Page personnelle de Laurent Ménard

Spectra of Erdös-Rényi random graphs

Below are histograms of the empirical spectral distribution of adjacency matrices of Erdös-Rényi random graphs with various average degree. The limit law is purely atomic iff the average degree is smaller than or equal to 1 (see the work of Bordenave Sen and Virag). Simulations are done with 50 matrices of size 5000.

Spectrum of 50 adjacency matrices of Erdös-Rényi graphs with 5000 vertices and average degree 0.5 Spectrum of 50 adjacency matrices of Erdös-Rényi graphs with 5000 vertices and average degree 1 Spectrum of 50 adjacency matrices of Erdös-Rényi graphs with 5000 vertices and average degree 1.5 Spectrum of 50 adjacency matrices of Erdös-Rényi graphs with 5000 vertices and average degree 2
Average degree 0.5 Average degree 1 Average degree 1.5 Average degree 2
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Spectrum of 50 adjacency matrices of Erdös-Rényi graphs with 5000 vertices and average degree 2.5 Spectrum of 50 adjacency matrices of Erdös-Rényi graphs with 5000 vertices and average degree 2.75 Spectrum of 50 adjacency matrices of Erdös-Rényi graphs with 5000 vertices and average degree 2.8 Spectrum of 50 adjacency matrices of Erdös-Rényi graphs with 5000 vertices and average degree 3
Average degree 2.5 Average degree 2.75 Average degree 2.8 Average degree 3
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Spectrum of 50 adjacency matrices of Erdös-Rényi graphs with 5000 vertices and average degree 4 Spectrum of 50 adjacency matrices of Erdös-Rényi graphs with 5000 vertices and average degree 5 Spectrum of 50 adjacency matrices of Erdös-Rényi graphs with 5000 vertices and average degree 10 Spectrum of 50 adjacency matrices of Erdös-Rényi graphs with 5000 vertices and average degree 20
Average degree 4 Average degree 5 Average degree 10 Average degree 20
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Brownian tree and Brownian excurion

Here are some pictures of uniform planar trees. The embeddings are obtained with GraphViz using a spring algorithm.

A uniform tree with 100 000 edges and its coding excursion: A uniform tree with 100 000 edes. An excursion with 200 000 steps.
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A uniform tree with 50 000 edges and its coding excursion: A uniform tree with 50 000 edes. An excursion with 100 000 steps.
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A uniform tree with 10 000 edges and its coding excursion: A uniform tree with 10 000 edes. An excursion with 20 000 steps.
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Large random maps

Here are some pictures of uniform simple triangulations. The embeddings are again obtained with GraphViz, I'll try to get better pictures when I have the time. If you want to play with these objects, I can provide files of the maps in the .dot format or any reasonable format listing edges and faces.

A uniform simple triangulation with 100 000 faces: A uniform tree with 100 000 edes.
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A uniform simple triangulation with 50 000 faces: A uniform tree with 50 000 edes.
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A uniform simple triangulation with 10 000 faces: A uniform tree with 10 000 edes.
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