Cutoff for non-backtracking random walks on sparse random graphs
Cutoff for non-backtracking random walks on sparse random graphs
复制标题
稀疏随机图上非回溯随机游走的截止
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发表时间：
2015
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影响因子：
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通讯作者：
J. Salez
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作者：
J. Salez
1 次约化 `-上同调（简称“LpR1”）是[有界价]图的一个有用的准等距不变量，其定义相对简单。在图上，从函数到顶点再到边上的函数，存在一个自然的梯度算子，通过查看边两端值的差异来定义。简而言之，这个上同调是在`（边）中梯度的函数与本身在`（顶点）中的函数的商。在本次演讲中，我将解释如何在等周轮廓的某些假设下，将“LpR1”识别为调和函数空间的子空间，并在调和函数空间上产生一些有趣的推论（例如，在点灯器图上）。事实上，“LpR1”的“有趣部分”可以用泊松边界的子空间来识别。运输成本自然成为证明中的关键因素。稀疏随机图上非回溯随机游走的截止作者：Justin Salez 摘要：如果遍历马尔可夫链的平稳性距离在可忽略不计的时间段内突然从接近 1 突然下降到接近 0，则据说该链会表现出截止。这种现象是在洗牌的背景下发现的（Aldous-Diaconis，1986），现在被认为在快速混合马尔可夫链中相当典型。在这里，我们考虑如果遍历马尔可夫链的平稳性距离在可忽略不计的时间段内突然从接近 1 突然下降到接近 0，则称其表现出截止性。这种现象是在洗牌的背景下发现的（Aldous-Diaconis，1986），现在被认为在快速混合马尔可夫链中相当典型。这里我们考虑
Reduced `-cohomology in degree 1 (for short "LpR1") is a useful quasiisometry invariant of graphs [of bounded valency] whose definition is relatively simple. On a graph, there is a natural gradient operator from functions to vertices to functions on edges defined by looking at the difference of the value on the extremities of the edge. Simply put, this cohomology is the quotient of functions with gradient in ` (of the edges) by functions who are themselves in ` (of the vertices). In this talk, I will explain how, under some assumptions on the isoperimetric profile one can identify "LpR1" with a subspace of the space of harmonic functions and yields some interesting corollaries on the space of harmonic functions (for example, on lamplighter graphs). In fact, the "interesting part" of "LpR1" can be identified with a subspace of the Poisson boundary. The transport cost comes up naturally as a key ingredient in the proof. Cutoff for non-backtracking random walks on sparse random graphs by Justin Salez Abstract: An ergodic Markov chain is said to exhibit cutoff if its distance to stationarity drops abruptly from near 1 to near 0 over a negligible time period. Discovered in the context of card shuffling (Aldous-Diaconis, 1986), this phenomenon is now believed to be rather typical among fast mixing Markov chains. Here we consider An ergodic Markov chain is said to exhibit cutoff if its distance to stationarity drops abruptly from near 1 to near 0 over a negligible time period. Discovered in the context of card shuffling (Aldous-Diaconis, 1986), this phenomenon is now believed to be rather typical among fast mixing Markov chains. Here we consider