# Nesting statistics in the $O(n)$ loop model on random planar maps

@article{Borot2016NestingSI, title={Nesting statistics in the \$O(n)\$ loop model on random planar maps}, author={Gaetan Borot and J{\'e}r{\'e}mie Bouttier and Bertrand Duplantier}, journal={arXiv: Mathematical Physics}, year={2016} }

In the O(n) loop model on random planar maps, we study the depth – in terms of the number of levels of nesting – of the loop configuration, by means of analytic combinatorics. We focus on the " refined " generating series of pointed disks or cylinders, which keep track of the number of loops separating the marked point from the boundary (for disks), or the two boundaries (for cylinders). For the general O(n) loop model, we show that these generating series satisfy functional relations obtained…

## 13 Citations

### NESTING STATISTICS IN THE O(n) LOOP MODEL ON RANDOM MAPS OF ARBITRARY TOPOLOGIES

- Mathematics
- 2016

We pursue the analysis of nesting statistics in the $O(n)$ loop model on random maps, initiated for maps with the topology of disks and cylinders in math-ph/1605.02239, here for arbitrary topologies.…

### The peeling process on random planar maps coupled to an O(n) loop model (with an appendix by Linxiao Chen)

- Mathematics
- 2018

We extend the peeling exploration introduced in arxiv:1506.01590 to the setting of Boltzmann planar maps coupled to a rigid $O(n)$ loop model. Its law is related to a class of discrete Markov…

### The exploration process of critical Boltzmann planar maps decorated by a triangular O(n) loop model

- MathematicsLatin American Journal of Probability and Mathematical Statistics
- 2022

. In this paper we investigate pointed ( q , g, n ) -Boltzmann loop-decorated maps with loops traversing only inner triangular faces. Using peeling exploration Budd (2018) modiﬁed to this setting we…

### Percolation probability and critical exponents for site percolation on the UIPT

- MathematicsCanadian Journal of Mathematics
- 2022

We derive three critical exponents for Bernoulli site percolation on the on the Uniform Infinite Planar Triangulation (UIPT). First we compute explicitly the probability that the root cluster is…

### Local convergence of large random triangulations coupled with an Ising model

- MathematicsTransactions of the American Mathematical Society
- 2020

We prove the existence of the local weak limit of the measure obtained by sampling random triangulations of size
n
n
decorated by an Ising configuration with a weight proportional to the…

### We call a random triangulation distributed according to this limiting law the Infinite Ising

- Mathematics, Physics
- 2018

We show that the uniform measure on triangulations of size n with an Ising configuration biased by the energy of the configuration converges weakly as n→∞ for the local topology. To do so, for any…

### The perimeter cascade in critical Boltzmann quadrangulations decorated by an $O(n)$ loop model

- Mathematics
- 2017

We study the branching tree of the perimeters of the nested loops in critical $O(n)$ model for $n \in (0,2)$ on random quadrangulations. We prove that after renormalization it converges towards an…

### Selected problems in enumerative combinatorics: permutation classes, random walks and planar maps

- Mathematics
- 2018

In this thesis we consider a number of enumerative combinatorial problems. We solve the problems of enumerating Eulerian orientations by edges and quartic Eulerian orientations counted by vertices.…

## References

SHOWING 1-10 OF 122 REFERENCES

### NESTING STATISTICS IN THE O(n) LOOP MODEL ON RANDOM MAPS OF ARBITRARY TOPOLOGIES

- Mathematics
- 2016

We pursue the analysis of nesting statistics in the $O(n)$ loop model on random maps, initiated for maps with the topology of disks and cylinders in math-ph/1605.02239, here for arbitrary topologies.…

### More on the O(n) model on random maps via nested loops: loops with bending energy

- Mathematics
- 2012

We continue our investigation of the nested loop approach to the O(n) model on random maps, by extending it to the case where loops may visit faces of arbitrary degree. This allows us to express the…

### A recursive approach to the O(n) model on random maps via nested loops

- Mathematics
- 2011

We consider the O(n) loop model on tetravalent maps and show how to rephrase it into a model of bipartite maps without loops. This follows from a combinatorial decomposition that consists in cutting…

### Loop models on random maps via nested loops: the case of domain symmetry breaking and application to the Potts model

- Mathematics
- 2012

We use the nested loop approach to investigate loop models on random planar maps where the domains delimited by the loops are given two alternating colors, which can be assigned different local…

### Scaling limits for the critical Fortuin-Kastelyn model on a random planar map II: local estimates and empty reduced word exponent

- Mathematics
- 2015

We continue our study of the inventory accumulation introduced by Sheffield (2011), which encodes a random planar map decorated by a collection of loops sampled from the critical Fortuin-Kasteleyn…

### The O(n) model on a random surface: critical points and large-order behaviour

- Mathematics, Physics
- 1992

### Random walk on random planar maps: Spectral dimension, resistance and displacement

- Mathematics
- 2017

We study simple random walk on the class of random planar maps which can be encoded by a two-dimensional random walk with i.i.d. increments or a two-dimensional Brownian motion via a…

### Scaling limits for the critical Fortuin-Kastelyn model on a random planar map III: finite volume case

- Mathematics
- 2015

We prove scaling limit results for the finite-volume version of the inventory accumulation model of Sheffield (2011), which encodes a random planar map decorated by a collection of loops sampled from…