Isoperimetric inequalities in random geometry Jean-François Le Gall, Thomas Lehéricy Université Paris-Sud Orsay Les Diablerets 2018 J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 1 / 24
Outline The Uniform Infinite Planar Triangulation (UIPT, Angel-Schramm 2003) and the Uniform Infinite Planar Quadrangulation (UIPQ, Krikun 2005) are standard models of discrete random geometry — there are other models which are expected to share the same qualitative properties. J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 2 / 24
Outline The Uniform Infinite Planar Triangulation (UIPT, Angel-Schramm 2003) and the Uniform Infinite Planar Quadrangulation (UIPQ, Krikun 2005) are standard models of discrete random geometry — there are other models which are expected to share the same qualitative properties. We are interested in isoperimetric bounds in these models: Given a set K which is a (simply connected) finite union of faces of the random lattice, how small can the size | ∂ K | of the boundary be if the volume | K | is fixed (and large) ? We get | ∂ K | ≥ | K | 1 / 4 up to logarithmic corrections. J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 2 / 24
Outline The Uniform Infinite Planar Triangulation (UIPT, Angel-Schramm 2003) and the Uniform Infinite Planar Quadrangulation (UIPQ, Krikun 2005) are standard models of discrete random geometry — there are other models which are expected to share the same qualitative properties. We are interested in isoperimetric bounds in these models: Given a set K which is a (simply connected) finite union of faces of the random lattice, how small can the size | ∂ K | of the boundary be if the volume | K | is fixed (and large) ? We get | ∂ K | ≥ | K | 1 / 4 up to logarithmic corrections. Similar (sharper) results for the Brownian plane, which is a continuous model expected to be the Gromov-Hausdorff scaling limit of the discrete models of random geometry (work in progress of A. Riera). J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 2 / 24
1. The Uniform Infinite Planar Quadrangulation Definition A planar map is a proper embedding of a finite connected graph into the two-dimensional sphere (considered up to orientation-preserving homeomorphisms of the sphere). Faces = connected components of Root the complement of edges vertex p -angulation: Root edge each face is bounded by p edges p = 3: triangulation p = 4: quadrangulation Rooted map: distinguished A rooted quadrangulation oriented edge with 7 faces J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 3 / 24
Key observation. Two planar maps M 1 and M 2 are identified if there is an (orientation-preserving) homeomorphism of the sphere that maps M 1 to M 2 . − → One is interested in the “shape” of the planar map, not in the details of the embedding (though there are important questions concerning canonical embeddings!). The same planar map Because of this, there are (for instance) finitely many quadrangulations with a fixed number n of faces: It makes sense to choose one of them uniformly at random. J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 4 / 24
The Uniform Infinite Planar Quadrangulation (UIPQ) Let Q n be uniformly distributed over { quadrangulations with n faces } . For every integer r ≥ 1, let B r ( Q n ) be the ball of radius r in Q n , defined as the union of all faces incident to a vertex at graph distance strictly less than r from the root vertex ρ . J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 5 / 24
The Uniform Infinite Planar Quadrangulation (UIPQ) Let Q n be uniformly distributed over { quadrangulations with n faces } . For every integer r ≥ 1, let B r ( Q n ) be the ball of radius r in Q n , defined as the union of all faces incident to a vertex at graph distance strictly less than r from the root vertex ρ . One can prove (Angel-Schramm 2003 for the UIPT, Krikun 2005) that ( d ) Q n n →∞ Q ∞ − → in the local limit sense, where Q ∞ is a (random) infinite quadrangulation called the UIPQ for Uniform Infinite Planar Quadrangulation. J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 5 / 24
