Planar Corner Growth Model

• The planar corner growth model is a directed growth process on a square lattice, modeling last-passage percolation with applications in KPZ universality.
• It is formulated via random vertex weights and equivalent recursions that yield deterministic limit shapes and explicit variational formulas.
• Exactly solvable cases using exponential or geometric weights provide KPZ scaling exponents, offering insights into geodesic behavior, fluctuations, and competition interfaces.

The planar corner growth model is a directed growth process on the square lattice, usually identified with directed last-passage percolation on $\mathbb Z^2$. Growth proceeds from a corner of the first quadrant under the rule that a site can be occupied only after its west and south predecessors are available, and the resulting random interface is encoded by maximization over up-right lattice paths with steps $e_1=(1,0)$ and $e_2=(0,1)$. In the literature this model appears in both general i.i.d. and exactly solvable exponential or geometric forms, and it serves as a standard $1+1$-dimensional KPZ model for the study of deterministic limit shapes, Busemann functions, stationary cocycles, semi-infinite geodesics, and competition interfaces (Georgiou et al., 2015, Seppäläinen, 2017).

1. Lattice definition and equivalent formulations

A basic formulation assigns a random weight $\omega_x$ to each vertex $x\in\mathbb Z^2$. For coordinatewise ordered points $x\le y$, the point-to-point last-passage time $G_{x,y}$ is the maximum total weight collected along an up-right path from $x$ to $y$. In one common convention, the terminal weight is excluded from the sum, while the growth cluster is then written as

$e_1=(1,0)$0

A second convention includes the terminal weight and leads to the recursion

$e_1=(1,0)$1

with $e_1=(1,0)$2 the site weight. These are equivalent descriptions of the same directed growth mechanism (Georgiou et al., 2015, Emrah, 2016).

The growth interpretation is that $e_1=(1,0)$3 becomes infected, red, or occupied only after both $e_1=(1,0)$4 and $e_1=(1,0)$5, together with its own waiting time. The induced time-$e_1=(1,0)$6 cluster can be written as

$e_1=(1,0)$7

so the planar corner growth model is simultaneously a random growth process, a directed polymer optimization problem at zero temperature, and a last-passage percolation model (Emrah, 2016, Yeliussizov, 2019).

Exactly solvable instances arise when the site weights are i.i.d. $e_1=(1,0)$8 or geometric. More general treatments assume only i.i.d., nondegenerate weights with finite $e_1=(1,0)$9 moment and a lower bound almost surely. In the continuous-weight case, finite maximizing paths are almost surely unique, and the collection of geodesics from the origin forms a geodesic tree (Georgiou et al., 2015, Georgiou et al., 2015).

A recurrent source of confusion is the coexistence of several indexing conventions. Some papers work on $e_2=(0,1)$0, others on $e_2=(0,1)$1, and some parameterize directions on the simplex $e_2=(0,1)$2. These are not different models; they are coordinate and normalization choices adapted to particular questions about shape, stationarity, or geodesics (Georgiou et al., 2015, Seppäläinen, 2018).

2. Deterministic asymptotics and shape functions

Under standard ergodicity and moment assumptions, the model has a deterministic limit shape

$e_2=(0,1)$3

and, in general i.i.d. settings, $e_2=(0,1)$4 is continuous, concave, symmetric, and $e_2=(0,1)$5-homogeneous. The point-to-line counterpart also has a deterministic limit and is related to $e_2=(0,1)$6 by convex duality: $e_2=(0,1)$7 This dual structure is central to stationary formulations and to the cocycle approach (Georgiou et al., 2015, Georgiou et al., 2014).

In the exponential corner growth model the shape function is explicit: $e_2=(0,1)$8 The corresponding point-to-line shape is

$e_2=(0,1)$9

These formulas make the exponential model the standard benchmark for exact KPZ analysis (Seppäläinen, 2017).

A key refinement is the stationary exponential model indexed by $1+1$0. In that construction the bulk weights are $1+1$1, the horizontal boundary increments are $1+1$2, and the vertical boundary increments are $1+1$3. The associated stationary last-passage field has linear shape

$1+1$4

This linear stationary shape is not the nonstationary limit shape itself; rather, it provides the stationary comparison object from which the nonstationary shape and Busemann structure are extracted (Seppäläinen, 2017).

The exact solvable inhomogeneous models show that strict curvature is not universal. When site weights are exponential with row and column parameters $1+1$5, or geometric with parameters $1+1$6, the shape is given by explicit variational formulas and can be strictly concave only inside a cone, while becoming linear outside it. This contrasts with the classical expectation for light-tailed i.i.d. models, where the shape is expected to be strictly concave in all directions (Emrah, 2016).

