On pinned fields, interlacements, and random walk on (Z/NZ)2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$({mathbb
On pinned fields, interlacements, and random walk on (Z/NZ)2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$({mathbb
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在 (Z/NZ)2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} 上的固定字段、交错和随机游走
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发表时间：
2017
影响因子：
2
通讯作者：
Pierre
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作者：
Pierre
我们在Z2documentclass[12pt]上定义了两个双向轨迹泊松{体}族，{最小}的{uspackagagamsmath}使用了{uspackagagamsfonts}使用了{uspackagagamssymb}使用了{uspackagagemathrsfs使用了}uspackageupgreek{ setlengththoddsidemargin} -69pt{ }egindocument{}{}{}$${mathbb {Z}}^2$$ enddocument，{可以看到，它}可以充分描述二维环面（Z/NZ）上随机行走留下的痕迹的局部图片2documentclass[12pt]minimal {usepackageamsmath} useppackagewasysym{ useppackageamsfonts} useppackageamssyb{ useppackageamssyb }useppackageamsfs{ useppackageupgreek }setlengthoddsidemargin{-69pt} egindocument {}{}{}{}{}{}$$({mathbb {Z}}/N {mathbb {Z}})^2$$ enddocument，{从均匀分布出发}，{运行到}order (NlogN)2documentclass[12pt]minimal uspackageamsmath uspackageamsfonts {uspackageamssymb} uspackageamssyb{ uspackagemathrsfs} uspackageupgreek{ setlengththoddsidemargin} -69pt {egindocument}{}{}{}{}{}{}$$(Nlog N)^2$$ enddocument{并}强制避免{固定点后者的局部极限}最近在Comets{ et al. （common Math Phys 343:129-164, 2016）中建立。在某种程度上，根据统计力学的精神，我们的构造是通过考虑一系列“有限体积”近似来进行的，这些近似由避开原点并在空间尺度N上终止的随机游走组成，要么使用狄利克雷边界条件，要么使用适当调整的质量。通过用N来调整这种漫步的强度u，可以看到职业场有一个非平凡的极限，对应于实际随机漫步的极限。因此，我们的构造在瞬态情况下产生了Sznitman （Ann Math 171(3): 2039-2087, 2010）中引入的随机交错模型的二维模拟。它还通过一个（固定的）Ray-Knight}类型同义{定理，将其链接到Z2documentclass[12pt]minimal中的固定自由字段useppackageamsmath }useppackagewasysym{ useppackageamsfonts} useppackageamssymb{ useppackageamssyb useppackageamssfs }useppackageupgreek {setlengthoddsidemargin}-69pt egindocument {}{}{}{}{}$${mathbb {Z}}^2$$ enddocument{。}
We define two families of Poissonian soups of bidirectional trajectories on Z2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${mathbb {Z}}^2$$end{document}, which can be seen to adequately describe the local picture of the trace left by a random walk on the two-dimensional torus (Z/NZ)2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$({mathbb {Z}}/N {mathbb {Z}})^2$$end{document}, started from the uniform distribution, run up to a time of order (NlogN)2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$(Nlog N)^2$$end{document} and forced to avoid a fixed point. The local limit of the latter was recently established in Comets et al. (Commun Math Phys 343:129–164, 2016). Our construction proceeds by considering, somewhat in the spirit of statistical mechanics, a sequence of “finite volume” approximations, consisting of random walks avoiding the origin and killed at spatial scale N, either using Dirichlet boundary conditions, or by means of a suitably adjusted mass. By tuning the intensity u of such walks with N, the occupation field can be seen to have a nontrivial limit, corresponding to that of the actual random walk. Our construction thus yields a two-dimensional analogue of the random interlacements model introduced in Sznitman (Ann Math 171(3):2039–2087, 2010) in the transient case. It also links it to the pinned free field in Z2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${mathbb {Z}}^2$$end{document}, by means of a (pinned) Ray–Knight type isomorphism theorem.