We study two variants of the notion of holes formed by planar simple random walk of time duration 2n and the areas associated with them. We prove in both cases that the number of holes of area greater than A(n), where {A(n)} is an increasing sequence, is, up to a logarithmic correction term, asymptotic to n⋅A(n)−1 for a range of large holes, thus confirming an observation by Mandelbrot. A consequence is that the largest hole has an area which is logarithmically asymptotic to n. We also discuss the different exponent of 5/3 observed by Mandelbrot for small holes.
References
Kesten, H. (1991). Relations between solutions to a discrete and continuous Dirichlet problem. In Random Walks, Brownian Motion, and Interacting Particle Systems. Progr. Probab. 28 309–321. Birkhäuser, Boston. MR1146455 0737.60060 10.1007/978-1-4612-0459-6 Kesten, H. (1991). Relations between solutions to a discrete and continuous Dirichlet problem. In Random Walks, Brownian Motion, and Interacting Particle Systems. Progr. Probab. 28 309–321. Birkhäuser, Boston. MR1146455 0737.60060 10.1007/978-1-4612-0459-6
Lawler, G. F. (1996). The dimension of the frontier of planar Brownian motion. Electron. Comm. Probab. 1 29–47. MR1386292 0857.60083 10.1214/ECP.v1-975 euclid.ecp/1453756496 Lawler, G. F. (1996). The dimension of the frontier of planar Brownian motion. Electron. Comm. Probab. 1 29–47. MR1386292 0857.60083 10.1214/ECP.v1-975 euclid.ecp/1453756496
Lawler, G. F. and Puckette, E. E. (1997). The disconnection exponent for simple random walk. Israel J. Math. 99 109–121. MR1469089 0958.60048 10.1007/BF02760678 Lawler, G. F. and Puckette, E. E. (1997). The disconnection exponent for simple random walk. Israel J. Math. 99 109–121. MR1469089 0958.60048 10.1007/BF02760678
Lawler, G. F. and Puckette, E. E. (2000). The intersection exponent for simple random walk. Combin. Probab. Comput. 9 441–464. MR1810151 0974.60088 10.1017/S0963548300004442 Lawler, G. F. and Puckette, E. E. (2000). The intersection exponent for simple random walk. Combin. Probab. Comput. 9 441–464. MR1810151 0974.60088 10.1017/S0963548300004442
Lawler, G. F., Schramm, O. and Werner, W. (2001). Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math. 187 237–273. MR1879850 1005.60097 10.1007/BF02392618 euclid.acta/1485891453 Lawler, G. F., Schramm, O. and Werner, W. (2001). Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math. 187 237–273. MR1879850 1005.60097 10.1007/BF02392618 euclid.acta/1485891453
Lawler, G. F., Schramm, O. and Werner, W. (2001). Values of Brownian intersection exponents. II. Plane exponents. Acta Math. 187 275–308. MR1879851 0993.60083 10.1007/BF02392619 euclid.acta/1485891454 Lawler, G. F., Schramm, O. and Werner, W. (2001). Values of Brownian intersection exponents. II. Plane exponents. Acta Math. 187 275–308. MR1879851 0993.60083 10.1007/BF02392619 euclid.acta/1485891454
Lawler, G. F., Schramm, O. and Werner, W. (2002). Analyticity of intersection exponents for planar Brownian motion. Acta Math. 189 179–201. MR1961197 1024.60033 10.1007/BF02392842 euclid.acta/1485891523 Lawler, G. F., Schramm, O. and Werner, W. (2002). Analyticity of intersection exponents for planar Brownian motion. Acta Math. 189 179–201. MR1961197 1024.60033 10.1007/BF02392842 euclid.acta/1485891523
Le Gall, J. F. (1991). On the connected components of the complement of a two-dimensional Brownian path. In Random Walks, Brownian Motion, and Interacting Particle Systems, Progr. Probab. 28 323–338. Birkhäuser, Boston. MR1146456 0748.60073 Le Gall, J. F. (1991). On the connected components of the complement of a two-dimensional Brownian path. In Random Walks, Brownian Motion, and Interacting Particle Systems, Progr. Probab. 28 323–338. Birkhäuser, Boston. MR1146456 0748.60073
Mandelbrot, B. B. (2002). Gaussian Self-Affinity and Fractals. Springer, New York. MR1878884 1007.01020 Mandelbrot, B. B. (2002). Gaussian Self-Affinity and Fractals. Springer, New York. MR1878884 1007.01020
Mountford, T. (1989). On the asymptotic number of small components created by planar Brownian motion. Stochastics Stochastics Rep. 28 177–188. MR1020270 0686.60087 10.1080/17442508908833591 Mountford, T. (1989). On the asymptotic number of small components created by planar Brownian motion. Stochastics Stochastics Rep. 28 177–188. MR1020270 0686.60087 10.1080/17442508908833591
Stewart, I. and Tall, D. (1983). Complex Analysis. Cambridge Univ. Press. MR0698076 0501.30001 Stewart, I. and Tall, D. (1983). Complex Analysis. Cambridge Univ. Press. MR0698076 0501.30001