3. Stationary cocycles and Busemann functions

A stationary $1+1$7 cocycle is a shift-covariant additive field

$1+1$8

that satisfies stationarity and the cocycle property

$1+1$9

Its decisive feature is the recovery identity

$\omega_x$0

which makes $\omega_x$1 a stationary boundary-gradient object compatible with the original environment. In general i.i.d. planar LPP, such cocycles were constructed from tandem queueing fixed points and used as boundary data for stationary versions of last-passage percolation (Georgiou et al., 2015, Georgiou et al., 2014).

These cocycles solve the variational formulas for the limit shapes. In particular, the point-to-line and point-to-point asymptotics can be written as infima over centered cocycles or cocycles with mean tilt, and the Busemann-type cocycles act as minimizers or correctors. This places the planar corner growth model in a framework analogous to homogenization, with cocycles furnishing stationary gradients of the asymptotic growth profile (Georgiou et al., 2015, Seppäläinen, 2017).

A Busemann function is an almost sure limit of passage-time differences to remote targets: $\omega_x$2 for sequences $\omega_x$3 tending to infinity in direction $\omega_x$4. In the general i.i.d. model, existence is proved at exposed differentiability points of the limit shape and on interiors of maximal linear segments with differentiable endpoints. In the exponential model, Busemann functions exist in every direction $\omega_x$5, and the nearest-neighbor increments have explicit distributions

$\omega_x$6

with mean equal to the shape gradient (Georgiou et al., 2015, Seppäläinen, 2017).

The exactly solvable exponential model admits a stronger all-directions description. The joint distribution of $\omega_x$7 is characterized through multiclass queueing fixed points; for fixed $\omega_x$8, the process $\omega_x$9 is càdlàg and equal in distribution to a pure-jump marked point process whose jump times on $x\in\mathbb Z^2$0 form a Poisson point process with intensity $x\in\mathbb Z^2$1. At each vertex there is a unique random threshold $x\in\mathbb Z^2$2 where the preferred predecessor switches, and

$x\in\mathbb Z^2$3

These formulas make the directional Busemann field an explicitly tractable stochastic process, not merely a collection of separate one-direction limits (Fan et al., 2018).

A further structural sharpening is the strong existence and uniqueness theorem for the tilt-indexed Busemann process. In the i.i.d. planar corner growth model, the Busemann process indexed by tilts in the super-differential of the limit shape can be realized as a measurable function of the underlying i.i.d. weights on the canonical probability space, and any two such realizations coincide almost surely. This removes the need for auxiliary extensions of the probability space and shows that the full tilt-indexed Busemann family is already encoded in the weight field itself (Janjigian et al., 11 Jul 2025).

4. Geodesics and the competition interface

Finite maximizing paths are geodesics. Under continuous weights, every vertex in $x\in\mathbb Z^2$4 has a unique geodesic from the origin, and these form a geodesic tree. Semi-infinite geodesics are infinite up-right paths whose every finite segment is maximizing. In the general i.i.d. model, stationary cocycles and Busemann functions yield existence of directional semi-infinite geodesics from every starting point, as well as leftmost and rightmost versions when continuity fails (Georgiou et al., 2015).

When the limit shape is differentiable at the relevant directions and the weight distribution is continuous, the directional theory becomes sharp: for each direction there is a unique semi-infinite geodesic from each site, and all such geodesics coalesce. Finite geodesics to targets $x\in\mathbb Z^2$5 converge locally to the corresponding Busemann geodesic. In the exponential model this program can be carried out by a comparatively soft argument based on the increment-stationary growth process, coupling, and planar monotonicity, rather than on detailed fluctuation estimates (Georgiou et al., 2015, Seppäläinen, 2018).

The competition interface is the up-right dual path separating the descendants of $x\in\mathbb Z^2$6 and $x\in\mathbb Z^2$7 in the geodesic tree rooted at the origin. In the continuous-weight setting it satisfies a law of large numbers,

$x\in\mathbb Z^2$8

where $x\in\mathbb Z^2$9 is a random direction characterized by the balancing of cocycle increments. The atoms of the law of $x\le y$0 are exactly the corners of the limit shape, and under differentiability assumptions on the endpoints of linear segments, $x\le y$1 is almost surely an exposed point of the shape (Georgiou et al., 2015).

