arXiv subjects

Evan Sorensen

Publications and source records attributed to Evan Sorensen.

17 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↗

arXiv2026

Invariant measures for half-space geometric LPP: classification and the one force--one solution principle

We prove a complete characterization of the extremal invariant measures for half-space geometric last-passage percolation with an arbitrary boundary parameter. This is the first result of its kind for a model in the KPZ universality class that has boundary effects and an unbounded domain. A description of a class of invariant measures was previously given in a work of Barraquand and Corwin, where it was conjectured that these should comprise all extremal invariant measures. To complete the classification, we prove a one force--one solution principle: when started in the distant past from an arbitrary initial condition with a given asymptotic slope at $\infty$, the recentered solution at time $0$ converges to a process which is distributed as the associated invariant measure with the specified slope. This limiting process is called the Busemann process, the first of its kind constructed for a half-space model. The Busemann process across all slopes is distributed as the joint invariant measure for geometric half-space LPP, recently constructed by Dauvergne and Zhang. There, it was conjectured that the constructed family of jointly invariant measures comprises all extremal jointly invariant measures; our analysis also confirms this conjecture. When the model has a strong (attractive) boundary, the collection of slopes for the invariant measures has a discontinuity, which does not arise in the full-space case. To handle this difficulty, we combine the control of the directions of semi-infinite geodesics with techniques from the theory of half-space Gibbsian line ensembles. Along the way, we classify the set of directions of semi-infinite geodesics for half-space geometric LPP, confirming a recent conjecture of Dauvergne and Zhang.

math.PR↗

arXiv2025

An upper bound on geodesic length in 2D critical first-passage percolation

We consider i.i.d. first-passage percolation (FPP) on the two-dimensional square lattice, in the critical case where edge-weights take the value zero with probability $\tfrac{1}{2}$. Critical FPP is unique in that the Euclidean lengths of geodesics are superlinear -- rather than linear -- in the distance between their endpoints. This fact was speculated by Kesten in 1986 but not confirmed until 2019 by Damron and Tang, who showed a lower bound on geodesic length that is polynomial with degree strictly greater than $1$. In this paper, we establish the first nontrivial upper bound. Namely, we prove that for a large class of critical edge-weight distributions, the shortest geodesic from the origin to a box of radius $R$ uses at most $R^{2+ε}π_3(R)$ edges with high probability, for any $ε> 0$. Here $π_3(R)$ is the polychromatic 3-arm probability from classical Bernoulli percolation; upon inserting its conjectural asymptotic, our bound converts to $R^{4/3 + ε}$. In any case, it is known that $π_3(R) \lesssim R^{-δ}$ for some $δ> 0$, so our bound gives an exponent strictly less than $2$. In the special case of Bernoulli($\tfrac{1}{2}$) edge-weights, we replace the additional factor of $R^ε$ with a constant and give an expectation bound.

math.PR↗

arXiv2025

Invariant measures and shocks in the KPZ fixed point

We construct a family of invariant measures from the perspective of a shock in the KPZ fixed point. These measures are parameterized by a positive number $\theta > 0$, and are supported on functions $f$ satisfying $\lim_{|x| \to \infty} \frac{f(x)}{|x|} = 2\theta$. Each can be described as the sum of a Brownian motion and an independent Bessel-$3$ process with drift. We show that these measures appear as the $L \to \infty$ limit of the (conjectural) stationary measures for the conjectural open KPZ fixed point on $[0,L]$, after recentering by an appropriately defined shock location. Furthermore, we show that, with respect to the standard, deterministic recentering at $x = 0$, all extremal invariant measures for the KPZ fixed point are Brownian motions with drift. To do this, we first show that any extremal invariant measures must be supported on functions having fixed asymptotic slopes at $\pm \infty$. Using a one-force-one-solution principle from the work of Busani, Sepp\"al\"ainen, and the second author, this rules out all other invariant measures except those having left slope $-2\theta$ and right slope $+2\theta$ for some $\theta > 0$. To handle this case, we derive the limiting fluctuations of the shock for a special choice of initial condition. Additionally, we derive the limiting fluctuations of the shock for the case of the invariant measure from the perspective of a shock, and for the case of initial data $f(x) = 2\theta|x|$.

math.PR↗

arXiv2025

The discrete periodic Pitman transform: invariances, braid relations, and Burke properties

We develop the theory of the discrete periodic Pitman transform, first introduced by Corwin, Gu, and the fifth author. We prove that the discrete periodic Pitman transform satisfies the same braid relations that are satisfied for the full-line Pitman transform shown by Biane, Bougerol, and O'Connell. This defines a group action of the infinite symmetric group on sequences of vectors in $\mathbb R^{\mathbb Z_N}$. We prove that, for polymers in a periodic environment, single-path and multi-path partition functions are preserved under the action of this transform on the weights in the polymer model. Combined with a new inhomogeneous Burke property for the periodic Pitman transform, we prove a multi-path invariance result for the periodic inverse-gamma polymer under permutations of the column parameters. In the limit to the full-line case, we obtain a multi-path extension of a recent invariance result of Bates, Emrah, Martin, Sepp\"al\"ainen, and the fifth author, in both positive and zero-temperature.

