论文标题
关键的一维多粒子DLA
The critical one-dimensional multi-particle DLA
论文作者
论文摘要
我们在其临界密度$λ= 1 $下研究一维多粒子扩散限制聚集（MDLA）。以前的工作已经证实，在超临界制度中，$ t $在时间$ t $的总$ x_t $是$ t^{1/2} $。本文确定了批评的增长率为$ t^{2/3} $。此外，我们得出缩放限制，证明 $$ \ big \ {t^{ - 2/3} x_ {st} \ big \ \} _ {s \ geq 0} \ overset {d} {\ rightarrow} \ rightarrow} \ big \ big \ { $ \ {z_t \} $是$（ - \ frac {1} {3}）$ - 由$ z_t =（3v_t）^{ - 2/3} $给出的自相似的扩散，其中$ v_t $是$ \ frac {8} {3} {3} {3} {3} {3} $ - bessel-bessel Crocess。 证明表明，在局部速度过程可以通过随机积分表示可以很好地近似，该表示本身可以通过具有连续边缘长度的关键分支过程来近似。从这些表示形式中，我们确定其无限漂移和差异，以表明该速度在sde $ dz_t = 2z_t^{5/2} db_t $中满足。为了使这些近似值，该过程的规律性属性是通过多尺度参数归纳建立的。
We study one-dimensional multi-particle Diffusion Limited Aggregation (MDLA) at its critical density $λ=1$. Previous works have verified that the size of the aggregate $X_t$ at time $t$ is $t^{1/2}$ in the subcritical regime and linear in the supercritical regime. This paper establishes the conjecture that the growth rate at criticiality is $t^{2/3}$. Moreover, we derive the scaling limit proving that $$\big\{ t^{-2/3}X_{st} \big\}_{s\geq 0} \overset{d}{\rightarrow} \Big\{ \int_0^s Z_u du \Big\}_{s\geq 0}, $$ where the speed process $\{Z_t\}$ is a $(-\frac{1}{3})$-self-similar diffusion given by $Z_t = (3V_t)^{-2/3}$, where $V_t$ is the $\frac{8}{3}$-Bessel process. The proof shows that locally the speed process can be well approximated by a stochastic integral representation which itself can be approximated by a critical branching process with continuous edge lengths. From these representations, we determine its infinitesimal drift and variance to show that the speed asymptotically satisfies the SDE $dZ_t = 2Z_t^{5/2}dB_t$. To make these approximations, regularity properties of the process are established inductively via a multiscale argument.
