A Bijection for Rooted Maps on Orientable Surfaces

Abstract

The enumeration of maps and the study of uniform random maps have been classical topics of combinatorics and statistical physics ever since the seminal work of Tutte in the sixties. Following the bijective approach initiated by Cori and Vauquelin in the eighties, we describe a bijection between rooted maps, or rooted bipartite quadrangulations, on a surface of genus g and some simpler objects that generalize plane trees. Thanks to a rerooting argument, our bijection allows to compute the generating series of rooted maps on a surface of genus g with respect to the number of edges, and to recover the asymptotic numbers of such maps.Our construction allows to keep track in a bipartite quadrangulation of the distances of all vertices to a random basepoint. This is an analog for higher genus surfaces of the basic result on which were built the recent advances in the comprehension of the intrinsec geometry of large random planar maps, hopefully opening the way to the study of a model of continuum random surfaces of genus g.

Tessellations of random maps of arbitrary genus

“…It also agrees with the recent results of Chapuy, Marcus & Schaeffer [9] who completed the exact enumeration of maps of genus g that was initiated in [31], by means of bijective methods. Although we rely on the same bijection as [9], our method is still different as we are really concerned with asymptotic rather than exact enumeration, and it crucially involves the probabilistic arguments of Sect. 4.…”

“…This result is equivalent to a known result for maps of genus g with n edges obtained by recursive decomposition methods by Bender & Canfield [4], since such maps are in bijection with bipartite quadrangulations of same genus with n faces [31,Proposition 1]. It also agrees with the recent results of Chapuy, Marcus & Schaeffer [9] who completed the exact enumeration of maps of genus g that was initiated in [31], by means of bijective methods. Although we rely on the same bijection as [9], our method is still different as we are really concerned with asymptotic rather than exact enumeration, and it crucially involves the probabilistic arguments of Sect.…”

“…The first is a counting result, giving the following asymptotic behavior for the cardinality of the set Q n g of bipartite quadrangulations of genus g with n faces, consistently with [4,9]:…”

“…We propose to study scaling limits of the latter in Section 4 (Theorem 5 and Propositions 3, 4), after showing how to encode them with appropriate processes in Section 3. Unlike the previously mentioned works, we do not restrict ourselves to the planar case, as the most natural framework for our bijection is to consider maps of arbitrary (but fixed) genus g, so our bijection is really a variant of the Marcus-Schaeffer bijection in arbitrary genus [31,9]. In fact, we will see (Sect.…”

Liouville quantum gravity and KPZ

“…These models have been very thoroughly studied in the physics literature, in part because of connections to string theory and conformal field theory [Pol81a, Pol81b, Pol87a, Pol89, Sei90, GM93, Dav94, Dav95, AJW95, AW95, DFGZJ95, Kle95, KH96, ADJ97, Eyn01, Dup06], and to random matrix theory and geometrical models; see, e.g., the references [BIPZ78, ADF85, KKM85, Dav85, BKKM86a, BKKM86b, Kaz86, DK88a, DK90, GK89, Kos89a, Kos89b, DDSW90, MSS91, KK92, EZ92, JM92, Kor92a, Kor92b, ABC93, Dur94, ADJ94, Dau95, EK95, KH95, BDKS95, AAMT96, Dup98, Dup99a, Dup99b, Dup99c, EB99, KZJ99, Kos00, Dup00, DFGG00, DB02, Dup04, Kos07, Kos09]. More recently, a purely combinatorial approach to discretized quantum gravity has been successful [Sch98, BFSS01, FSS04, BDFG02, BS03, AS03, BDFG03a, BDFG03b, DFG05, BDFG07, Mie09, LG07, MM07, Ber07, Ber08a, Ber08b, Ber08c, BG08a, MW08, Mie08, BG08b, LG08, BG09, LM09, BB09], as well as the so-called topological expansion involving higher-genus random surfaces [CMS09,Cha09,Cha10,EO07,EO08,Eyn09].…”