Local limits of bipartite maps with prescribed face degrees in high genus

Abstract

We prove the local convergence of uniform bipartite maps with prescribed face degrees in the high genus regime. Unlike in the previous work [6] on the subject, we do not make any assumption on the tail of the face degrees, except that they remain finite in the limit.

Large expanders in high genus unicellular maps

“…By the Euler formula, the high genus implies that these maps have negative average discrete curvature (or, equivalently, that the average degree is higher than in planar maps). They must therefore have hyperbolic features, some of whose have been identified in previous works [ACCR13,Ray15,BL21,BL22,Lou22].…”

“…The local behaviour of high genus maps around their root is now well understood ([ACCR13] in the unicellular case, [BL21,BL22] in the general case), and some global properties have been tackled: the planarity radius [Lou22] (see also [Ray15] for unicellular maps) and the diameter ( [Ray15] for unicellular maps, still open for other models).…”

Random graphs embeddable in order‐dependent surfaces

“…In Lemma 15, suppose that 𝒢 has a growth constant, and let R n ∈ u 𝒢 . Then by part (a) of the lemma if H is weakly attachable to 𝒢 then pend(R n , H) > (1 − 𝜀) 𝛼 H n wvhp; (11) and by part (b),…”

“…Initially we will apply these results with H as a single vertex, so we are talking about the number 𝓁(R n ) of leaves in R n . The following lemma will allow us for example to apply (11) and ( 12) when g is 'small', see also Lemma 40.…”

“…There have been recent results on uniformly random m-edge maps of various kinds in orientable surfaces where the Euler genus g = g(m) of the surface grows with m, typically linearly with m. These results tend to be phrased in terms of the genus, which is the number of handles and which is thus half the Euler genus, but we shall stick to Euler genus. These kinds of maps include unicellular maps [33,34], triangulations [10], quadrangulations [39], bipartite maps [11], and more general maps [10,11,39].…”