关于Hausdorff度量下超空间中非独立随机变量的大数定律

Shuffle group laws. Applications in free probability

Commutative shuffle products are known to be intimately related to universal formulas for products, exponentials and logarithms in group theory as well as in the theory of free Lie algebras, such as, for instance, the Baker-Campbell-Hausdorff formula or the analytic expression of a Lie group law in exponential coordinates in the neighbourhood of the identity. Non-commutative shuffle products happen to have similar properties with respect to pre-Lie algebras. However, the situation is more complex since in the non-commutative framework three exponential-type maps and corresponding logarithms ar…

Diagonal Minkowski classes, zonoid equivalence, and stable laws

We consider the family of convex bodies obtained from an origin symmetric convex body $K$ by multiplication with diagonal matrices, by forming Minkowski sums of the transformed sets, and by taking limits in the Hausdorff metric. Support functions of these convex bodies arise by an integral transform of measures on the family of diagonal matrices, equivalently, on Euclidean space, which we call $K$-transform. In the special case, if $K$ is a segment not lying on any coordinate hyperplane, one obtains the family of zonoids and the cosine transform. In this case two facts are known: the vector sp…

Continuum models of directed polymers on disordered diamond fractals in the critical case

We construct and study a family random continuum polymer measures $\mathbf{M}_{r}$ corresponding to limiting partition function laws recently derived in a weak-coupling regime of polymer models on hierarchical graphs with marginally relevant disorder. The continuum polymers, which we refer to as directed paths, are identified with isometric embeddings of the unit interval $[0,1]$ into a compact diamond fractal with Hausdorff dimension two, and there is a natural 'uniform' probability measure, $μ$, over the space of directed paths, $Γ$. Realizations of the random path measures $\mathbf{M}_{r}$ …

Fréchet均值集推广形式的强大数定律

Strong Laws of Large Numbers for Generalizations of Fréchet Mean Sets

本文针对三类广义Fréchet均值集，在极少假设下证明了其经验版本在外极限和单侧Hausdorff距离下的收敛性，建立了相应的强大数定律。

Scaling limits of the three-dimensional uniform spanning tree and associated random walk

We show that the law of the three-dimensional uniform spanning tree (UST) is tight under rescaling in a space whose elements are measured, rooted real trees, continuously embedded into Euclidean space. We also establish that the relevant laws actually converge along a particular scaling sequence. The techniques that we use to establish these results are further applied to obtain various properties of the intrinsic metric and measure of any limiting space, including showing that the Hausdorff dimension of such is given by $3/β$, where $β\approx 1.624\dots$ is the growth exponent of three-dimens…