Brownian Excursion Research Articles

The distance profile of rooted and unrooted simply generated trees

AbstractIt is well known that the height profile of a critical conditioned Galton–Watson tree with finite offspring variance converges, after a suitable normalisation, to the local time of a standard Brownian excursion. In this work, we study the distance profile, defined as the profile of all distances between pairs of vertices. We show that after a proper rescaling the distance profile converges to a continuous random function that can be described as the density of distances between random points in the Brownian continuum random tree. We show that this limiting function a.s. is Hölder continuous of any order $\alpha<1$ , and that it is a.e. differentiable. We note that it cannot be differentiable at 0, but leave as open questions whether it is Lipschitz, and whether it is continuously differentiable on the half-line $(0,\infty)$ . The distance profile is naturally defined also for unrooted trees contrary to the height profile that is designed for rooted trees. This is used in our proof, and we prove the corresponding convergence result for the distance profile of random unrooted simply generated trees. As a minor purpose of the present work, we also formalize the notion of unrooted simply generated trees and include some simple results relating them to rooted simply generated trees, which might be of independent interest.

On excursions inside an excursion

The distribution of ranked heights of excursions of a Brownian bridge is given by Pitman and Yor (2001). In this work, we consider excursions of a Brownian excursion above a random level x, where x is the value of the excursion at an independent uniform time on [0,1]. We study the maximum heights of these excursions as Pitman and Yor did for excursions of a Brownian bridge. In particular, the probability functions and the moments of the sum of the jth highest maximum over an excursion above x and the absolute value of the kth lowest minimum over an excursion below x, including j=1 and k=2, are computed in this paper.

Equilibrium Mean-Field-Like Statistical Models with KPZ Scaling

We have considered three different “one-body” statistical systems involving Brownian excursions, which possess for fluctuations Kardar–Parisi–Zhang scaling with the critical exponent $$\nu = \tfrac{1}{3}$$ . In all models imposed external constraints push the underlying stochastic process to a large deviation regime. Specifically, we have considered fluctuations for: (i) Brownian excursions on non-uniform finite trees with linearly growing branching originating from the mean-field approximation of the Dumitriu–Edelman representation of matrix models, (ii) (1+1)D “magnetic” Dyck paths within the strip of finite width, (iii) inflated ideal polymer ring with fixed gyration radius. In the latter problem cutting off the long-ranged spatial fluctuations and leaving only the “typical” modes for stretched paths, we ensure the KPZ-like scaling for bond fluctuations. To the contrary, summing up all normal modes, we get the Gaussian behavior. In all considered models, KPZ fluctuations emerge in presence of two complementary conditions: (i) the trajectories are pushed to a large deviation region of a phase space, and (ii) the trajectories are leaning on an impenetrable boundary.

Area fluctuations on a subinterval of Brownian excursion

Area fluctuations of a Brownian excursion are described by the Airy distribution, which has found applications in different areas of physics, mathematics and computer science. Here we generalize this distribution to describe the area fluctuations on a subinterval of a Brownian excursion. In the first version of the problem (model 1) no additional conditions are imposed. In the second version (model 2) we study the distribution of the area fluctuations on a subinterval given the excursion area on the whole interval. Both versions admit convenient path-integral formulations. In model 1 we obtain an explicit expression for the Laplace transform of the area distribution on the subinterval. In both models we focus on large deviations of the area by evaluating the tails of the area distributions, sometimes taking into account pre-exponential factors. When conditioning on very large areas in model 2, we uncover two singularities in the rate function of the subinterval area fraction. They can be interpreted as dynamical phase transitions of the second and third order.

Double Jump Phase Transition in a Soliton Cellular Automaton

AbstractIn this paper, we consider the soliton cellular automaton introduced in [ 26] with a random initial configuration. We give multiple constructions of a Young diagram describing various statistics of the system in terms of familiar objects like birth-and-death chains and Galton–Watson forests. Using these ideas, we establish limit theorems showing that if the 1st $n$ boxes are occupied independently with probability $p\in (0,1)$, then the number of solitons is of order $n$ for all $p$ and the length of the longest soliton is of order $\log n$ for $p<1/2$, order $\sqrt{n}$ for $p=1/2$, and order $n$ for $p>1/2$. Additionally, we uncover a condensation phenomenon in the supercritical regime: for each fixed $j\geq 1$, the top $j$ soliton lengths have the same order as the longest for $p\leq 1/2$, whereas all but the longest have order $\log n$ for $p>1/2$. As an application, we obtain scaling limits for the lengths of the $k^{\textrm{th}}$ longest increasing and decreasing subsequences in a random stack-sortable permutation of length $n$ in terms of random walks and Brownian excursions.

