arXiv subjects

Ofer Busani

Publications and source records attributed to Ofer Busani.

At least 19 recordsLinked to original sources

arXiv2026

Classification of the eternal solutions and multiple coalescing shocks in the KPZ fixed point

We give a complete classification of the eternal solutions for the KPZ fixed point. Each of these is a (possibly infinite) max-plus convolution of the known eternal solutions, called Busemann functions. Specifically, we show that the space of eternal solutions is homeomorphic to a certain space of upper semicontinuous functions that encodes the weights of each of the Busemann functions. As a result, we show that the Busemann process gives a spectral decomposition of eternal solutions of the KPZ fixed point analogous to that for Hamilton-Jaccobi equations. The resulting evolution of the KPZ fixed point exhibits a shock at each of the boundaries between the different Busemann functions. Moving forward in time, the shocks coalesce, while moving backwards in time, additional shocks can form. We describe several geometric properties of this tree of shocks. This completes the study of eternal solutions initiated in earlier work of the second and third authors with Sepp\"al\"ainen and continued in the previous work of the authors and in recent work of Rassoul-Agha and Sweeney.

math.PR↗

arXiv2025

Exceptional force, uncountably many solutions in the KPZ fixed point

We give a complete characterization of all eternal solutions $b(x,t)$ of the KPZ fixed point satisfying the asymptotic slope condition $\lim_{|x| \to \infty} \frac{b(x,0)}{x} = 2\xi$. For fixed $\xi$, there is exactly one eternal solution with probability one. However, in the second and third authors' work with Sepp\"al\"ainen, it was shown that there exists a random, countably infinite set of slopes, for which there exist at least two eternal solutions. These correspond to two non-coalescing families of infinite geodesics in the same direction for the directed landscape. We denote the two eternal solutions as $b^{\xi-}$ and $b^{\xi +}$. In the present paper, we show that, for the exceptional slopes, there are in fact uncountably many eternal solutions. To give the characterization, we show that these eternal solutions are in bijection with a certain set of bi-infinite competition interfaces. Each bi-infinite interface separates the plane into two connected components -- a left component and a right component. A general eternal solution with slope $\xi$ is equal to $b^{\xi-}$ on the left component and equal to $b^{\xi +}$ on the right component. For these bi-infinite interfaces in the exceptional directions, we uncover new geometric phenomena that is not present for directed landscape geodesics. Additionally, we show that this set of eternal solutions appears as the Busemann limits $\mathcal{L}(\mathbf {v_n};\mathbf {p}) - \mathcal{L}(\mathbf{v_n};\mathbf {q})$ for sequences $\mathbf {v_n}$ going to $-\infty$ in direction $\xi$.

math.PR↗

arXiv2025

Hydrodynamics and relaxation limit for multilane exclusion process and related hyperbolic systems

We investigate the hydrodynamic behavior and local equilibrium of the multilane exclusion process, whose invariant measures were studied in our previous paper \cite{mlt1a}. The dynamics on each lane follows a hyperbolic time scaling, whereas the interlane dynamics has an arbitrary time scaling. We prove the following: (i) the hydrodynamic behavior of the global density (i.e. summed over all lanes) is governed by a scalar conservation law; (ii) the latter, as well as the limit of individual lanes, is the relaxation limit of a weakly coupled hyperbolic system of balance laws that approximates the particle system. For the hydrodynamic limit, to highlight new phenomena arising in our model, a precise computation of the flux function, with the transitions between different possible shapes (and a physical interpretation thereof), is given for the two-lane model.

math.PR↗

arXiv2024

Partial yet definite emergence of the Kardar-Parisi-Zhang class in isotropic spin chains