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 $\lim_{|x| \to \infty} \frac{b(x,0)}{x} = 2ξ$. For fixed $ξ$, there is exactly one eternal solution with probability one. However, in the second and third authors' work with Seppäläinen, 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 $b^{ξ-}$ and $b^{ξ+}$. 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 $ξ$ is equal to $b^{ξ-}$ on the left component and equal to $b^{ξ+}$ 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 $ξ$.

math.PR↗

arXiv2025

Permutation invariance in last-passage percolation and the distribution of the Busemann process

In i.i.d. exponential last-passage percolation, we describe the joint distribution of Busemann functions, over all edges and over all directions, in terms of a joint last-passage problem in a finite inhomogeneous environment. More specifically, the Busemann increments within a $k\times\ell$ grid, and associated to $d$ different directions, are equal in distribution to a particular collection of last-passage increments inside a $(k+d-1)\times(\ell+d-1)$ grid. The joint Busemann distribution was previously described along a horizontal line by Fan and the fourth author, using certain queuing maps. By contrast, our new description explicitly gives the joint distribution for any collection of edges (not just along a horizontal line) using only finitely many random variables. Our result thus provides an exact and accessible way to sample from the joint distribution. In the proof, we rely on one-directional marginal distributions of the inhomogeneous Busemann functions recently studied by Janjigian and the second and fourth authors. The second ingredient of our proof is a novel joint invariance of inhomogeneous last-passage times under permutations of the inhomogeneity parameters. Our proof of the invariance is different from earlier proofs of such results, using the Burke property instead of the RSK correspondence, and leading to an explicit coupling of the weights before and after the permutation of the parameters.

math.PR↗

arXiv2025

Jointly invariant measures for the Kardar-Parisi-Zhang Equation

We give an explicit description of the jointly invariant measures for the KPZ equation. These are couplings of Brownian motions with drift, and can be extended to a process defined for all drift parameters simultaneously. We term this process the KPZ horizon (KPZH). As a corollary of this description, we resolve a recent conjecture of Janjigian, and the second and third authors by showing the existence of a random, countably infinite dense set of directions at which the Busemann process of the KPZ equation is discontinuous. This signals instability and shows the failure of the one force--one solution principle and the existence of at least two extremal semi-infinite polymer measures in the exceptional directions. As the inverse temperature parameter $β$ for the KPZ equation goes to $\infty$, the KPZH converges to the stationary horizon (SH) first introduced by Busani, and studied further by Busani and the third and fourth authors. As $β\searrow 0$, the KPZH converges to a coupling of Brownian motions that differ by linear shifts, which is a jointly invariant measure for the Edwards-Wilkinson fixed point.

math.PR↗

arXiv2024

Periodic Pitman transforms and jointly invariant measures

We construct explicit jointly invariant measures for the periodic KPZ equation (and therefore also the stochastic Burgers' and stochastic heat equations) for general slope parameters and prove their uniqueness via a one force--one solution principle. The measures are given by polymer-like transforms of independent Brownian bridges. We describe several properties and limits of these measures, including an extension to a continuous process in the slope parameter that we term the periodic KPZ horizon. As an application of our construction, we prove a Gaussian process limit theorem with an explicit covariance function for the long-time height function fluctuations of the periodic KPZ equation when started from varying slopes. In connection with this, we conjecture a formula for the fluctuations of cumulants of the endpoint distribution for the periodic continuum directed random polymer. To prove joint invariance, we address the analogous problem for a semi-discrete system of SDEs related to the periodic O'Connell-Yor polymer model and then perform a scaling limit of the model and jointly invariant measures. For the semi-discrete system, we demonstrate a bijection that maps our systems of SDEs to another system with product invariant measure. Inverting the map on this product measure yields our invariant measures. This map relates to a periodic version of the discrete geometric Pitman transform that we introduce and probe. As a by-product of this, we show that the jointly invariant measures for a periodic version of the inverse-gamma polymer are the same as those for the O'Connell-Yor polymer.

math.PR↗

arXiv2024

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 Virág. 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↗

arXiv2024

Viscous shock fluctuations in KPZ

We study ``V-shaped'' solutions to the KPZ equation, those having opposite asymptotic slopes $\theta$ and $-\theta$, with $\theta>0$, at positive and negative infinity, respectively. Answering a question of Janjigian, Rassoul-Agha, and Sepp\"al\"ainen, we show that the spatial increments of V-shaped solutions cannot be statistically stationary in time. This completes the classification of statistically time-stationary spatial increments for the KPZ equation by ruling out the last case left by those authors. To show that these V-shaped time-stationary measures do not exist, we study the location of the corresponding ``viscous shock,'' which, roughly speaking, is the location of the bottom of the V. We describe the limiting rescaled fluctuations, and in particular show that the fluctuations of the shock location are not tight, for both stationary and flat initial data. We also show that if the KPZ equation is started with V-shaped initial data, then the long-time limits of the time-averaged laws of the spatial increments of the solution are mixtures of the laws of the spatial increments of $x\mapsto B(x)+\theta x$ and $x\mapsto B(x)-\theta x$, where $B$ is a standard two-sided Brownian motion.