Dini derivatives and regularity for exchangeable increment processes

Let X X be an exchangeable increment (EI) process whose sample paths are of infinite variation. We prove that for any fixed t t almost surely, lim sup h → 0 ± ( X t + h − X t ) / h = ∞ and lim sup h → 0 ± ( X t + h − X t ) / h = − ∞ . \begin{equation*} \limsup _{h\to 0 \pm } \left (X_{t+h}-X_t\right )/h=\infty \quad \text {and}\quad \limsup _{h\to 0\pm } \left (X_{t+h}-X_t\right )/h=-\infty . \end{equation*} This extends a celebrated result of Rogozin for Lévy processes obtained in 1968 and completes the known picture for finite-variation EI processes. Applications are numerous. For example, we deduce that both half-lines ( − ∞ , 0 ) (-\infty , 0) and ( 0 , ∞ ) (0,\infty ) are visited immediately for infinite variation EI processes (called upward and downward regularity). We also generalize the zero-one law of Millar for Lévy processes by showing continuity of X X when it reaches its minimum in the infinite variation EI case; an analogous result for all EI processes links right and left continuity at the minimum with upward and downward regularity. We also consider results of Durrett, Iglehart, and Miller on the weak convergence of conditioned Brownian bridges to the normalized Brownian excursion considered in \cite{MR0436353} and broadened to a subclass of Lévy processes and EI processes by Chaumont and the second author. We prove it here for all infinite variation EI processes. We furthermore obtain a description of the convex minorant known for Lévy processes found in [Ann. Prob. 40 (2012), pp. 1636–1674] and extend it to non-piecewise linear EI processes. Our main tool to study the Dini derivatives is a change of measure for EI processes which extends the Esscher transform for Lévy processes.

Invariance Principle for a Potts Interface Along a Wall

We consider nearest-neighbor two-dimensional Potts models, with boundary conditions leading to the presence of an interface along the bottom wall of the box. We show that, after a suitable diffusive scaling, the interface weakly converges to the standard Brownian excursion.

The First Passage Sets of the 2D Gaussian Free Field: Convergence and Isomorphisms

In a previous article, we introduced the first passage set (FPS) of constant level $-a$ of the two-dimensional continuum Gaussian free field (GFF) on finitely connected domains. Informally, it is the set of points in the domain that can be connected to the boundary by a path along which the GFF is greater than or equal to $-a$. This description can be taken as a definition of the FPS for the metric graph GFF, and it justifies the analogy with the first hitting time of $-a$ by a one-dimensional Brownian motion. In the current article, we prove that the metric graph FPS converges towards the continuum FPS in the Hausdorff metric. This allows us to show that the FPS of the continuum GFF can be represented as a union of clusters of Brownian excursions and Brownian loops, and to prove that Brownian loop soup clusters admit a non-trivial Minkowski content in the gauge $r\mapsto |\log r|^{1/2}r^2$. We also show that certain natural interfaces of the metric graph GFF converge to SLE$_4$ processes.

Airy distribution: Experiment, large deviations, and additional statistics

The Airy distribution (AD) describes the probability distribution of the area under a Brownian excursion. The AD is prominent in several areas of physics, mathematics and computer science. Here we use a dilute colloidal system to directly measure, for the first time, the AD in experiment. We also show how two different techniques of theory of large deviations - the Donsker-Varadhan formalism and the optimal fluctuation method - manifest themselves in the AD. We advance the theory of the AD by calculating, at large and small areas, the position distribution of a Brownian excursion conditioned on a given area, and measure its mean in the experiment. For large areas, we uncover two singularities in the large deviation function, which can be interpreted as dynamical phase transitions of third order. For small areas the position distribution coincides with the Ferrari-Spohn distribution, and we identify the reason for this coincidence.