Integrable spin chains with a continuous non-Abelian symmetry, such as the one-dimensional isotropic Heisenberg model, show superdiffusive transport with little theoretical understanding. Although recent studies reported a surprising connection to the Kardar-Parisi-Zhang (KPZ) universality class in that case, this view was most recently questioned by discrepancies in full counting statistics. Here, by combining extensive numerical simulations of classical and quantum integrable isotropic spin chains with a framework developed by exact studies of the KPZ class, we characterize various two-point quantities that remain hitherto unexplored in spin chains, and find full agreement with KPZ scaling laws without adjustable parameters. This establishes the partial emergence of the KPZ class in integrable isotropic spin chains. Moreover, we reveal that the KPZ scaling laws are intact in the presence of an energy current, under the appropriate Galilean boost required by the propagation of spacetime correlation.

cond-mat.stat-mech↗

arXiv2023

Non-existence of three non-coalescing infinite geodesics with the same direction in the directed landscape

It is believed that for metric-like models in the KPZ class the following property holds: with probability one, starting from any point, there are at most two semi-infinite geodesics with the same direction that do not coalesce. Until now, such a result was only proved for one model - exponential LPP (Coupier 11') using its inherent connection to the totally asymmetric exclusion process. We prove that the above property holds for the directed landscape, the universal scaling limit of models in the KPZ class. Our proof reduces the problem to one on line ensembles and therefore paves the way to show similar results for other metric-like models in the KPZ class. Finally, combining our result with the ones in (Busani, Seppalainen,Sorensen 22', Bhatia 23') we obtain the full qualitative geometric description of infinite geodesics in the directed landscape.

math.PR↗

arXiv2023

Scaling limit of multi-type invariant measures via the directed landscape

This paper studies the large scale limits of multi-type invariant distributions and Busemann functions of planar stochastic growth models in the Kardar-Parisi-Zhang (KPZ) class. We identify a set of sufficient hypotheses for convergence of multi-type invariant measures of last-passage percolation (LPP) models to the stationary horizon (SH), which is the unique multi-type stationary measure of the KPZ fixed point. Our limit theorem utilizes conditions that are expected to hold broadly in the Kardar-Parisi-Zhang class, including convergence of the scaled last-passage process to the directed landscape. We verify these conditions for the six exactly solvable models whose scaled bulk versions converge to the directed landscape, as shown by Dauvergne and Vir\'ag. We also present a second, more general, convergence theorem with potential future applications to polymer models and particle systems. Our paper is the first to show convergence to the SH without relying on information about the structure of the multi-type invariant measures of the prelimit models. These results are consistent with the conjecture that the SH is the universal scaling limit of multi-type invariant measures in the KPZ class.

math.PR↗

arXiv2022

Scaling limit of the TASEP speed process

We show that the multi-type stationary distribution of the totally asymmetric simple exclusion process (TASEP) scales to a nontrivial limit around the Bernoulli measure of density $1/2$. This is obtained by showing that the TASEP speed process, introduced by Amir, Angel and Valk\'o, scales around the speed $v=0$ to the stationary horizon (SH), a function-valued stochastic process recently introduced and studied by the authors, SH is believed to be the universal scaling limit of Busemann processes in the KPZ universality class. Our results add to the evidence for this universality by connecting SH with multiclass particle configurations. Previously SH has been associated with the exponential corner growth model, Brownian last-passage percolation, and the directed landscape.

math.PR↗

arXiv2022

On the exponent governing the correlation decay of the Airy$_1$ process

We study the decay of the covariance of the Airy$_1$ process, $\mathcal{A}_1$, a stationary stochastic process on $\mathbb{R}$ that arises as a universal scaling limit in the Kardar-Parisi-Zhang (KPZ) universality class. We show that the decay is super-exponential and determine the leading order term in the exponent by showing that $\textrm{Cov}(\mathcal{A}_1(0),\mathcal{A}_1(u))= e^{-(\frac{4}{3}+o(1))u^3}$ as $u\to\infty$. The proof employs a combination of probabilistic techniques and integrable probability estimates. The upper bound uses the connection of $\mathcal{A}_1$ to planar exponential last passage percolation and several new results on the geometry of point-to-line geodesics in the latter model which are of independent interest; while the lower bound is primarily analytic, using the Fredholm determinant expressions for the two point function of the Airy$_1$ process together with the FKG inequality.

