Achlioptas过程相变是连续的
Achlioptas process phase transitions are continuous
摘要：
人们普遍认为，随机图过程的某些简单修改会导致不连续的相变。特别地，从$n$顶点上的空图开始，假设在每一步中随机均匀地选择两对顶点，但只有一对顶点被连接，即，一个最小化要连接的组件的大小的乘积。明确Achlioptas和其他人的早期信念，在2009年，Achlioptas，D 'Souza和Spencer [Science 323（2009）1453-1455]证实存在$δ>0$（实际上，$δ\ge1/2$），使得在该过程的$o(n)$步骤中，最大分量的阶以高概率从$o(n)$“跳”到至少$δn$，这种现象被称为“爆炸渗透”。“我们给出了一个简单的证明，证明情况并非如此。我们的结果适用于所有的“Achlioptas过程”，更一般地说，任何过程中，一个固定数量的独立的随机顶点被选择在每一步，和（至少）这些顶点之间的一个边缘添加到当前图，根据任何（在线）规则。我们还证明了在一类这样的过程中的巨分量的重标度大小的极限的存在性和连续性，解决了一些问题。然而，有趣的问题仍然存在，特别是对于上面描述的产品规则。
英文摘要：
It is widely believed that certain simple modifications of the random graph process lead to discontinuous phase transitions. In particular, starting with the empty graph on $n$ vertices, suppose that at each step two pairs of vertices are chosen uniformly at random, but only one pair is joined, namely, one minimizing the product of the sizes of the components to be joined. Making explicit an earlier belief of Achlioptas and others, in 2009, Achlioptas, D'Souza and Spencer [Science 323 (2009) 1453-1455] conjectured that there exists a $δ>0$ (in fact, $δ\ge1/2$) such that with high probability the order of the largest component "jumps" from $o(n)$ to at least $δn$ in $o(n)$ steps of the process, a phenomenon known as "explosive percolation." We give a simple proof that this is not the case. Our result applies to all "Achlioptas processes," and more generally to any process where a fixed number of independent random vertices are chosen at each step, and (at least) one edge between these vertices is added to the current graph, according to any (online) rule. We also prove the existence and continuity of the limit of the rescaled size of the giant component in a class of such processes, settling a number of conjectures. Intriguing questions remain, however, especially for the product rule described above.
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