1} \big )\). Then \({\mathbb {E}}(X_{k,{\lfloor nt \rfloor }}(a)X_{k,{\lfloor ns \rfloor }}(0)| \mathscr {F}_{k+1,{\lfloor nt \rfloor }}) = \frac{1}{2\pi c } \int _{-\pi }^\pi g_\vartheta ( Z_1) g_\vartheta ( Z_2 ) \mathrm {d}\vartheta -c\), and we have to show that

$$\begin{aligned} \bigg | \frac{1}{2\pi c } \int _{-\pi }^\pi g_\vartheta ( Z_1) g_\vartheta ( Z_2 ) \mathrm {d}\vartheta \!-\! \frac{1}{2\pi c} \int _{-\pi }^\pi h_\vartheta ( W_1 ) h_\vartheta ( W_2 ) \mathrm {d}\vartheta \bigg | \!\le \! \frac{C(t) c \varepsilon }{(\sigma +({\lfloor ns \rfloor }-k)c)^3} .\nonumber \\ \end{aligned}$$
(31)

Trivially, the l.h.s. is bounded above by

$$\begin{aligned} \frac{1}{2\pi c} \int _{-\pi }^\pi ( |g_\vartheta ( Z_2 )| \cdot | g_\vartheta ( Z_1) - h_\vartheta ( W_1)| + |h_\vartheta ( W_1)| \cdot | g_\vartheta ( Z_2 ) - h_\vartheta ( W_2 )| ) \mathrm {d}\vartheta . \end{aligned}$$

We bound each term separately. Recall the definition of \(E(m,\varepsilon )\), from which it follows that

$$\begin{aligned} \begin{aligned} \max _{\vartheta \in [-\pi , \pi )} \{ |Z_{k,{\lfloor nt \rfloor }} (\vartheta )| \vee |Z_{k,{\lfloor ns \rfloor }} (\vartheta )| \}&\le e^{\sigma + ({\lfloor nt \rfloor }-k)c} (1+2\varepsilon ) \le 2e^{t+2} = C(t) , \\ \min _{\vartheta \in [-\pi , \pi )} \{ |Z_{k,{\lfloor nt \rfloor }} (\vartheta )| \wedge |Z_{k,{\lfloor ns \rfloor }} (\vartheta )| -1 \}&\ge e^{\sigma + ({\lfloor ns \rfloor }-k)c} (1-2\varepsilon )\\&\ge \frac{\sigma + ({\lfloor ns \rfloor }-k)c }{2} \end{aligned} \end{aligned}$$

as long as n is large enough and \(\sigma \gg \varepsilon \). We combine the above estimates with the bounds in Corollary 2, to get

$$\begin{aligned} | g_\vartheta (Z_2) |\le & {} \bigg | \log \frac{F(e^{-i\vartheta } Z_2 )}{e^{-i\vartheta } Z_2 } - c \frac{e^{-i\vartheta } Z_2 +1}{e^{-i\vartheta } Z_2 -1} \bigg | + c \bigg | \frac{e^{-i\vartheta } Z_2 +1}{e^{-i\vartheta } Z_2 -1} \bigg | \nonumber \\\le & {} \frac{C c^{3/2} |Z_2|^2}{(|Z_2|-1)^3} + \frac{2c |Z_2|}{|Z_2|-1} \nonumber \\\le & {} \frac{C(t)c^{3/2}}{(\sigma + ({\lfloor ns \rfloor }-k)c)^3} + \frac{C(t)c}{\sigma + ({\lfloor ns \rfloor }-k)c} \le \frac{2C(t)c}{\sigma + ({\lfloor ns \rfloor }-k)c}\quad \quad \quad \end{aligned}$$
(32)

for n large enough. Similarly, we find

$$\begin{aligned} |h_\vartheta (W_1 )| \le c \Bigg ( 1+\frac{2}{|W_1|-1}\Bigg ) \le c \Bigg ( 1 + \frac{2}{\sigma + ({\lfloor ns \rfloor }-k)c} \Bigg ) \le \frac{C(t)c}{\sigma + ({\lfloor ns \rfloor }-k)c} \end{aligned}$$
(33)

on the event \(E(m,\varepsilon )\) for, say, \(C(t) = t+3\) and n large enough.

In order to bound the remaining terms, observe that, by definition, on \(E(m,\varepsilon )\) we have

$$\begin{aligned} \max _{\vartheta \in [-\pi , \pi )} \{ | Z_{k,{\lfloor ns \rfloor }} (\vartheta ) - e^{i\vartheta + \sigma + ({\lfloor ns \rfloor }-k)c} | \vee | Z_{k,{\lfloor nt \rfloor }} (\vartheta ) - e^{i\vartheta + \sigma + ({\lfloor nt \rfloor }-k)c} | \} \le C(t) \, \varepsilon , \end{aligned}$$

from which we get \(|Z_1 - W_1| \le C(t) \varepsilon \) and \(|Z_2 - W_2 | \le C(t) \varepsilon \). Combining this with Corollary 2 we finally obtain

$$\begin{aligned} \begin{aligned} | g(Z_1) - h(W_1)|&\le |g(Z_1) - h(Z_1)| + |h(Z_1) - h(W_1)|\\&\le \frac{C c^{3/2} |Z_1|^2}{(\sigma + ({\lfloor ns \rfloor }-k)c)^3} + \frac{2c |Z_1 - W_1|}{(|Z_1|-1)(|W_1|-1)}\\&\le \frac{C(t)c^{3/2}}{(\sigma + ({\lfloor ns \rfloor }-k)c)^3} + \frac{C(t)c\varepsilon }{(\sigma + ({\lfloor ns \rfloor }-k)c)^2}\\&\le \frac{2C(t)c\varepsilon }{(\sigma + ({\lfloor ns \rfloor }-k)c)^2} \end{aligned} \end{aligned}$$

for n large enough. Similarly one shows that the same bound holds for \(| g(Z_2) - h(W_2)|\). Putting now this together with (32) and (33) gives (31).

The second statement of the lemma follows by the same arguments, and the proof is omitted.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Silvestri, V. Fluctuation results for Hastings–Levitov planar growth. Probab. Theory Relat. Fields 167, 417–460 (2017). https://doi.org/10.1007/s00440-015-0688-7

Download citation

Mathematics Subject Classification

  • 60F17
  • 60G60
  • 30C85