Internal DLA and the Stefan problem
Internal DLA and the Stefan problem
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DLA 内部和 Stefan 问题
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
J. Quastel
中科院分区：
文献类型：
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作者：
Janko Gravner;J. Quastel
广义内部扩散有限聚集是晶格上的随机增长模型，其中有限数量的位点充当粒子的泊松源，然后以有吸引力的零范围相互作用执行对称随机游走，直到到达第一个被少于 $alpha$ 粒子访问过的位点，此时它们停止。粒子被冻结的位置构成了占据集。我们证明，在适当的情况下，颗粒密度具有流体力学极限，即单相 Stefan 问题。然后用它来研究占用集的渐近行为。在二维中，当原点处有一个源且 $alpha=1$ 时，游走是独立的，我们特别得到占据集是渐近半径为 $Ksqrt{t}$ 的圆盘，其中 $K$ 是 $exp (-K^2 /4) = pi K^2$ 的解，解决了 Lawler、Bramson 和 Griffeath 的猜想。
Generalized internal diffusion limited aggregation is a stochastic growth model on the lattice in which a finite number of sites act as Poisson sources of particles which then perform symmetric random walks with an attractive zero-range interaction until they reach the first site which has been visited by fewer than $alpha$ particles, at which point they stop. Sites on which particles are frozen constitute the occupied set. We prove that in appropriate regimes the particle density has a hydrodynamic limit which is the one-phase Stefan problem. This is then used to study the asymptotic behavior of the occupied set. In two dimensions when the walks are independent with one source at the origin and $alpha=1$, we obtain in particular that the occupied set is asymptotically a disc of radius $Ksqrt{t}$, where $K$ is the solution of $exp (-K^2 /4) = pi K^2$, settling a conjecture of Lawler, Bramson and Griffeath.