arXiv subjects

Jacopo Borga

Publications and source records attributed to Jacopo Borga.

28 records · Page 2Linked to original sources

arXiv2021

Random Permutations -- A geometric point of view

We look at geometric limits of large random non-uniform permutations. We mainly consider two theories for limits of permutations: permuton limits, introduced by Hoppen, Kohayakawa, Moreira, Rath, and Sampaio to define a notion of scaling limits for permutations; and Benjamini-Schramm limits, introduced by the author to define a notion of local limits for permutations. The models of random permutations that we consider are mainly constrained models, that is, uniform permutations belonging to a given subset of the set of all permutations. We often identify this subset using pattern-avoidance, focusing on: permutations avoiding a pattern of length three, substitution-closed classes, (almost) square permutations, permutation families encoded by generating trees, and Baxter permutations. We explore some universal phenomena for the models mentioned above. For Benjamini-Schramm limits we explore a concentration phenomenon for the limiting objects. For permuton limits we deepen the study of some known universal permutons, called biased Brownian separable permutons, and we introduce some new ones, called Baxter permuton and skew Brownian permutons. In addition, for (almost) square permutations, we investigate the occurrence of a phase transition for the limiting permutons. On the way, we establish various combinatorial results both for permutations and other related objects. Among others, we give a complete description of the feasible region for consecutive patterns as the cycle polytope of a specific graph; and we find new bijections relating Baxter permutations, bipolar orientations, walks in cones, and a new family of discrete objects called coalescent-walk processes.

math.PR↗

arXiv2021

Splitting groups with cubic Cayley graphs of connectivity two

A group $G$ splits over a subgroup $C$ if $G$ is either a free product with amalgamation $A \underset{C}{\ast} B$ or an HNN-extension $G=A \underset{C}{\ast} (t)$. We invoke Bass-Serre theory and classify all infinite groups which admit cubic Cayley graphs of connectivity two in terms of splittings over a subgroup.

math.CO↗

arXiv2021

Quenched law of large numbers and quenched central limit theorem for multi-player leagues with ergodic strengths

We propose and study a new model for competitions, specifically sports multi-player leagues where the initial strengths of the teams are independent i.i.d. random variables that evolve during different days of the league according to independent ergodic processes. The result of each match is random: the probability that a team wins against another team is determined by a function of the strengths of the two teams in the day the match is played. Our model generalizes some previous models studied in the physical and mathematical literature and is defined in terms of different parameters that can be statistically calibrated. We prove a quenched -- conditioning on the initial strengths of the teams -- law of large numbers and a quenched central limit theorem for the number of victories of a team according to its initial strength. To obtain our results, we prove a theorem of independent interest. For a stationary process $ξ=(ξ_i)_{i\in \mathbb{N}}$ satisfying a mixing condition and an independent sequence of i.i.d. random variables $(s_i)_{i\in \mathbb{N}}$, we prove a quenched -- conditioning on $(s_i)_{i\in\mathbb{N}}$ -- central limit theorem for sums of the form $\sum_{i=1}^{n}g\left(ξ_i,s_i\right)$, where $g$ is a bounded measurable function. We highlight that the random variables $g\left(ξ_i,s_i\right)$ are not stationary conditioning on $(s_i)_{i\in\mathbb{N}}$.

math.PR↗

arXiv2021

Asymptotic normality of consecutive patterns in permutations encoded by generating trees with one-dimensional labels

We consider uniform random permutations drawn from a family enumerated through generating trees. We develop a new general technique to establish a central limit theorem for the number of consecutive occurrences of a fixed pattern in such permutations. We propose a technique to sample uniform permutations in such families as conditioned random colored walks. Building on that, we derive the behavior of the consecutive patterns in random permutations studying properties of the consecutive increments in the corresponding random walks. The method applies to families of permutations with a one-dimensional-labeled generating tree (together with some technical assumptions) and implies local convergence for random permutations in such families. We exhibit ten different families of permutations, most of them being permutation classes, that satisfy our assumptions. To the best of our knowledge, this is the first work where generating trees - which were introduced to enumerate combinatorial objects - have been used to establish probabilistic results.