The eternal multiplicative coalescent encoding via excursions of Lévy-type processes

The multiplicative coalescent is a mean-field Markov process in which any pair of blocks coalesces at rate proportional to the product of their masses. In Aldous and Limic (Electron. J. Probab. 3 (1998) Paper no. 3) each extreme eternal version of the multiplicative coalescent was described in three different ways, one of which matched its (marginal) law to that of the ordered excursion lengths above past minima of a certain Lévy-type process. Using a modification of the breadth-first-walk construction from Aldous (Ann. Probab. 25 (1997) 812–854) and Aldous and Limic (Electron. J. Probab. 3 (1998) Paper no. 3), and some new insight from the thesis by Uribe Bravo (Markovian bridges, Brownian excursions, and stochastic fragmentation and coalescence (2007) UNAM), this work settles an open problem (3) from Aldous (Ann. Probab. 25 (1997) 812–854) in the more general context of Aldous and Limic (Electron. J. Probab. 3 (1998) Paper no. 3). Informally speaking, each eternal version is entirely encoded by its Lévy-type process, and contrary to Aldous’ original intuition, the time for the multiplicative coalescent does correspond to the linear increase in the constant part of the drift of the Lévy-type process. In the “standard multiplicative coalescent” context of Aldous (Ann. Probab. 25 (1997) 812–854), this result was first announced by Armendáriz in 2001, while its first published proof is due to Broutin and Marckert (Probab. Theory Related Fields 166 (2016) 515–552), who simultaneously account for the process of excess (or surplus) edge counts.

On the Brownian separable permuton

AbstractThe Brownian separable permuton is a random probability measure on the unit square, which was introduced by Bassino, Bouvel, Féray, Gerin and Pierrot (2016) as the scaling limit of the diagram of the uniform separable permutation as size grows to infinity. We show that, almost surely, the permuton is the pushforward of the Lebesgue measure on the graph of a random measure-preserving function associated to a Brownian excursion whose strict local minima are decorated with independent and identically distributed signs. As a consequence, its support is almost surely totally disconnected, has Hausdorff dimension one, and enjoys self-similarity properties inherited from those of the Brownian excursion. The density function of the averaged permuton is computed and a connection with the shuffling of the Brownian continuum random tree is explored.

Decomposition of Brownian loop-soup clusters

We study the structure of Brownian loop-soup clusters in two dimensions. Among other things, we obtain the following decomposition of the clusters with critical intensity: If one conditions a loop-soup cluster on its outer boundary \partial (which is known to be an SLE _4 -type loop), then the union of all excursions away from \partial by all the Brownian loops in the loop-soup that touch \partial is distributed exactly like the union of all excursions of a Poisson point process of Brownian excursions in the domain enclosed by \partial . A related result that we derive and use is that the couplings of the Gaussian Free Field (GFF) with CLE _4 via level lines (by Miller–Sheffield), of the square of the GFF with loop-soups via occupation times (by Le Jan), and of the CLE _4 with loop-soups via loop-soup clusters (by Sheffield and Werner) can be made to coincide. An instrumental role in our proof of this fact is played by Lupu’s description of CLE _4 as limits of discrete loop-soup clusters.

Fluctuations of random Motzkin paths

It is known that after scaling a random Motzkin path converges in distribution to a Brownian excursion. We prove that the fluctuations of the counting processes of the ascent steps, the descent steps and the level steps converge jointly to linear combinations of two independent processes: a Brownian motion and a Brownian excursion. The proofs rely on the Laplace transforms and an integral representation based on an identity connecting non-crossing pair partitions and joint moments of an explicit non-homogeneous Markov process.

Geometrical optics of constrained Brownian excursion: from the KPZ scaling to dynamical phase transitions

We study a Brownian excursion on the time interval , conditioned to stay above a moving wall such that , and . For a whole class of moving walls, typical fluctuations of the conditioned Brownian excursion are described by the Ferrari–Spohn (FS) distribution and exhibit the Kardar–Parisi–Zhang (KPZ) dynamic scaling exponents 1/3 and 2/3. Here we use the optimal fluctuation method (OFM) to study atypical fluctuations, which turn out to be quite different. The OFM provides their simple description in terms of optimal paths, or rays, of the Brownian motion. We predict two singularities of the large deviation function, which can be interpreted as dynamical phase transitions, and they are typically of third order. Transitions of a fractional order can also appear depending on the behavior of in a close vicinity of . Although the OFM does not describe typical fluctuations, it faithfully reproduces the near tail of the FS distribution and therefore captures the KPZ scaling. If the wall function is not parabolic near its maximum, typical fluctuations (which we probe in the near tail) exhibit a more general scaling behavior with a continuous one-parameter family of scaling exponents.

