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We provide necessary and sufficient conditions for convergence towards a unique IPVT on any proper pointed measured metric space. The conditions are that the volume function, when composed with \(\log\), is regularly varying and that the limit of the uniform probability measure on a large ball exists in the horocompactification. As an application we prove convergence towards a unique IPVT for higher rank symmetric spaces, which solves an open problem of [FMW23]. Versions of this theorem are provided for graphs and edge-measured graphs, where a natural parameter \(\xi\) appears. We prove independence on \(\xi\) in a specific sense under mild assumptions, which answers an open problem of [DACELÜ23]. As a main example, we show that the latter holds for the IPVT of Diestel-Leader graphs. We also focus on further properties of this example, in particular, that its IPVT cells are distinguishable, providing the first Cayley graph with this property.
We determine analytically for all \(k\in\{0,1,\ldots,d-1\}\) the \(k\)-volume densities of a Poisson-Voronoi tessellation of intensity \(\lambda>0\) in the \(d\)-dimensional hyperbolic space of constant curvature \(-1\). This largely extends previous results of Isokawa in dimensions two and three. As applications, we provide closed form expressions for all face volume densities and all typical face volumes of the ideal Poisson-Voronoi tessellation (IPVT), which is the low-intensity limit as \(\lambda\downarrow 0\) of the hyperbolic Poisson-Voronoi tessellation. As a main tool we develop a new Blaschke-Petkantschin-type formula in hyperbolic space.
We prove concentration bounds for random Euclidean combinatorial optimization problems with \( p \) costs. For bipartite matching and for the (mono and bipartite) traveling salesperson problem in dimension \( d\geq 3\), we obtain concentration at the natural energy scale \( n^{1-p/d} \) for \( 1\leq p<d^2/2 \). Our method combines a Poincaré inequality with a robust geometric mechanism providing uniform bounds on the edges of optimizers. We also formulate a conjectural \(p \!\to\! q\) transfer principle for the \( p \)-optimal matching which, if true, would extend the concentration range to all \( p\geq 1 \).
We study Poisson--Voronoi percolation and its discrete analogue Bernoulli--Voronoi percolation in spaces with a non-amenable product structure. We develop a new method of proving smallness of the uniqueness threshold \(p_u(\lambda)\) at small intensities \(\lambda>0\) based on the unbounded borders phenomenon of their underlining ideal Poisson--Voronoi tessellation. We apply our method to several concrete examples in both the discrete and the continuum setting, including \(k\)-fold graph products of \(d\)-regular trees for \(k\ge2,d\ge3\) and products of hyperbolic spaces \(\mathbb H_{d_1}\times \ldots \times \mathbb H_{d_k}\) for \(k\ge2, d_i\ge2\), complementing a recent result of the second and fourth author for symmetric spaces of connected higher rank semisimple real Lie groups with property (T). We also provide new examples of non-amenable Cayley graphs with the FIID sparse unique infinite cluster property, answering positively a recent question of Pete and Rokob.
We construct and study the ideal Poisson-Voronoi tessellation of the product of two hyperbolic planes \( \mathbb{H}_{2}\times \mathbb{H}_{2}\) endowed with the \(L^{1}\) norm. We prove that its law is invariant under all isometries of this space and study some geometric features of its cells. Among other things, we prove that the set of points at equal separation to any two corona points is unbounded almost surely. This is analogous to a recent result of Frączyk-Mellick-Wilkens for higher rank symmetric spaces.
We exhibit an uncountable family of extremal inhomogeneous Gibbs measures of the low–temperature Ising model on regular tilings of the hyperbolic plane. These states arise as low temperature perturbations of local ground states having a sparse enough set of frustrated edges, the sparseness being measured in terms of the isoperimetric constant of the graph. This result is implied by an extension of the article [5] about spin models on regular trees to Ising models on a more general class of regular non-amenable graphs. We moreover argue how we can deduce the extremality of an uncountable subset of the Series–Sinai states [24] at low temperature.
We fully characterize the set of finite shapes with minimal perimeter on hyperbolic latticesgiven by regular tilings of the hyperbolic plane whose tiles are regular \(p\)-gons meeting at vertices of degree \(q\), with \(1/p + 1/q < 1/2\). In particular, we prove that the ratio between the perimeter and the area (i.e., the number of vertices) of this set of minimal shapes converges to the isoperimetric constant computed in Häggström-Jonasson-Lyons. In fact, our regular balls which are constructed via layers and not combinatorial balls, will realize the isoperimetric constant for any fixed number of vertices.
We identify the local limit of massive spanning forests on the complete graph. This generalizes a well-known theorem of Grimmett on the local limit of uniform spanning trees on the complete graph.
We study the limit in low intensity of Poisson-Voronoi tessellations in hyperbolic spaces \( \mathbb{H}_{d} \) for \( d \geq 2 \). In contrast to the Euclidean setting, a limiting nontrivial ideal tessellation \( \mathcal{V}_{d} \) appears as the intensity tends to \(0\). The tessellation \( \mathcal{V}_{d} \) is a natural, isometry-invariant decomposition of \( \mathbb{H}_{d} \) into countably many unbounded polytopes, each with a unique end. We study its basic properties, in particular, the geometric features of its cells.
We show how decimated Gibbs measures having unbroken continuous symmetry due to the Mermin–Wagner theorem, despite their discrete equivalents exhibiting phase transition, can still become non-Gibbsian. The mechanism rests on the occurrence of a spin-flop transition with a broken discrete symmetry, once the model is constrained by the decimated spins in a suitably chosen "bad" configuration.
