Let $\unicode[STIX]{x1D707}$ be the projection on $[0,1]$ of a Gibbs measure on $\unicode[STIX]{x1D6F4}=\{0,1\}^{\mathbb{N}}$ (or more generally a Gibbs capacity) associated with a Hölder potential. The thermodynamic and multifractal properties of $\unicode[STIX]{x1D707}$ are well known to be linked via the multifractal formalism. We study the impact of a random sampling procedure on this structure. More precisely, let $\{{I_{w}\}}_{w\in \unicode[STIX]{x1D6F4}^{\ast }}$ stand for the collection of dyadic subintervals of $[0,1]$ naturally indexed by the finite dyadic words. Fix $\unicode[STIX]{x1D702}\in (0,1)$, and a sequence $(p_{w})_{w\in \unicode[STIX]{x1D6F4}^{\ast }}$ of independent Bernoulli variables of parameters $2^{-|w|(1-\unicode[STIX]{x1D702})}$. We consider the (very sparse) remaining values $\widetilde{\unicode[STIX]{x1D707}}=\{\unicode[STIX]{x1D707}(I_{w}):w\in \unicode[STIX]{x1D6F4}^{\ast },p_{w}=1\}$. We study the geometric and statistical information associated with $\widetilde{\unicode[STIX]{x1D707}}$, and the relation between $\widetilde{\unicode[STIX]{x1D707}}$ and $\unicode[STIX]{x1D707}$. To do so, we construct a random capacity $\mathsf{M}_{\unicode[STIX]{x1D707}}$ from $\widetilde{\unicode[STIX]{x1D707}}$. This new object fulfills the multifractal formalism, and its free energy is closely related to that of $\unicode[STIX]{x1D707}$. Moreover, the free energy of $\mathsf{M}_{\unicode[STIX]{x1D707}}$ generically exhibits one first order and one second order phase transition, while that of $\unicode[STIX]{x1D707}$ is analytic. The geometry of $\mathsf{M}_{\unicode[STIX]{x1D707}}$ is deeply related to the combination of approximation by dyadic numbers with geometric properties of Gibbs measures. The possibility to reconstruct $\unicode[STIX]{x1D707}$ from $\widetilde{\unicode[STIX]{x1D707}}$ by using the almost multiplicativity of $\unicode[STIX]{x1D707}$ and concatenation of words is discussed as well.

中文翻译：

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吉布斯加权树中的随机稀疏采样和相变

让$\unicode[STIX]{x1D707}$成为投影$[0,1]$吉布斯测度$\unicode[STIX]{x1D6F4}=\{0,1\}^{\mathbb{N}}$（或更一般地，吉布斯容量）与 Hölder 势相关。的热力学和多重分形性质$\unicode[STIX]{x1D707}$众所周知，它们是通过多重分形形式联系起来的。我们研究了随机抽样程序对该结构的影响。更准确地说，让$\{{I_{w}\}}_{w\in \unicode[STIX]{x1D6F4}^{\ast }}$代表二元子区间的集合$[0,1]$自然地由有限的二元词索引。使固定$\unicode[STIX]{x1D702}\in (0,1)$, 和一个序列$(p_{w})_{w\in \unicode[STIX]{x1D6F4}^{\ast }}$参数的独立伯努利变量$2^{-|w|(1-\unicode[STIX]{x1D702})}$. 我们考虑（非常稀疏的）剩余值$\widetilde{\unicode[STIX]{x1D707}}=\{\unicode[STIX]{x1D707}(I_{w}):w\in \unicode[STIX]{x1D6F4}^{\ast },p_{ w}=1\}$. 我们研究与相关的几何和统计信息$\widetilde{\unicode[STIX]{x1D707}}$，以及之间的关系$\widetilde{\unicode[STIX]{x1D707}}$和$\unicode[STIX]{x1D707}$. 为此，我们构建了一个随机容量$\mathsf{M}_{\unicode[STIX]{x1D707}}$从$\widetilde{\unicode[STIX]{x1D707}}$. 这个新物体满足多重分形形式，它的自由能与$\unicode[STIX]{x1D707}$. 此外，自由能$\mathsf{M}_{\unicode[STIX]{x1D707}}$通常表现出一个一阶和一个二阶相变，而$\unicode[STIX]{x1D707}$是解析的。几何学$\mathsf{M}_{\unicode[STIX]{x1D707}}$与二进数近似与吉布斯测度的几何特性的结合密切相关。重建的可能性$\unicode[STIX]{x1D707}$从$\widetilde{\unicode[STIX]{x1D707}}$通过使用几乎可乘性$\unicode[STIX]{x1D707}$并且还讨论了单词的连接。