math.PR↗

arXiv2023

The stationary horizon as the central multi-type invariant measure in the KPZ universality class

The Kardar-Parisi-Zhang (KPZ) universality class describes a large class of 2-dimensional models of random growth, which exhibit universal scaling exponents and limiting statistics. The last ten years has seen remarkable progress in this area, with the formal construction of two interrelated limiting objects, now termed the KPZ fixed point and the directed landscape (DL). This dissertation focuses on a third central object, termed the stationary horizon (SH). The SH was first introduced (and named) by Busani as the scaling limit of the Busemann process in exponential last-passage percolation. Shortly after, in the author's joint work with Seppäläinen, it was independently constructed in the context of Brownian last-passage percolation. In this dissertation, we give an alternate construction of the SH, directly from the description of its finite-dimensional distributions and without reference to Busemann functions. From this description, we give several exact distributional formulas for the SH. Next, we show the significance of the SH as a key object in the KPZ universality class by showing that the SH is the unique coupled invariant distribution for the DL. A major consequence of this result is that the SH describes the Busemann process for the DL. From this connection, we give a detailed description of the collection of semi-infinite geodesics in the DL, from all initial points and in all directions. As a further evidence of the universality of the SH, we show that it appears as the scaling limit of the multi-species invariant measures for the totally asymmetric simple exclusion process (TASEP). This dissertation is adapted from two joint works with Seppäläinen and two joint works with Busani and Seppäläinen.

math.PR↗

arXiv2023

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 $Ξ$ 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 $Ξ$ direction. In $Ξ$ directions, the Busemann difference profile is distributed like Brownian local time. We describe the point process of directions $ξ\inΞ$ and spatial locations where the $ξ\pm$ Busemann functions separate.

math.PR↗

arXiv2023

Global structure of semi-infinite geodesics and competition interfaces in Brownian last-passage percolation

In Brownian last-passage percolation (BLPP), the Busemann functions $\mathcal B^θ(\mathbf x,\mathbf y)$ are indexed by two points $\mathbf x,\mathbf y \in \mathbb Z \times \mathbb R$, and a direction parameter $θ> 0$. We derive the joint distribution of Busemann functions across all directions. The set of directions where the Busemann process is discontinuous, denoted $Θ$, provides detailed information about the uniqueness and coalescence of semi-infinite geodesics. The uncountable set of initial points in BLPP gives rise to new phenomena not seen in discrete models. For example, in every direction $θ> 0$, there exists a countably infinite set of initial points $\mathbf x$ such that there exist two $θ$-directed geodesics that split but eventually coalesce. Further, we define the competition interface in BLPP and show that the set of initial points whose competition interface is nontrivial has Hausdorff dimension $\frac{1}{2}$. From each of these exceptional points, there exists a random direction $θ\in Θ$ for which there exists two $θ$-directed semi-infinite geodesics that split immediately and never meet again. Conversely, when $θ\in Θ$, from every initial point $\mathbf x \in \mathbb Z \times \mathbb R$, there exists two $θ$-directed semi-infinite geodesics that eventually separate. Whenever $θ\notin Θ$, all $θ$-directed semi-infinite geodesics coalesce.

math.PR↗

arXiv2023

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 Valkó, 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↗

arXiv2021

Busemann process and semi-infinite geodesics in Brownian last-passage percolation

We prove the existence of semi-infinite geodesics for Brownian last-passage percolation (BLPP). Specifically, on a single event of probability one, there exist semi-infinite geodesics started from every space-time point and traveling in every asymptotic direction. Properties of these geodesics include uniqueness for a fixed initial point and direction, non-uniqueness for fixed direction but random initial points, and coalescence of all geodesics traveling in a common, fixed direction. Along the way, we prove that for fixed northeast and southwest directions, there almost surely exist no bi-infinite geodesics in the given directions. The semi-infinite geodesics are constructed from Busemann functions. Our starting point is a result of Alberts, Rassoul-Agha and Simper that established Busemann functions for fixed points and directions. Out of this, we construct the global process of Busemann functions simultaneously for all initial points and directions, and then the family of semi-infinite Busemann geodesics. The uncountable space of the semi-discrete setting requires extra consideration and leads to new phenomena, compared to discrete models.

math.PR↗

arXiv2018

On Zero-Sector Reducing Operators

We prove a Jensen-disc type theorem for polynomials $p\in\mathbb{R}[z]$ having all their zeros in a sector of the complex plane. This result is then used to prove the existence of a collection of linear operators $T\colon\mathbb{R}[z]\to\mathbb{R}[z]$ which map polynomials with their zeros in a closed convex sector $|\arg z| \leq θ<π/2$ to polynomials with zeros in a smaller sector $|\arg z| \leq γ<θ$. We, therefore, provide the first example of a zero-sector reducing operator.

math.CV↗