Iterated Foldings of Discrete Spaces and Their Limits: Candidates for the Role of Brownian Map in Higher Dimensions

IF 0.9 3区 数学 Q3 MATHEMATICS, APPLIED
Luca Lionni, Jean-François Marckert
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引用次数: 9

Abstract

In this last decade, an important stochastic model emerged: the Brownian map. It is the limit of various models of random combinatorial maps after rescaling: it is a random metric space with Hausdorff dimension 4, almost surely homeomorphic to the 2-sphere, and possesses some deep connections with Liouville quantum gravity in 2D. In this paper, we present a sequence of random objects that we call \(D\hbox {th}\)-random feuilletages (denoted by \(\mathbf{r}[{D}]\)), indexed by a parameter \(D\ge 0\) and which are candidate to play the role of the Brownian map in dimension D. The construction relies on some objects that we name iterated Brownian snakes, which are branching analogues of iterated Brownian motions, and which are moreover limits of iterated discrete snakes. In the planar \(D=2\) case, the family of discrete snakes considered coincides with some family of (random) labeled trees known to encode planar quadrangulations. Iterating snakes provides a sequence of random trees \((\mathbf{t}^{(j)}, j\ge 1)\). The \(D\hbox {th}\)-random feuilletage \(\mathbf{r}[{D}]\) is built using \((\mathbf{t}^{(1)},\ldots ,\mathbf{t}^{(D)})\): \(\mathbf{r}[{0}]\) is a deterministic circle, \(\mathbf{r}[{1}]\) is Aldous’ continuum random tree, \(\mathbf{r}[{2}]\) is the Brownian map, and somehow, \(\mathbf{r}[{D}]\) is obtained by quotienting \(\mathbf{t}^{(D)}\) by \(\mathbf{r}[{D-1}]\). A discrete counterpart to \(\mathbf{r}[{D}]\) is introduced and called the \(D\)th random discrete feuilletage with \(n+D\) nodes (\(\mathbf{R}_{n}[D]\)). The proof of the convergence of \(\mathbf{R}_{n}[D]\) to \(\mathbf{r}[{D}]\) after appropriate rescaling in some functional space is provided (however, the convergence obtained is too weak to imply the Gromov-Hausdorff convergence). An upper bound on the diameter of \(\mathbf{R}_{n}[D]\) is \(n^{1/2^{D}}\). Some elements allowing to conjecture that the Hausdorff dimension of \(\mathbf{r}[{D}]\) is \(2^D\) are given.

离散空间的迭代折叠及其极限:高维布朗映射作用的候选项
在过去的十年里,一个重要的随机模型出现了:布朗图。它是随机组合映射的各种模型在重新标度后的极限:它是一个Hausdorff维数为4的随机度量空间,几乎肯定与2球同纯,并且与二维中的Liouville量子引力有一些深刻的联系。在本文中,我们提出了一系列随机对象,我们称之为\(D\hbox {th}\) -random feuilletages(用\(\mathbf{r}[{D}]\)表示),由参数\(D\ge 0\)索引,它们是d维布朗映射的候选对象。构造依赖于我们称之为迭代布朗蛇的一些对象,它们是迭代布朗运动的分支类似物,而且是迭代离散蛇的极限。在平面\(D=2\)情况下,所考虑的离散蛇族与已知编码平面四边形的一些(随机)标记树族一致。迭代蛇提供了一系列随机树\((\mathbf{t}^{(j)}, j\ge 1)\)。\(D\hbox {th}\) -random feuletage \(\mathbf{r}[{D}]\)是利用\((\mathbf{t}^{(1)},\ldots ,\mathbf{t}^{(D)})\)构建的:\(\mathbf{r}[{0}]\)是一个确定性圆,\(\mathbf{r}[{1}]\)是Aldous的连续随机树,\(\mathbf{r}[{2}]\)是Brownian map, \(\mathbf{r}[{D-1}]\)以某种方式将\(\mathbf{t}^{(D)}\)除以得到\(\mathbf{r}[{D}]\)。引入了一个与\(\mathbf{r}[{D}]\)相对应的离散模型,称为\(D\)具有\(n+D\)节点(\(\mathbf{R}_{n}[D]\))的第1个随机离散模型。给出了\(\mathbf{R}_{n}[D]\)到\(\mathbf{r}[{D}]\)在一定的函数空间中经过适当的缩放后的收敛性的证明(但所得到的收敛性太弱,不能暗示Gromov-Hausdorff收敛性)。\(\mathbf{R}_{n}[D]\)直径的上界是\(n^{1/2^{D}}\)。给出了一些可以推测\(\mathbf{r}[{D}]\)的Hausdorff维数为\(2^D\)的元素。
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来源期刊
Mathematical Physics, Analysis and Geometry
Mathematical Physics, Analysis and Geometry 数学-物理:数学物理
CiteScore
2.10
自引率
0.00%
发文量
26
审稿时长
>12 weeks
期刊介绍: MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas. The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process. The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed. The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.
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