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Timo Seppäläinen

Publications and source records attributed to Timo Seppäläinen.

At least 19 recordsLinked to original sources
arXiv2026

The Green's function of the parabolic Anderson model and the continuum directed polymer

We build a regular version of the field $Z_β(t,x|s,y)$ which describes the Green's function, or fundamental solution, of the parabolic Anderson model (PAM) with white noise forcing on $\mathbb{R}^{1+1}$: $\partial_t Z_β(t,x | s,y) =$ $\frac{1}{2}\partial_{xx} Z_β(t,x|s,y) + βZ_β(t,x | s,y)W(t,x)$, $Z_β(s,x | s,y) = δ(x-y)$ for all $-\infty < s \leq t < \infty$, all $x,y \in \mathbb{R}$, and all $β\in \mathbb{R}$ simultaneously. Through the superposition principle, our construction gives a pointwise coupling of all solutions to the PAM with initial or terminal conditions satisfying sharp growth assumptions, for all initial and terminal times. Using this coupling, we show that the PAM with a (sub-)exponentially growing initial condition admits conserved quantities given by the limits $\displaystyle \lim_{x\to \pm\infty} x^{-1}\log Z_β(t,x)$, in addition to proving many new basic properties of solutions to the PAM with general initial conditions. These properties are then connected to the existence, regularity, and continuity of the quenched continuum polymer measures. Through the polymer connection, we also show that the kernel $(x,y) \mapsto Z_β(t,x | s,y)$ is strictly totally positive for all $t>s$ and $β\in \mathbb{R}$.

math.PR↗
arXiv2026

Rooted Gibbs-DLR Measures in Planar Directed Polymers

We study rooted Gibbs-DLR measures in the directed polymer model on $\mathbb{Z}^2$ with an ergodic disorder distribution which satisfies an additional mild hypothesis. We prove that the set of extremal rooted Gibbs-DLR measures is closed and totally ordered, characterize extremality in terms of path coalescence, and show that each fully supported extremal rooted Gibbs measure canonically generates a globally consistent and coalescing family of extremal rooted Gibbs measures indexed by all lattice sites. These families are, moreover, totally ordered. Building on this structure, we prove strong existence and strong uniqueness of the associated Busemann process, together with an $L^1$ continuity theorem for the shift-covariant Busemann cocycles which is joint in the inverse temperature, the tilt parameter, and the random environment. This yields, as a corollary, in-probability continuity of the generated extremal Gibbs measures corresponding to directions of differentiability under bounded i.i.d. perturbations of the weights. In positive temperature, it shows in-probability convergence of the generated extremal Gibbs measures. At zero temperature, it also yields quenched subsequential large deviation principles for the corresponding positive-temperature rooted Gibbs-DLR measures on path space. The rate functions are determined by a zero-temperature Busemann cocycle and vanish precisely on the infinite geodesics generated by that cocycle.

math.PR↗
arXiv2026

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

Anomalous geodesics in the inhomogeneous corner growth model

We study Busemann functions, semi-infinite geodesics, and competition interfaces in the exactly solvable last-passage percolation with inhomogeneous exponential weights. New phenomena concerning geodesics arise due to inhomogeneity. These include novel Busemann functions associated with flat regions of the limit shape and thin rectangles, undirected semi-infinite geodesics, non-trivial axis-directed geodesics, intervals with no geodesic directions, and isolated geodesic directions. We further observe a new dichotomy for competition interfaces and second-class customers in a series of memoryless continuous-time queues with inhomogeneous service rates: a second class customer either becomes trapped or proceeds through the queue at strictly positive speed.

math.PR↗
arXiv2025

Temporal correlation in the inverse-gamma polymer

Understanding the decay of correlations in time for (1+1)-dimensional polymer models in the KPZ universality class has been a challenging topic. Following numerical studies by physicists, concrete conjectures were formulated by Ferrari and Spohn (Ferrari-Spohn '16) in the context of planar exponential last passage percolation. These have mostly been resolved by various authors. In the context of positive temperature lattice models, however, these questions have remained open. We consider the time correlation problem for the exactly solvable inverse-gamma polymer in $\mathbb{Z}^2$. We establish, up to constant factors, upper and lower bounds on the correlation between free energy functions for two polymers rooted at the origin (droplet initial condition) when the endpoints are either close together or far apart. We find the same exponents as predicted in (Ferrari-Spohn '16). Our arguments rely on the understanding of stationary polymers, coupling, and random walk comparison. We use recently established moderate deviation estimates for the free energy. In particular, we do not require asymptotic analysis of complicated exact formulae.

math.PR↗
arXiv2025

Intertwining the Busemann process of the directed polymer model

We study the Busemann process and competition interfaces of the planar directed polymer model with i.i.d.\ weights on the vertices of the planar square lattice, in both the general case and the solvable inverse-gamma case. We prove new regularity properties of the Busemann process without reliance on unproved assumptions on the shape function. For example, each nearest-neighbor Busemann function is strictly monotone and has the same random set of discontinuities in the direction variable. When all Busemann functions on a horizontal line are viewed together, the Busemann process intertwines with an evolution that obeys a version of the geometric Robinson-Schensted-Knuth correspondence. When specialized to the inverse-gamma case, this relationship enables an explicit distributional description: the Busemann function on a nearest-neighbor edge has independent increments in the direction variable, and its distribution comes from an inhomogeneous planar Poisson process. The distribution of the asymptotic competition interface direction of the inverse-gamma polymer is discrete and supported on the Busemann discontinuities which -- unlike in zero-temperature last-passage percolation -- are dense. Further implications follow for the eternal solutions and the failure of the one force -- one solution principle of the discrete stochastic heat equation solved by the polymer partition function.

