--- title: Stationary Last Passage Percolation url: https://www.emergentmind.com/topics/stationary-last-passage-percolation-lpp type: topic --- # Stationary Last Passage Percolation Searching arXiv for relevant papers on stationary last passage percolation and closely related variants. Stationary last passage percolation (LPP) is the directed LPP regime in which the nearest-neighbor increment field, rather than the raw passage-time field itself, is spatially stationary. In the planar corner growth model this stationarity can be realized by boundary conditions, by stationary cocycles, or by Busemann functions, and in the exponential case it is the LPP representation of stationary TASEP at density \(\rho\) [2004.12987]. Across full-space, half-space, strip, and cylinder geometries, stationary LPP furnishes the canonical KPZ stationary regime: passage times have \(N^{1/3}\) fluctuations, geodesics have \(N^{2/3}\) transversal scale, and the corresponding finite-\(N\) tail behavior, variational structure, and scaling limits are encoded by cocycles, Airy objects, and, in integrable cases, Fredholm Pfaffian or contour-integral formulas [1404.7786][1812.00309]. ## 1. Directed LPP and the meaning of stationarity In the corner growth model on \(\mathbb{Z}^2\) with admissible steps \(e_1=(1,0)\) and \(e_2=(0,1)\), the last-passage time from \(x\) to \(y\) is \[ G_{x,y}=\max_{\pi\in\Pi_{x\to y}}\sum_{z\in\pi}\omega_z, \] where \(\omega=\{\omega_x\}\) is an i.i.d. vertex-weight field and \(\Pi_{x\to y}\) denotes up-right paths from \(x\) to \(y\) [1510.00859]. Under the standing assumptions \(E[|\omega_0|^p]<\infty\) for some \(p>2\), \(\mathrm{Var}(\omega_0)>0\), and \(\omega_0\ge c\) a.s., the shape theorem gives a deterministic limit shape \(g\colon \mathbb{R}_+^2\to\mathbb{R}\) that is continuous, concave, homogeneous of degree \(1\), and symmetric in the coordinates [1404.7786]. Stationarity does not mean that \(G_{x,y}\) itself is translation-invariant. The stationary object is the increment field. In the cocycle formulation, a stationary \(L^1\)-cocycle \(B\) is a measurable function on pairs of lattice points satisfying stationarity under translations, the cocycle property \[ B(x,y)+B(y,z)=B(x,z), \] and integrability on coordinate edges [1404.7786]. If \(B\) recovers the potential, \[ \omega_0=\min\{B(0,e_1),B(0,e_2)\}, \] then it serves as a stationary boundary condition for an LPP process in a quadrant, and the increments \[ \mathcal G_{x,v}-\mathcal G_{x+e_i,v}=B(x,x+e_i) \] are stationary in \(x\) and independent of the target \(v\) [1510.00859]. This distinction is central. A common misconception is to identify stationary LPP with a globally shift-invariant last-passage field. The established framework is instead shift-invariance of edge increments, equivalently of a cocycle or Busemann field. This is the notion that persists in non-integrable models, where the raw passage-time field retains macroscopic growth governed by the shape \(g\) [1404.7786]. ## 2. Boundary constructions, cocycles, and Busemann functions Two complementary constructions dominate the subject. In the exponential corner growth model, stationary LPP with density \(\rho\in(0,1)\) can be realized by boundary weights on the positive axes: \[ \omega'_{i e_1}\sim \mathrm{Exp}(1-\rho),\qquad \omega'_{j e_2}\sim \mathrm{Exp}(\rho), \] with i.i.d. \(\mathrm{Exp}(1)\) bulk weights and all variables independent [2004.12987]. The associated stationary passage time \(G^1_{\mathrm{stat}}(v)\) is the maximum over up-right paths from \(0\) to \(v\). The same model also has a point-to-line representation: one puts i.i.d. \(\mathrm{Exp}(1)\) weights in the region \(\{x+y>0\}\), zero below the line \(\{x+y=0\}\), and introduces a random boundary field \(T\) on \(\{x+y=0\}\) built from independent \(\mathrm{Exp}(1-\rho)\) and \(\mathrm{Exp}(\rho)\) variables [2004.12987]. The two representations are equivalent in law, and a coupling via the Burke property gives equality of the passage-time fields in the positive quadrant. In the general i.i.d. corner growth model, Georgiou, Rassoul-Agha, and Seppäläinen constructed stationary cocycles from queueing fixed