We extend proofs of non-Gibbsianness of decimated Gibbs measures at low temperatures to include long-range as well as vector-spin interactions. Our main tools consist in a two-dimensional use of "equivalence of boundary conditions" in the long-range case and an extension of global specifications for two-dimensional vector spins.
We consider the ferromagnetic n.n. Ising model on Cayley trees submitted to a modified majority rule transformation with overlapping cells already known to lead to non-Gibbsian measures. We describe the renormalized measures within the generalized Gibbs framework and prove that they are almost Gibbs at any temperature.
We consider the assignment problem between two sets of \( N \) random points on a smooth, two-dimensional manifold \( \Omega \) of unit area. It is known that the average cost scales as \( E_\Omega(N) \sim 1/2\pi \ln N \) with a correction that is at most of order \( \sqrt{\ln N \ln \ln N} \). In this paper, we show that, within the linearization approximation of the field-theoretical formulation of the problem, the first \( \Omega \)-dependent correction is on the constant term, and can be exactly computed from the spectrum of the Laplace–Beltrami operator on \( \Omega \). We perform the explicit calculation of this constant for various families of surfaces, and compare our predictions with extensive numerics.
We consider models of assignment for random \( N \) blue points and \( N \) red points on an interval of length \( 2N \), in which the cost for connecting a blue point in \( x \) to a red point in \( y\) is the concave function \(|x-y|^p\) , for \( 0<p<1 \). Contrarily to the convex case \( p>1\), where the optimal matching is trivially determined, here the optimization is non-trivial. The purpose of this paper is to introduce a special configuration, that we call the _Dyck matching_, and to study its statistical properties. We compute exactly the average cost, in the asymptotic limit of large \( N \), together with the first subleading correction. The scaling is remarkable: it is of order \( N \) for \( p < \frac{1}{2} \), order \( N\ln N \) for \( p = \frac{1}{2}\), and \( N^{\frac{1}{2}+p} \) for \( p > \frac{1}{2} \), and it is universal for a wide class of models. We conjecture that the average cost of the Dyck matching has the same scaling in \( N \) as the cost of the optimal matching, and we produce numerical data in support of this conjecture. We hope to produce a proof of this claim in future work.
We consider the random Euclidean assignment problem on the line between two sets of \( N \) random points, independently generated with the same probability density function \( \varrho \). The cost of the matching is supposed to be dependent on a power \(p>1\) of the Euclidean distance of the matched pairs. We discuss an integral expression for the average optimal cost for \(N \gg 1 \) that generalizes a previous result obtained for \(p=2\). We also study the possible divergence the given expression due to the vanishing of the probability density function. The provided regularization recipe allows us to recover the proper scaling law for the cost in the divergent cases, and possibly some of the involved coefficients. The possibility that the support of \( \varrho \) is a disconnected interval is also analysed. We exemplify the proposed procedure and we compare our predictions with the results of numerical simulations.
We discuss the optimal matching solution for both the assignment problem and the matching problem in one dimension for a large class of convex cost functions. We consider the problem in a compact set with the topology both of the interval and of the circumference. Afterwards, we assume the points positions to be random variables identically and independently distributed on the considered domain. We analytically obtain the average optimal cost in the asymptotic regime of very large number of points \( N \) and some correlation functions for a cost function in the form \( c(z)=z^p \), both in the \(p>1\) case and in the \( p<0 \) case. The scaling of the optimal mean cost with the number of points is \( N^{-p/2} \) for the assignment and \( N^{-p} \) for the matching when \( p>1 \), whereas in both cases it is a constant when \( p<0 \). Finally, our predictions are compared with the results of numerical simulations.
We analytically derive, in the context of the replica formalism, the first finite-size corrections to the average optimal cost in the random assignment problem for a quite generic distribution law for the costs. We show that, when moving from a power-law distribution to a \(\Gamma\) distribution, the leading correction changes both in sign and in its scaling properties. We also examine the behavior of the corrections when approaching a \(\delta\)-function distribution. By using a numerical solution of the saddle-point equations, we provide predictions that are confirmed by numerical simulations.
(chronological order)
Sergio Caracciolo, Gabriele Sicuro, Enrico M. Malatesta, Vittorio Erba, Andrea Sportiello, Dario Benedetto, Emanuele Caglioti, Arnaud Le Ny, Aernout C.D. van Enter, Nicolas Curien, Nathanaël Enriquez, Russell Lyons, Meltem Ünel, Paul Melotti, Vanessa Jacquier, Wioletta M. Ruszel, Loren Coquille, Jan Grebík, Ali Khezeli, Konstantin Recke, Amanda Wilkens, Francesco Mattesini, Dario Trevisan, Christoph Thäle
Statistical properties of the Euclidean random assignment problem
2020, Université Paris-Saclay. Manuscript: - -
Co-Directors: William Jalby, Olivier Rivoire and Andrea Sportiello
Jury: Michel Ledoux (président), Charles Bordenave (rapporteur), Massimiliano Gubinelli (rapporteur), Guilhem Semerjian (examinateur), Lenka Zdeborová (examinatrice), Sergio Caracciolo (membre invité)
On two linear assignment problems: random assignment and Euclidean bipartite matching
2016, University of Milan. Manuscript: - -
Supervisor: Sergio Caracciolo
External rapporteur: Gabriele Sicuro
La teoria di Schwarz-Christoffel e il Biliardo Quantistico Poligonale
2012, University of Milan. Manuscript (in italian): - -
Supervisor: Luca Guido Molinari