The nonexistence of bi-infinite geodesics is one of the main global rigidity results. For a fixed direction, the increment-stationary approach yields absence of bi-infinite directed geodesics in the exponential model. A stronger theorem shows that, with i.i.d. $x\le y$2 weights, there are almost surely no nontrivial bi-infinite geodesics at all; the only bi-infinite geodesics are the trivial axis lines. The proof combines stationary couplings, coarse graining, exit-point estimates, and planarity (Seppäläinen, 2018, Balázs et al., 2019).

5. Fluctuations, coalescence scales, and KPZ geometry

In the stationary exponential corner growth model, the longitudinal fluctuation exponent is $x\le y$3 and the transversal exponent is $x\le y$4. The variance of $x\le y$5 under characteristic scaling is of order $x\le y$6, corresponding to $x\le y$7-order fluctuations of passage times, while the geodesic deviates from its characteristic line on the $x\le y$8 scale. These are the standard KPZ exponents in $x\le y$9 dimensions, proved in this setting by coupling and variance identities rather than solely by asymptotic Fredholm determinant analysis (Seppäläinen, 2017).

In exactly solvable inhomogeneous exponential and geometric models, fluctuations remain KPZ inside the strictly concave cone. For slopes $G_{x,y}$0, the rescaled last-passage time converges to the Tracy-Widom GUE law, and the upper tail above the shape satisfies

$G_{x,y}$1

Thus disorder in row and column parameters can alter the macroscopic geometry by creating linear regions, while preserving Tracy-Widom fluctuations in the curved regime (Emrah, 2016).

Coalescence times of semi-infinite geodesics exhibit the KPZ-predicted $G_{x,y}$2 exponent. In the exponential model, when two $G_{x,y}$3-directed geodesics start $G_{x,y}$4 apart, the natural coalescence distance is of order $G_{x,y}$5. This is quantified by four tail estimates. For small $G_{x,y}$6, the probability of slow coalescence outside a box is bounded by

$G_{x,y}$7

and for large $G_{x,y}$8, the fast-coalescence probability inside the box satisfies

$G_{x,y}$9

These estimates are derived from stationary LPP, Busemann functions, exit-point control, and Pimentel’s duality (Seppäläinen et al., 2019).

The resulting picture is that the same directional Busemann field governs both geodesic geometry and fluctuation theory. This suggests a unified structure: shape curvature determines the macroscopic direction, stationary increments produce Busemann limits, and the KPZ exponents dictate the scales on which geodesics separate and merge.

6. Inhomogeneous, combinatorial, and higher-dimensional extensions

A major extension replaces i.i.d. weights by row- and column-dependent laws. In one exactly solvable family, the exponential model uses rates $x$0 and the geometric model uses parameters $x$1, with $x$2 and $x$3 drawn from ergodic distributions. The shape functions are then

$x$4

in the exponential case, and an analogous formula in the geometric case. These models retain enough solvability for full limit-shape theory and Tracy-Widom asymptotics in the curved cone, while allowing linear sectors absent from the homogeneous exponential benchmark (Emrah, 2016).

A related inhomogeneous exponential theory treats additively separable finite-array rates $x$5. Its first-order asymptotics are given by the explicit centering

$x$6

and under vague convergence of the parameter arrays the rescaled cluster converges to a deterministic limit shape. The boundary of that shape can contain flat segments adjacent to the axes, while the cluster can develop visible axis spikes and persistent crevices. A vertical spike occurs iff $x$7, a horizontal spike iff $x$8, and flat segments adjacent to the axes are characterized by the finiteness of

$x$9

The same variational structure yields macroscopic flux and particle-profile formulas for step-initial TASEP with particle and hole disorder (Emrah et al., 2019).

The geometric corner growth model also has a precise combinatorial incarnation in boxed plane partitions and lozenge tilings. For i.i.d. geometric weights

$y$0

the joint law of a column of last-passage times is identified with a $y$1-measure on partitions: $y$2 Under the plane-partition and lozenge-tiling bijections, corner counts have generating functions given by normalized Schur polynomials, and asymptotic consequences include limit shapes, Tracy-Widom GUE fluctuations, and GUE-minors limits in fixed-width regimes (Yeliussizov, 2019).

The term “corner growth model” is also used in higher-dimensional settings. In a three-dimensional corner growth model formulated as an SOS process inside a corner, a projection onto the $y$3 plane yields an RSOS model on a triangular lattice with periodic boundary conditions, enabling a high-precision velocity estimate

$y$4

in disagreement with the conjectured $y$5. This higher-dimensional result is not a planar corner growth theorem, but it highlights how dimension changes both the natural stationary representation and the asymptotic-shape problem (Singh et al., 2012).