更改

芯片构型的稳定遵循一种“最小作用原理”的形式：每个顶点在稳定过程中不超过必要的崩塌量。

芯片构型的稳定遵循一种“最小作用原理”的形式：每个顶点在稳定过程中不超过必要的崩塌量。

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）chip configurations的翻译存疑==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）

 

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）chip configurations的翻译存疑==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）

This can be formalized as follows.  Call a sequence of topples ''legal'' if it only topples unstable vertices, and ''stabilizing'' if it results in a stable configuration.  The standard way of stabilizing the sandpile is to find a maximal legal sequence; i.e., by toppling so long as it is possible.  Such a sequence is obviously stabilizing, and the Abelian property of the sandpile is that all such sequences are equivalent up to permutation of the toppling order; that is, for any vertex <math>v</math>, the number of times <math>v</math> topples is the same in all legal stabilizing sequences.  According to the least action principle, a '''minimal stabilizing''' sequence is also equivalent up to permutation of the toppling order to a legal (and still stabilizing) sequence.  In particular, the configuration resulting from a minimal stabilizing sequence is the same as results from a maximal legal sequence.

This can be formalized as follows.  Call a sequence of topples ''legal'' if it only topples unstable vertices, and ''stabilizing'' if it results in a stable configuration.  The standard way of stabilizing the sandpile is to find a maximal legal sequence; i.e., by toppling so long as it is possible.  Such a sequence is obviously stabilizing, and the Abelian property of the sandpile is that all such sequences are equivalent up to permutation of the toppling order; that is, for any vertex <math>v</math>, the number of times <math>v</math> topples is the same in all legal stabilizing sequences.  According to the least action principle, a '''minimal stabilizing''' sequence is also equivalent up to permutation of the toppling order to a legal (and still stabilizing) sequence.  In particular, the configuration resulting from a minimal stabilizing sequence is the same as results from a maximal legal sequence.

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）<math>\mathbf{n}</math> is an integral vector (not necessarily non-negative)这一句话的翻译存疑。

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）<math>\mathbf{n}</math> is an integral vector (not necessarily non-negative)这一句话的翻译存疑。

=== Scaling limits缩放限制===

===缩放限制===

 

[[File:Scaling sandpile identity.gif|thumb|Animation of the sandpile identity on square grids of increasing size. Black color denotes vertices with 0 grains, green is for 1, purple is for 2, and gold is for 3.]]

[[File:Scaling sandpile identity.gif|thumb|Animation of the sandpile identity on square grids of increasing size. Black color denotes vertices with 0 grains, green is for 1, purple is for 2, and gold is for 3.]]

[[File:Scaling sandpile identity.gif|thumb|沙堆标识在方形网格不断增加的动画。 黑色表示沙粒数为0的顶点，绿色表示沙粒数为1，紫色表示沙粒数为2，金色表示沙粒数为3。]]

[[File:Scaling sandpile identity.gif|thumb|沙堆标识在方形网格不断增加的动画。 黑色表示沙粒数为0的顶点，绿色表示沙粒数为1，紫色表示沙粒数为2，金色表示沙粒数为3。]]

动画显示了对应网格尺寸<math>N\geq 1</math>不断增大，不同大小的<math>N\times N</math>正方形网格上的沙堆群标识的重复构型，从而重新缩放构型，使其最终具有相同的物理尺度。从视觉上看，更大网格上的标识似乎变得越来越详细，并且“收敛到一个连续的图像”。从数学上讲，这表明基于弱收敛的概念（或其他一些广义的收敛概念），正方形网格上沙堆恒等式存在标度极限。事实上，Wesley-Pegden和Charles-Smart已经证明了循环沙堆结构标度极限的存在性。<ref name=Pegden2016>{{cite arxiv |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Stability of patterns in the Abelian sandpile.|eprint=1708.09432 | date=2017  | ref=Pegden2017|class=math.AP }}</ref>

动画显示了对应网格尺寸<math>N\geq 1</math>不断增大，不同大小的<math>N\times N</math>正方形网格上的沙堆群标识的重复构型，从而重新缩放构型，使其最终具有相同的物理尺度。从视觉上看，更大网格上的标识似乎变得越来越详细，并且“收敛到一个连续的图像”。从数学上讲，这表明基于弱收敛的概念（或其他一些广义的收敛概念），正方形网格上沙堆恒等式存在标度极限。事实上，Wesley-Pegden和Charles-Smart已经证明了循环沙堆结构标度极限的存在性。<ref name=Pegden2016>{{cite arxiv |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Stability of patterns in the Abelian sandpile.|eprint=1708.09432 | date=2017  | ref=Pegden2017|class=math.AP }}</ref>

.<ref name=Pegden2013>{{cite journal |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Convergence of the Abelian sandpile |journal=Duke Mathematical Journal |date=2013 |volume=162 |issue=4 |pages=627–642 |doi=10.1215/00127094-2079677 |ref=Pegden2013|arxiv=1105.0111 |s2cid=13027232 }}</ref>  In further joint work with Lionel Levine, they use the scaling limit to explain the fractal structure of the sandpile on square grids.<ref name=Levine2016>{{cite journal |last1=Levine |first1=Lionel |last2=Pegden |first2=Wesley |title=Apollonian structure in the Abelian sandpile |journal=Geometric and Functional Analysis |date=2016 |volume=26 |issue=1 |pages=306–336 |doi=10.1007/s00039-016-0358-7 |ref=Levine2016|hdl=1721.1/106972 |s2cid=119626417 |hdl-access=free }}</ref>

.<ref name=Pegden2013>{{cite journal |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Convergence of the Abelian sandpile |journal=Duke Mathematical Journal |date=2013 |volume=162 |issue=4 |pages=627–642 |doi=10.1215/00127094-2079677 |ref=Pegden2013|arxiv=1105.0111 |s2cid=13027232 }}</ref>  In further joint work with Lionel Levine, they use the scaling limit to explain the fractal structure of the sandpile on square grids.<ref name=Levine2016>{{cite journal |last1=Levine |first1=Lionel |last2=Pegden |first2=Wesley |title=Apollonian structure in the Abelian sandpile |journal=Geometric and Functional Analysis |date=2016 |volume=26 |issue=1 |pages=306–336 |doi=10.1007/s00039-016-0358-7 |ref=Levine2016|hdl=1721.1/106972 |s2cid=119626417 |hdl-access=free }}</ref>

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）whereby the configurations are rescaled to always have the same physical dimension翻译存疑

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）whereby the configurations are rescaled to always have the same physical dimension翻译存疑 ==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）

 

== Generalizations and related models归纳与相关模型==

== Generalizations and related models归纳与相关模型==