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

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↗
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

Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique

We develop a new probabilistic method for deriving deviation estimates in directed planar polymer and percolation models. The key estimates are for exit points of geodesics as they cross transversal down-right boundaries. These bounds are of optimal cubic-exponential order. We derive them in the context of last-passage percolation with exponential weights with near-stationary boundary conditions. As a result, the probabilistic coupling method is empowered to treat a variety of problems optimally, which could previously be achieved only via inputs from integrable probability. As applications in the bulk setting, we obtain upper bounds of cubic-exponential order for transversal fluctuations of geodesics, and cube-root upper bounds with a logarithmic correction for distributional Busemann limits and competition interface limits. Several other applications are already in the literature.

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

Coalescence and total-variation distance of semi-infinite inverse-gamma polymers

We show that two semi-infinite positive temperature polymers coalesce on the scale predicted by KPZ (Kardar-Parisi-Zhang) universality. The two polymer paths have the same asymptotic direction and evolve in the same environment, independently until coalescence. If they start at distance $k$ apart, their coalescence occurs on the scale $k^{3/2}$. It follows that the total variation distance of two semi-infinite polymer measures decays on this same scale. Our results are upper and lower bounds on probabilities and expectations that match, up to constant factors and occasional logarithmic corrections. Our proofs are done in the context of the solvable inverse-gamma polymer model, but without appeal to integrable probability. With minor modifications, our proofs give also bounds on transversal fluctuations of the polymer path. Since the free energy of a directed polymer is a discretization of a stochastically forced viscous Hamilton-Jacobi equation, our results suggest that the hyperbolicity phenomenon of such equations obeys the KPZ exponent.

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↗
arXiv2023

Geodesic length and shifted weights in first-passage percolation

We study first-passage percolation through related optimization problems over paths of restricted length. The path length variable is in duality with a shift of the weights. This puts into a convex duality framework old observations about the convergence of the normalized Euclidean length of geodesics due to Hammersley and Welsh, Smythe and Wierman, and Kesten, and leads to new results about geodesic length and the regularity of the shape function as a function of the weight shift. For points far enough away from the origin, the ratio of the geodesic length and the $\ell^1$ distance to the endpoint is uniformly bounded away from one. The shape function is a strictly concave function of the weight shift. Atoms of the weight distribution generate singularities, that is, points of nondifferentiability, in this function. We generalize to all distributions, directions and dimensions an old singularity result of Steele and Zhang for the planar Bernoulli case. When the weight distribution has two or more atoms, a dense set of shifts produce singularities. The results come from a combination of the convex duality, the shape theorems of the different first-passage optimization problems, and modification arguments.

math.PR↗
arXiv2022

Ergodicity and synchronization of the Kardar-Parisi-Zhang equation

The Kardar-Parisi-Zhang (KPZ) equation on the real line is well-known to admit Brownian motion with a linear drift as a stationary distribution (modulo additive constants). We show that these solutions are attractive, a result known as a one force--one solution (1F1S) principle or synchronization: the solution to the KPZ equation started in the distant past from an initial condition with a given slope will converge almost surely to a Brownian motion with that drift, which shows in particular that these invariant measures are totally ergodic. Our proof constructs the Busemann process for the equation, which gives the natural jointly stationary coupling of all of these stationary solutions. Synchronization then holds simultaneously (on a single full probability event) for all but an at most countable random set of asymptotic slopes. This set of exceptional slopes of instability for which synchronization fails is either almost surely empty or almost surely dense. Along the way, we prove a shape theorem which implies almost sure stochastic homogenization of the KPZ equation, for which the Busemann process gives the process of correctors. We also show that the forward and backward point-to-point and point-to-line continuum polymers converge to semi-infinite continuum polymers whose transitions are Doob transforms via Busemann functions of the transitions of the finite length polymers.

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↗
arXiv2021

Geometry of geodesics through Busemann measures in directed last-passage percolation

We consider planar directed last-passage percolation on the square lattice with general i.i.d. weights and study the geometry of the full set of semi-infinite geodesics in a typical realization of the random environment. The structure of the geodesics is studied through the properties of the Busemann functions viewed as a stochastic process indexed by the asymptotic direction. Our results are further connected to the ergodic program for and stability properties of random Hamilton-Jacobi equations. In the exactly solvable exponential model, our results specialize to give the first complete characterization of the uniqueness and coalescence structure of the entire family of semi-infinite geodesics for any model of this type. Furthermore, we compute statistics of locations of instability, where we discover an unexpected connection to simple symmetric random walk.

math.PR↗