The Uniform Infinite Planar Quadrangulation (UIPQ) Let Q n be uniformly distributed over { quadrangulations with n faces } . For every integer r ≥ 1, let B r ( Q n ) be the ball of radius r in Q n , defined as the union of all faces incident to a vertex at graph distance strictly less than r from the root vertex ρ . One can prove (Angel-Schramm 2003 for the UIPT, Krikun 2005) that ( d ) Q n n →∞ Q ∞ − → in the local limit sense, where Q ∞ is a (random) infinite quadrangulation called the UIPQ for Uniform Infinite Planar Quadrangulation. The convergence holds in the sense of local limits: for every r and for every fixed (finite) planar map M , P ( B r ( Q n ) = M ) − n →∞ P ( B r ( Q ∞ ) = M ) . → The distribution of what one sees in a fixed neighborhood of the root vertex of Q n “stabilizes” when n → ∞ J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 5 / 24
An artistic view of the UIPQ J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 6 / 24
Balls and hulls in the UIPQ The ball B r ( Q ∞ ) is the union of all faces incident to a vertex at graph distance strictly less than r from the root vertex ρ . The hull B • r ( Q ∞ ) is obtained by “filling in” the bounded holes in the ball B r ( Q ∞ ) . The ball B 2 ( Q ∞ ) ρ J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 7 / 24
Balls and hulls in the UIPQ The ball B r ( Q ∞ ) is the union of all faces incident to a vertex at graph distance strictly less than r from the root vertex ρ . The hull B • r ( Q ∞ ) is obtained by “filling in” the bounded holes in the ball B r ( Q ∞ ) . The hull B • 2 ( Q ∞ ) ρ J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 8 / 24
Volume and boundary sizes of hulls To get a set with a “large” volume but a “small” boundary size, one may first think of a hull B • r of large radius r . The volume | B • r | is the number of faces in B • r . The perimeter | ∂ B • r | is the number of edges in ∂ B • r . ∞ distance It turns out that, for r large, ρ from | B • r | ≈ r 4 | ∂ B • r | ≈ r 2 Hence | ∂ B • r | ≈ | B • r | 1 / 2 r ( Remark. If one replaces hulls by balls one has: ∂B • r | B r | ≈ r 4 | ∂ B r | ≈ r 3 ) B • hull of r r radius ρ J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 9 / 24
More precise asymptotics (Curien-LG 2015) For every λ > 0, � − 3 / 2 r | ] = 3 3 / 2 cosh ( λ 1 / 4 ) � r →∞ E [ e − λ c 1 r − 4 | B • cosh 2 ( λ 1 / 4 ) + 2 lim r →∞ E [ e − λ c 2 r − 2 | ∂ B • r | ] = ( 1 + λ ) − 3 / 2 lim (where c 1 and c 2 are known constants — the same results hold for the UIPT with different constants c 1 , c 2 ) J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 10 / 24
Smaller separating cycles ∞ distance ρ from separating Krikun (2005) constructed cycle cycles that separate the hull B • r from infinity, whose r length is roughly linear in r . He conjectured that one ∂B • r cannot do better than linear. (of size ∼ r 2 ) B • hull of r r radius ρ Theorem (LG-Lehéricy) Let L r be the minimal length of a cycle separating B • r from ∞ . P ( L r ≤ ε r ) ≤ C δ ε δ , for every δ < 2 P ( L r ≥ u r ) ≤ C exp ( − c u ) . J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 11 / 24
2. Isoperimetric bounds If Γ r is a cycle separating the hull B • r from ∞ with minimal length , and if A r is the (finite) region bounded by Γ r , one has | A r | ≥ | B • r | ≈ r 4 | ∂ A r | = | Γ r | ≈ r so that in that case | ∂ A r | ≤ | A r | 1 / 4 . This is essentially the worse possible situation if one considers regions containing the root ρ . Theorem (LG-Lehéricy) Let K be the class of all simply connected regions that are finite unions of faces of the UIPQ and contain the root ρ . For every δ > 0 , | ∂ A | inf | A | 1 / 4 ( 1 + log | A | ) 3 / 4 + δ > 0 , almost surely . A ∈K J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 12 / 24
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