Large fluctuations of the area under a constrained Brownian excursion

We study large fluctuations of the area under a Brownian excursion on the time interval , constrained to stay away from a moving wall x0(t) such that and x0(|t| < T) > 0. We focus on wall functions described by a family of generalized parabolas , where . Using the optimal fluctuation method (OFM), we calculate the large deviation function (LDF) of the area at long times. The OFM provides a simple description of the area fluctuations in terms of optimal paths, or rays, of the Brownian motion. We show that the LDF has a jump in the third derivative with respect to at a critical value of . This singularity results from a qualitative change of the optimal path, and it can be interpreted as a third-order dynamical phase transition.Although the OFM is not applicable for typical (small) area fluctuations, we argue that it correctly captures their power-law scaling of with T, with an exponent that depends continuously on and on k. We also consider the cosine wall to illustrate a different possible behavior of the optimal path and of the scaling of typical fluctuations. For some wall functions additional phase transitions, which result from a coexistence of multiple OFM solutions, should be possible.

Visibility in the vacant set of the Brownian interlacements and the Brownian excursion process

We consider the Brownian interlacements model in Euclidean space, introduced by Sznitman (2013). We give estimates for the asymptotics of the visibility in the vacant set. We also consider visibility inside the vacant set of the Brownian excursion process in the unit disc and show that it undergoes a phase transition regarding visibility to infinity as in Benjamini et al. (2009). Additionally, we determine the critical value and that there is no visibility to infinity at the critical intensity.

On the trajectory of an individual chosen according to supercritical Gibbs measure in the branching random walk

Consider a branching random walk on the real line. Madaule (2016) showed the renormalized trajectory of an individual selected according to the critical Gibbs measure converges in law to a Brownian meander. Besides, Chen (2015) proved that the renormalized trajectory leading to the leftmost individual at time n converges in law to a standard Brownian excursion. In this article, we prove that the renormalized trajectory of an individual selected according to a supercritical Gibbs measure also converges in law toward the Brownian excursion. Moreover, refinements of this results enables to express the probability for the trajectories of two individuals selected according to the Gibbs measure to have split before time t, partially answering a question of Derrida and Spohn (1988).

Patterns in random permutations avoiding the pattern 321

We consider a random permutation drawn from the set of 321‐avoiding permutations of length n and show that the number of occurrences of another pattern σ has a limit distribution, after scaling by nm + ℓ where m is the length of σ and ℓ is the number of blocks in it. The limit is not normal, and can be expressed as a functional of a Brownian excursion.

Inversions in Split Trees and Conditional Galton–Watson Trees

We studyI(T), the number of inversions in a treeTwith its vertices labelled uniformly at random, which is a generalization of inversions in permutations. We first show that the cumulants ofI(T) have explicit formulas involving thek-total common ancestors ofT(an extension of the total path length). Then we considerXn, the normalized version ofI(Tn), for a sequence of treesTn. For fixedTn's, we prove a sufficient condition forXnto converge in distribution. As an application, we identify the limit ofXnfor completeb-ary trees. ForTnbeing split trees [16], we show thatXnconverges to the unique solution of a distributional equation. Finally, whenTn's are conditional Galton–Watson trees, we show thatXnconverges to a random variable defined in terms of Brownian excursions. By exploiting the connection between inversions and the total path length, we are able to give results that significantly strengthen and broaden previous work by Panholzer and Seitz [46].

Excursion theory for Brownian motion indexed by the Brownian tree

We develop an excursion theory for Brownian motion indexed by the Brownian tree, which in many respects is analogous to the classical Itô theory for linear Brownian motion. Each excursion is associated with a connected component of the complement of the zero set of the tree-indexed Brownian motion. Each such connected component is itself a continuous tree, and we introduce a quantity measuring the length of its boundary. The collection of boundary lengths coincides with the collection of jumps of a continuous-state branching process with branching mechanism \psi(u)=\sqrt{8/3}\,u^{3/2} . Furthermore, conditionally on the boundary lengths, the different excursions are independent, and we determine their conditional distribution in terms of an excursion measure \mathbb M_0 which is the analog of the Itô measure of Brownian excursions. We provide various descriptions of M0, and we also determine several explicit distributions, such as the joint distribution of the boundary length and the mass of an excursion under \mathbb M_0 . We use the Brownian snake as a convenient tool for defining and analysing the excursions of our tree-indexed Brownian motion.