在高维度的统一跨度树的 GHP 缩放限制
Eleanor Archer1,2, Asaf Nachmias1, Matan Shalev1
1Department of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel.
Communications in mathematical physics
|March 7, 2024
概括
布朗连续随机树是高维图上统一跨度树的缩放极限. 这一发现扩展到诸如直径和高度之类的相关量,揭示了它们的连续类比.
科学领域:
- 概率理论的概率理论是什么
- 图形理论是指图形的理论.
- 随机过程是指随机的过程.
背景情况:
- 统一跨度树 (UST) 是统计物理学和概率论中的基本对象.
- 了解像图表上的 UST 这样的离散结构的缩放极限对于将离散和连续模型相结合至关重要.
- 高维图,包括托里,超立方体和扩展图,作为研究非对称行为的重要测试台.
研究的目的:
- 在各种高维图上建立布朗连续随机树作为 UST 的格罗莫夫-豪斯多夫-普罗霍罗夫缩放极限.
- 推断有关联的USTT数量与连续性的对应数量的趋同的相关结果.
- 提供一个严格的数学框架,以理解高维度的USTs的大规模几何.
主要方法:
- 利用概率理论和几何分析的技术.
- 将格罗莫夫 - 豪斯多夫 - 普罗霍罗夫收的概念应用于随机度量空间.
- 在特定类型的高维图 (d-dimensional torus,超立方,扩展器图) 上分析 UST 的结构.
主要成果:
- 布朗连续的随机树被证明是USTs在高维的托里,超立方体和扩展图上的缩放极限.
- 对于这些 UST 的重新缩放的直径和高度,分布的趋同被显示为它们的连续类型.
- 简单的随机步行在这些USTs的行为也被证明是汇聚到他们的连续性对应物.
结论:
- 这项研究建立了在高维图上的离散UST和连续的布朗连续随机树之间的深层联系.
- 结果提供了对高维度极限中的随机树的通用几何性质的见解.
- 这项工作为进一步研究其他随机图结构的缩放极限提供了基础.
相关概念视频
01:19Survival Tree
85
Survival trees are a non-parametric method used in survival analysis to model the relationship between a set of covariates and the time until an event of interest occurs, often referred to as the "time-to-event" or "survival time." This method is particularly useful when dealing with censored data, where the event has not occurred for some individuals by the end of the study period, or when the exact time of the event is unknown.
Building a Survival Tree
Constructing a...
Building a Survival Tree
Constructing a...
85
01:12Maximum Size of Aggregate
115
The maximum size of aggregate is defined as the aperture of the sieve retaining 15 percent or more of the particles present in the aggregate sample. The aggregate's maximum size impacts the concrete's water requirement, workability, and strength. Larger aggregates reduce the surface area needing cement paste coverage, which can lower water needs, thereby allowing a decrease in the water-to-cement ratio when the desired workability and richness of the mix are to be maintained, which can...
115
01:26Scaling
245
In designing and analyzing filters, resonant circuits, or circuit analysis at large, working with standard element values like 1 ohm, 1 henry, or 1 farad can be convenient before scaling these values to more realistic figures. This approach is widely utilized by not employing realistic element values in numerous examples and problems; it simplifies mastering circuit analysis through convenient component values. The complexity of calculations is thereby reduced, with the understanding that...
245
01:19Routh-Hurwitz Criterion II
246
In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
The first scenario occurs when a singular zero appears in the first column of the Routh table. This situation creates a division by zero issues. To resolve this, a small positive or negative number, denoted as epsilon (∈), is substituted for the zero. The stability analysis proceeds by assuming a sign for ∈. If ∈ is positive, any sign change in the first...
The first scenario occurs when a singular zero appears in the first column of the Routh table. This situation creates a division by zero issues. To resolve this, a small positive or negative number, denoted as epsilon (∈), is substituted for the zero. The stability analysis proceeds by assuming a sign for ∈. If ∈ is positive, any sign change in the first...
246
01:15Routh-Hurwitz Criterion I
239
Consider an electrical power grid, where stability is essential to prevent blackouts. The Routh-Hurwitz criterion is a valuable tool for assessing system stability under varying load conditions or faults. By analyzing the closed-loop transfer function, the Routh-Hurwitz criterion helps determine whether the system remains stable.
To apply the Routh-Hurwitz criterion, a Routh table is constructed. The table's rows are labeled with powers of the complex frequency variable s, starting from the...
To apply the Routh-Hurwitz criterion, a Routh table is constructed. The table's rows are labeled with powers of the complex frequency variable s, starting from the...
239
01:12Theorems of Pappus and Guldinus: Problem Solving
740
Pappus and Guldinus's theorems are powerful mathematical principles that are used for finding the surface area and volume of composite shapes. For example, consider a cylindrical storage tank with a conical top. Finding the surface area or volume can be challenging for such complex shapes. These theorems are particularly useful in calculating the volume and surface area of such systems. Here, the cylindrical storage tank with a conical top can be broken down into two simple shapes: a...
740