Ali Khezeli

• PhD of Mathematics
• Assistant Professor at Institute for Research in Fundamental Sciences

In this work, we define the notion of unimodular random measured metric spaces as a common generalization of various other notions. This includes the discrete cases like unimodular graphs and stationary point processes, as well as the non-discrete cases like stationary random measures and the continuum metric spaces arising as scaling limits of gra...

In this article, we show that every stationary random measure on R^d that is essentially free (i.e., has no symmetries a.s.) admits a point process as a factor. As a result, we improve the results of Last and Thorisson (2022) on the existence of a factor balancing allocation between ergodic pairs of stationary random measures Φ and Ψ with equal int...

It is known that the size of the largest common subtree (i.e., the maximum agreement subtree) of two independent random binary trees with $n$ given labeled leaves is of order between $n^{0.366}$ and $n^{1/2}$. We improve the lower bound to order $n^{0.4464}$ by constructing a common subtree recursively and by proving a lower bound for its asymptoti...

In this work, a unimodular random planar triangulation is constructed that has no invariant circle packing. This disputes a problem asked in [arXiv:1910.01614]. A natural weaker problem is the existence of point-stationary circle packings for a graph, which are circle packings that satisfy a certain mass transport principle. It is shown that the an...

In this work, it is proved that the set of boundedly-compact pointed metric spaces, equipped with the Gromov–Hausdorff topology, is a Polish space. The same is done for the Gromov–Hausdorff–Prokhorov topology. This extends previous works which consider only length spaces or discrete metric spaces. This is a measure theoretic requirement to study ra...

In this work, a metric is presented on the set of boundedly-compact pointed metric spaces that generates the Gromov-Hausdorff topology. A similar metric is defined for measured metric spaces that generates the Gromov-Hausdorff-Prokhorov topology. This extends previous works which consider only length spaces or discrete metric spaces. Completeness a...

In this paper, a general approach is presented for generalizing the Gromov-Hausdorff metric to consider metric spaces equipped with some additional structure. A special case is the Gromov-Hausdorff-Prokhorov metric which considers measured metric spaces. This abstract framework also unifies several existing generalizations which consider metric spa...

This paper is the third part of a series of three. The notions of unimodular discrete spaces and their unimodular (Minkowski and Hausdorff) dimensions were introduced in Part I. The connections of these dimensions to the growth rate were discussed in Part II. In this paper, complements to the mathematical framework of unimodular dimensions (packing...

The notions of unimodular Minkowski and Hausdorff dimensions are defined in [F. Baccelli, M.-O. Haji-Mirsadeghi, A. Khezeli, preprint (2018)] for unimodular random discrete metric spaces. The present paper is focused on the connections between these notions and the polynomial growth rate of the underlying space. It is shown that bounding the dimens...

This work introduces two new notions of dimension, namely the unimodular Minkowski and Hausdorff dimensions, which are inspired from the classical analogous notions. These dimensions are defined for unimodular discrete spaces, introduced in this work, which provide a common generalization to stationary point processes under their Palm version and u...

In this paper, we present a result similar to the shift-coupling result of Thorisson (1996) in the context of random graphs and networks. The result is that a given random rooted network can be obtained by changing the root of another given one if and only if the distributions of the two agree on the invariant sigma-field. Several applications of t...

This paper is centered on covariant dynamics on unimodular random graphs and random networks (marked graphs), namely maps from the set of vertices to itself which are preserved by graph or network isomorphisms. Such dynamics are referred to as vertex-shifts here. The first result of the paper is a classification of vertex-shifts on unimodular rando...

In this paper, we present a result similar to the shift-coupling result of Thorisson (1996) in the context of random graphs and networks. The result is that a given random rooted network can be obtained by changing the root of another given one if and only if the distributions of the two agree on the invariant sigma-field. Several applications of t...

This paper is centered on covariant dynamics on unimodular random graphs and random networks (marked graphs), namely maps from the set of vertices to itself which are preserved by graph or network isomorphisms. Such dynamics are referred to as vertex-shifts here. The first result of the paper is a classification of vertex-shifts on unimodular rand...

We give an algorithm to construct a translation-invariant transport kernel between ergodic stationary random measures $\Phi$ and $\Psi$ on $\mathbb R^d$, given that they have equal intensities. Our algorithm is deterministic given realizations $\varphi$ and $\psi$ of the measures. The existence of such a transport kernel was proved by Thorisson and...