math.PR↗

arXiv2022

The stationary horizon and semi-infinite geodesics in the directed landscape

The stationary horizon (SH) is a stochastic process of coupled Brownian motions indexed by their real-valued drifts. It was first introduced by the first author as the diffusive scaling limit of the Busemann process of exponential last-passage percolation. It was independently discovered as the Busemann process of Brownian last-passage percolation by the second and third authors. We show that SH is the unique invariant distribution and an attractor of the KPZ fixed point under conditions on the asymptotic spatial slopes. It follows that SH describes the Busemann process of the directed landscape. This gives control of semi-infinite geodesics simultaneously across all initial points and directions. The countable dense set $\Xi$ of directions of discontinuity of the Busemann process is the set of directions in which not all geodesics coalesce and in which there exist at least two distinct geodesics from each initial point. This creates two distinct families of coalescing geodesics in each $\Xi$ direction. In $\Xi$ directions, the Busemann difference profile is distributed like Brownian local time. We describe the point process of directions $\xi\in\Xi$ and spatial locations where the $\xi\pm$ Busemann functions separate.

math.PR↗

arXiv2021

Diffusive scaling limit of the Busemann process in Last Passage Percolation

In exponential last passage percolation, we consider the rescaled Busemann process $x\mapsto N^{-1/3}B^\rho_{0,[xN^{2/3}]e_1} \,\, (x\in\mathbb{R})$, as a process parametrized by the scaled density $\rho=1/2+\frac{\mu}{4} N^{-1/3}$, and taking values in $C(\mathbb{R})$. We show that these processes, as $N\rightarrow \infty$, have a c\`adl\`ag scaling limit $G=(G_\mu)_{\mu\in \mathbb{R}}$, parametrized by $\mu$ and taking values in $C(\mathbb{R})$. The limiting process $G$, which can be thought of as the Busemann process under the KPZ scaling, can be described as an ensemble of "sticky" lines of Brownian regularity. We believe $G$ is the universal scaling limit of Busemann processes in the KPZ universality class. Our proof provides insight into this limiting behaviour by highlighting a connection between the joint distribution of Busemann functions obtained by Fan and Sepp\"al\"ainen in arXiv:1808.09069, and a sorting algorithm of random walks introduced by O'Connell and Yor [32]

math.PR↗

arXiv2021

Invariant measures for multilane exclusion process

We consider the simple exclusion process on Z x {0, 1}, that is, an ''horizontal ladder'' composed of 2 lanes, depending on 6 parameters. Particles can jump according to a lane-dependent translation-invariant nearest neighbour jump kernel, i.e. ''horizontally'' along each lane, and ''vertically'' along the scales of the ladder. We prove that generically, the set of extremal invariant measures consists of (i) translation-invariant product Bernoulli measures; and, modulo translations along Z: (ii) at most two shock measures (i.e. asymptotic to Bernoulli measures at $\pm$$\infty$) with asymptotic densities 0 and 2; (iii) at most one (outside degenerate cases) shock measure with a density jump of magnitude 1. We fully determine this set for a range of parameter values. In fact, outside degenerate cases, there is at most one shock measure of type (iii). Our results can be generalized in several directions using the same approach and answer certain open questions formulated in \cite{ligd} as a step towards the process on $\mathbb{Z}^2$.

math.PR↗

arXiv2020

Non-existence of bi-infinite polymer Gibbs measures

We show that nontrivial bi-infinite polymer Gibbs measures do not exist in typical environments in the inverse-gamma (or log-gamma) directed polymer model on the planar square lattice. The precise technical result is that, except for measures supported on straight-line paths, such Gibbs measures do not exist in almost every environment when the weights are independent and identically distributed inverse-gamma random variables. The proof proceeds by showing that when two endpoints of a point-to-point polymer distribution are taken to infinity in opposite directions but not parallel to lattice directions, the midpoint of the polymer path escapes. The proof is based on couplings, planar comparison arguments, and a recently discovered joint distribution of Busemann functions.