math.PR↗

arXiv2020

Almost square permutations are typically square

A record in a permutation is a maximum or a minimum, from the left or from the right. The entries of a permutation can be partitioned into two types: the ones that are records are called external points, the others are called internal points. Permutations without internal points have been studied under the name of square permutations. Here, we explore permutations with a fixed number of internals points, called almost square permutations. Unlike with square permutations, a precise enumeration for the total number of almost square permutations of size $n+k$ with exactly $k$ internal points is not known. However, using a probabilistic approach, we are able to determine the asymptotic enumeration. This allows us to describe the permuton limit of almost square permutations with $k$ internal points, both when $k$ is fixed and when $k$ tends to infinity along a negligible sequence with respect to the size of the permutation. Finally, we show that our techniques are quite general by studying the set of $321$-avoiding permutations of size $n$ with exactly $k$ additional internal points ($k$ fixed). In this case we obtain an interesting asymptotic enumeration in terms of the Brownian excursion area. As a consequence, we show that the points of a uniform permutation in this set concentrate on the diagonal and the fluctuations of these points converge in distribution to a biased Brownian excursion.

math.PR↗

arXiv2020

The feasible region for consecutive patterns of permutations is a cycle polytope

We study proportions of consecutive occurrences of permutations of a given size. Specifically, the feasible limits of such proportions on large permutations form a region, called feasible region. We show that this feasible region is a polytope, more precisely the cycle polytope of a specific graph called overlap graph. This allows us to compute the dimension, vertices and faces of the polytope. Finally, we prove that the limits of classical occurrences and consecutive occurrences are independent, in some sense made precise in the extended abstract. As a consequence, the scaling limit of a sequence of permutations induces no constraints on the local limit and vice versa.

math.CO↗

arXiv2020

Scaling and local limits of Baxter permutations through coalescent-walk processes

Baxter permutations, plane bipolar orientations, and a specific family of walks in the non-negative quadrant are well-known to be related to each other through several bijections. We introduce a further new family of discrete objects, called coalescent-walk processes, that are fundamental for our results. We relate these new objects with the other previously mentioned families introducing some new bijections. We prove joint Benjamini--Schramm convergence (both in the annealed and quenched sense) for uniform objects in the four families. Furthermore, we explicitly construct a new fractal random measure of the unit square, called the coalescent Baxter permuton and we show that it is the scaling limit (in the permuton sense) of uniform Baxter permutations. To prove the latter result, we study the scaling limit of the associated random coalescent-walk processes. We show that they converge in law to a continuous random coalescent-walk process encoded by a perturbed version of the Tanaka stochastic differential equation. This result has connections (to be explored in future projects) with the results of Gwynne, Holden, Sun (2016) on scaling limits (in the Peanosphere topology) of plane bipolar triangulations. We further prove some results that relate the limiting objects of the four families to each other, both in the local and scaling limit case.

math.PR↗

arXiv2020

A decorated tree approach to random permutations in substitution-closed classes

We establish a novel bijective encoding that represents permutations as forests of decorated (or enriched) trees. This allows us to prove local convergence of uniform random permutations from substitution-closed classes satisfying a criticality constraint. It also enables us to reprove and strengthen permuton limits for these classes in a new way, that uses a semi-local version of Aldous' skeleton decomposition for size-constrained Galton--Watson trees.

math.PR↗

arXiv2019

Square permutations are typically rectangular

We describe the limit (for two topologies) of large uniform random square permutations, i.e., permutations where every point is a record. The starting point for all our results is a sampling procedure for asymptotically uniform square permutations. Building on that, we first describe the global behavior by showing that these permutations have a permuton limit which can be described by a random rectangle. We also explore fluctuations about this random rectangle, which we can describe through coupled Brownian motions. Second, we consider the limiting behavior of the neighborhood of a point in the permutation through local limits. As a byproduct, we also determine the random limits of the proportion of occurrences (and consecutive occurrences) of any given pattern in a uniform random square permutation.

math.PR↗

arXiv2019

Local convergence for permutations and local limits for uniform $ρ$-avoiding permutations with $|ρ|=3$

We set up a new notion of local convergence for permutations and we prove a characterization in terms of proportions of \emph{consecutive} pattern occurrences. We also characterize random limiting objects for this new topology introducing a notion of "shift-invariant" property (corresponding to the notion of unimodularity for random graphs). We then study two models in the framework of random pattern-avoiding permutations. We compute the local limits of uniform $ρ$-avoiding permutations, for $|ρ|=3,$ when the size of the permutations tends to infinity. The core part of the argument is the description of the asymptotics of the number of consecutive occurrences of any given pattern. For this result we use bijections between $ρ$-avoiding permutations and rooted ordered trees, local limit results for Galton--Watson trees, the Second moment method and singularity analysis.

math.PR↗