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Within the Kardar–Parisi–Zhang universality class, the space-time Airy sheet is conjectured to be the canonical scaling limit for last passage percolation models. In recent work  of Dauvergne, Ortmann, and Virág, this object was constructed and, upon a parabolic correction, shown to be the limit of one such model: Brownian last passage percolation. The limit object without parabolic correction, called the directed landscape, admits geodesic paths between any two space-time points and with . In this article, we examine fractal properties of the set of these paths. Our main results concern exceptional endpoints admitting disjoint geodesics. First, we fix two distinct starting locations and , and consider geodesics traveling and . We prove that the set of for which these geodesics coalesce only at time 1 has Hausdorff dimension one-half. Second, we consider endpoints and between which there exist two geodesics intersecting only at times 0 and 1. We prove that the set of such also has Hausdorff dimension one-half. The proofs require several inputs of independent interest, including (i) connections to the so-called difference weight profile studied in ; and (ii) a tail estimate on the number of disjoint geodesics starting and ending in small intervals. The latter result extends the analogous estimate proved for the prelimiting model in .
For fixed functions , consider the rotationally invariant probability density on of the form
We show that when n is large, the Euclidean norm of a random vector distributed according to satisfies a thin-shell property, in that its distribution is highly likely to concentrate around a value minimizing a certain variational problem. Moreover, we show that the fluctuations of this modulus away from have the order and are approximately Gaussian when n is large.
We apply these observations to rotationally invariant random simplices: the simplex whose vertices consist of the origin as well as independent random vectors distributed according to , ultimately showing that the logarithmic volume of the resulting simplex exhibits highly Gaussian behavior. Our class of measures includes the Gaussian distribution, the beta distribution and the beta prime distribution on , provided a generalizing and unifying setting for the objects considered in Grote-Kabluchko-Thäle [Limit theorems for random simplices in high dimensions, ALEA 16, 141–177 (2019)].
Finally, the volumes of random simplices may be related to the determinants of random matrices, and we use our methods with this correspondence to show that if is an random matrix whose entries are independent standard Gaussian random variables, then there are explicit constants and an absolute constant such that
sharpening the bound in Nguyen and Vu [Random matrices: Law of the determinant, Ann. Probab. 42(1) (2014), 146–167].
We develop a stochastic calculus that makes it easy to capture a variety of predictable transformations of semimartingales such as changes of variables, stochastic integrals, and their compositions. The framework offers a unified treatment of real-valued and complex-valued semimartingales. The proposed calculus is a blueprint for the derivation of new relationships among stochastic processes with specific examples provided below.
In this paper we consider an infinite time horizon risk-sensitive optimal stopping problem for a Feller–Markov process with an unbounded terminal cost function. We show that in the unbounded case an associated Bellman equation may have multiple solutions and we give a probabilistic interpretation for the minimal and the maximal one. Also, we show how to approximate them using finite time horizon problems. The analysis, covering both discrete and continuous time case, is supported with illustrative examples.
Let O be chosen uniformly at random from the group of orthogonal matrices. Denote by the upper-left corner of O, which we refer to as a truncation of O. In this paper we prove two conjectures of Forrester, Ipsen and Kumar (2020) on the number of real eigenvalues of the product matrix , where the matrices are independent copies of . When L grows in proportion to N, we prove that
We also prove the conjectured form of the limiting real eigenvalue distribution of the product matrix. Finally, we consider the opposite regime where L is fixed with respect to N, known as the regime of weak non-orthogonality. In this case each matrix in the product is very close to an orthogonal matrix. We show that as and compute the constant explicitly. These results generalise the known results in the one matrix case due to Khoruzhenko, Sommers and Życzkowski (2010).
We prove a quenched functional central limit theorem for a one-dimensional random walk driven by a simple symmetric exclusion process. This model can be viewed as a special case of the random walk in a balanced random environment, for which the weak quenched limit is constructed as a function of the invariant measure of the environment viewed from the walk. We bypass the need to show the existence of this invariant measure. Instead, we find the limit of the quadratic variation of the walk and give an explicit formula for it.
The non-linear sewing lemma constructs flows of rough differential equations from a broad class of approximations called almost flows. We consider a class of almost flows that could be approximated by solutions of ordinary differential equations, in the spirit of the backward error analysis. Mixing algebra and analysis, a Taylor formula with remainder and a composition formula are central in the expansion analysis. With a suitable algebraic structure on the non-smooth vector fields to be integrated, we recover in a single framework several results regarding high-order expansions for various kinds of driving paths. We also extend the notion of driving rough path. We introduce as an example a new family of branched rough paths, called aromatic rough paths modeled after aromatic Butcher series.
We study dynamic random conductance models on in which the environment evolves as a reversible Markov process that is stationary under space-time shifts. We prove under a second moment assumption that two conditionally independent random walks in the same environment collide infinitely often almost surely. These results apply in particular to random walks on dynamical percolation.
Consider a population evolving from year to year through three seasons: spring, summer and winter. Every spring starts with N dormant individuals waking up independently of each other according to a given distribution. Once an individual is awake, it starts reproducing at a constant rate. By the end of spring, all individuals are awake and continue reproducing independently as Yule processes during the whole summer. In the winter, N individuals chosen uniformly at random go to sleep until the next spring, and the other individuals die. We show that because an individual that wakes up unusually early can have a large number of surviving descendants, for some choices of model parameters the genealogy of the population will be described by a Λ-coalescent. In particular, the beta coalescent can describe the genealogy when the rate at which individuals wake up increases exponentially over time. We also characterize the set of all Λ-coalescents that can arise in this framework.