math.PR↗

arXiv2020

Bounds on the running maximum of a random walk with small drift

We derive a lower bound for the probability that a random walk with i.i.d.\ increments and small negative drift $\mu$ exceeds the value $x>0$ by time $N$. When the moment generating functions are bounded in an interval around the origin, this probability can be bounded below by $1-O(x|\mu| \log N)$. The approach is elementary and does not use strong approximation theorems.

math.PR↗

arXiv2020

Universality of the geodesic tree in last passage percolation

In this paper we consider the geodesic tree in exponential last passage percolation. We show that for a large class of initial conditions around the origin, the line-to-point geodesic that terminates in a cylinder of width $o(N^{2/3})$ and length $o(N)$ agrees in the cylinder, with the stationary geodesic sharing the same end point. In the case of the point-to-point model, we consider width $\delta N^{2/3}$ and length up to $\delta^{3/2} N/(\log(\delta^{-1}))^3$ and provide lower and upper bound for the probability that the geodesics agree in that cylinder.

math.PR↗

arXiv2020

Local stationarity of exponential last passage percolation

We consider point to point last passage times to every vertex in a neighbourhood of size $\delta N^{\frac{2}{3}}$, distance $N$ away from the starting point. The increments of these last passage times in this neighbourhood are shown to be jointly equal to their stationary versions with high probability that depends on $\delta$ only. With the help of this result we show that 1) the $\text{Airy}_2$ process is locally close to a Brownian motion in total variation; 2) the tree of point to point geodesics starting from every vertex in a box of side length $\delta N^{\frac{2}{3}}$ going to a point at distance $N$ agree inside the box with the tree of infinite geodesics going in the same direction; 3) two geodesics starting from $N^{\frac{2}{3}}$ away from each other, to a point at distance $N$ will not coalesce too close to either endpoints on the macroscopic scale. Our main results rely on probabilistic methods only.

math.PR↗

arXiv2019

The TAZRP speed process

In [AAV] Amir, Angel and Valk{\'o} studied a multi-type version of the totally asymmetric simple exclusion process (TASEP) and introduced the TASEP speed process, which allowed them to answer delicate questions about the joint distribution of the speed of several second-class particles in the TASEP rarefaction fan. In this paper we introduce the analogue of the TASEP speed process for the totally asymmetric zero-range process (TAZRP), and use it to obtain new results on the joint distribution of the speed of several second-class particles in the TAZRP with a reservoir. There is a close link from the speed process to questions about stationary distributions of multi-type versions of the TAZRP; for example we are able to give a precise description of the contents of a single site in equilibrium for a multi-type TAZRP with continuous labels.

math.PR↗

arXiv2019

Non-existence of bi-infinite geodesics in the exponential corner growth model

This paper gives a self-contained proof of the non-existence of nontrivial bi-infinite geodesics in directed planar last-passage percolation with exponential weights. The techniques used are couplings, coarse graining, and control of geodesics through planarity and estimates derived from increment-stationary versions of the last-passage percolation process.

math.PR↗

arXiv2017

Continuous time random walk as a random walk in a random environment

We show that for a weakly dense subset of the domain of attraction of a positive stable random variable of index $0<\alpha<1$($DOA\left(\alpha\right))$ the functional stable convergence is a time-changed renewal convergence of distribution of finite mean. Applied to Continuous Time Random Walk(CTRW) \'a la Montroll and Wiess we show that CTRW with renewal times in a weakly dense set of $DOA\left(\alpha\right)$ can be realized as random walk in a random environment. We find the quenched limit and give a bound on the error of the approximation.

math.PR↗