points [1404.7786]. On an extended space they obtained a family \(B^\xi_\pm\) indexed by directions \(\xi\in\mathcal U=\{te_1+(1-t)e_2:t\in[0,1]\}\), with the properties that each \(B^\xi_\pm\) is a stationary \(L^1\)-cocycle recovering the potential, the mean vectors satisfy \[ h_\pm(\xi)=h(B^\xi_\pm)=-\nabla g(\xi\pm), \] and the cocycles are monotone and one-sided continuous in direction [1404.7786]. Uniqueness across tilt holds along linear segments of \(g\), and at differentiability points one has \(B^\xi_+=B^\xi_-\). Busemann functions are the asymptotic incarnation of these cocycles. For sequences \(v_n\to\infty\) in direction \(\xi\), the directional Busemann function is \[ B^\xi(x,y)=\lim_{n\to\infty}\big(G_{x,v_n}-G_{y,v_n}\big), \] when the limit exists [1510.00859]. The queueing construction proves existence of such limits along sectors where the shape has the requisite endpoint differentiability, identifies them with the cocycles \(B^\xi_\pm\), and yields \[ \nabla g(\xi)=\big(E[B^\xi(0,e_1)],E[B^\xi(0,e_2)]\big) \] at differentiability points [1510.00859]. The same stationary objects control geodesics. In the exponential model, the minimum-gradient rule for the Busemann field generates infinite geodesics in the characteristic direction \(\xi(\alpha)\), and local stationarity results show that finite geodesic trees near a macroscopic endpoint coincide with the corresponding stationary infinite-geodesic tree with high probability [2001.03961]. This suggests a precise interpretation of stationarity as the local asymptotic geometry seen from a characteristic observer. ## 3. Characteristic directions, shape, and KPZ scaling For exponential LPP the shape function is explicit: \[ g(x,y)=(\sqrt{x}+\sqrt{y})^2. \] In the stationary model with density \(\rho\), the characteristic direction is \[ v_N=((1-\rho)^2N,\rho^2N), \] and \[ g(v_N)=N \] by direct substitution [2004.12987]. This is the natural centering for stationary passage times. The same density-direction relation appears in Busemann form as \[ \xi(\alpha)=\Big(\frac{(1-\alpha)^2}{(1-\alpha)^2+\alpha^2},\frac{\alpha^2}{(1-\alpha)^2+\alpha^2}\Big), \] which parametrizes the characteristic direction attached to the cocycle \(B^\alpha\) in exponential LPP [2001.03961]. The KPZ stationary scaling exponents are explicit in this regime. Passage-time fluctuations are of order \(N^{1/3}\), while transversal fluctuations of geodesics and the stationary exit time are of order \(N^{2/3}\) [2004.12987]. The corresponding moderate-deviation tails match the Tracy–Widom exponents: \[ \text{upper tail}\sim \exp(-c s^{3/2}),\qquad \text{lower tail}\sim \exp(-c s^3). \] The same asymptotic exponents are consistent with the Baik–Rains distribution governing the stationary one-point limit, and the variance along the characteristic direction is of order \(N^{2/3}\) [2004.12987]. A geometric explanation for the \(N^{2/3}\) exit scale is encoded in the axis expansion used by Bhatia. If \(x\) denotes the exit coordinate from the boundary, then the expected total weight of a path that exits at \(x\) is lower than \(N\) by a term of order \(x^2/N\), via the expansion \[ g_{\mathrm{axis}}(x)+g(v_N-(x,1)) = N-\frac{h(x)}{N}-N\,O\big((x/N)^3\big), \] with \(g_{\mathrm{axis}}\) and \(h\) given explicitly and the \(O(\cdot)\) term positive [2004.12987]. Consequently, exit locations of order \(N^{2/3}\) carry an order-\(N^{1/3}\) penalty, which is exactly the moderate-deviation scale for stationary passage times. The probabilistic local-stationarity theorem sharpens this picture. In a box of side \(M=cN^{2/3}\) adjacent to a macroscopic target \(N\xi\), the vector of nearest-neighbor increments of point-to-point LPP is within total variation \(Cc^{3/8}\) of the stationary increment field with matching density [2001.03961]. In this sense, stationary LPP is not merely an invariant model; it is the local tangent model for curved point-to-point exponential LPP at characteristic scale. ## 4. Moderate deviations, exit times, and optimal exponents Bhatia established a set of sharp finite-\(N\) results for planar stationary exponential LPP along the characteristic direction [2004.12987]. Let \(Z^{0\to v_N}\) be the signed exit time of the stationary geodesic from the boundary axes. Then for every \(\rho\in(0,1)\) there exist \(N_0(\rho),C(\rho),c(\rho)>0\) such that \[ P^\rho\big(|Z^{0\to v_N}|\ge rN^{2/3}\big)\le C e^{-cr^3} \] for all \(r>0\) and \(N\ge N_0\). The exponent \(3\) is the optimal KPZ right-tail exponent for exit times [2004.12987]. For the stationary passage time itself, the paper proves two-sided upper-tail moderate deviations with the optimal \(3/2\) exponent. For \(y\in(0,\delta_1 N^{2/3})\), \[ P^\rho\big(G^1_{\mathrm{stat}}(v_N)-N\ge yN^{1/3}\big)\le C e^{-c y^{3/2}}, \] and also \[ P^\rho\big(G^1_{\mathrm{stat}}(v_N)-N\ge yN^{1/3}\big)\ge C e^{-c y^{3/2}} \] with constants depending on \(\rho\) and \(\delta_1\) [2004.12987]. For the lower tail, the uniform upper bound \[ P^\rho\big(G^1_{\mathrm{stat}}(v_N)-N\le -yN^{1/3}\big)\le C e^{-c y^3} \] holds for all \(\rho\in(0,1)\), while the matching lower bound \[ P^{1/2}\big(G^1_{\mathrm{stat}}(v_N)-N\le -yN^{1/3}\big)\ge C e^{-c y^3} \] is proved at \(\rho=\tfrac12\) [2004.12987]. The proof strategy is comparison-based rather than formula-based. Point-to-point exponential LPP is tied to Laguerre/Wishart random matrices, which supply the needed moderate deviation bounds for \(G(0,(m,n))\) with the same \(3/2\) and \(3\) exponents, and point-to-line estimates over \(N^{2/3}\)-scale segments provide the corresponding flat-geometry input [2004.12987]. Stationary LPP itself lacks a direct random-matrix representation, so the stationary theorems are obtained by combining those integrable inputs with shape concavity, independence, and the geometry of boundary exit. The paper explicitly positions this as complementary to parallel exact MGF methods. One limitation is intrinsic to the current method. For \(\rho=\tfrac12\), the point-to-line boundary field \(T\) is a centered random walk, so a Brownian invariance-principle argument yields the lower-tail lower bound with exponent \(3\). For general \(\rho\neq \tfrac12\), \(T\) has nonzero drift, and the matching stationary lower-tail lower bound remains open in this approach [2004.12987]. ## 5. Other stationary geometries: half-space, strip, and cylinder Stationary LPP has several non-equivalent geometric realizations beyond the quarter-plane. In half-space exponential LPP, the domain is \[ \mathcal D=\{(i,j)\in\mathbb Z^2:1\le j\le i\}, \] with diagonal weights \(\mathrm{Exp}(\tfrac12+\alpha)\), first-row weights \(\mathrm{Exp}(\tfrac12-\alpha)\), bulk weights \(\mathrm{Exp}(1)\), and \(\alpha\in(-\tfrac12,\tfrac12)\) [1905.08582]. The increments along down-right paths are stationary, and the critical window near the origin yields a new two-parameter family of one-point distributions \(F^{(\delta,u)}_{0,\mathrm{half}}\), depending on the scaled diagonal strength and the scaled distance from the characteristic line [1905.08582]. Far from the characteristic, this family converges to the Baik–Rains stationary family, so the half-space stationary theory is not a trivial restriction of the full-space case. Half-space geometric LPP admits an even richer two-parameter stationary measure. For \(\sqrt q\in(0,1)\), \(s\in(\sqrt q,1)\), and \(r\in(0,1/s)\), the stationary law is encoded by a process \[ I_{r,s}(x)=\max\Big\{R_2(x),\max_{1\le k\le x}\big(R_1(k)+R_2(x-k+1)-Y\big)\Big\}, \] where \(R_1\) and \(R_2\) are geometric random walks and \(Y\sim \mathrm{Geom}(rs)\) [2507.11887]. The paper derives explicit Pfaffian formulas for the diagonal distribution and a critical \(N^{1/3}\)-scale asymptotic law depending on the pair \((\tilde r,\tilde s)\). In the product stationary specializations \(r=s\) and \(rs=1\), the model reduces to the previously known one-parameter product stationary theory [2507.11887]. On a diagonal strip of width \(N\), Barraquand, Corwin, and Yang identified the stationary measure for geometric LPP as the marginal of a two-layer Gibbs measure with explicit reweighting by the maximal gap between two geometric random walks [2306.05983]. In the homogeneous case, this stationary measure is unique and ergodic for the increment process along the horizontal path [2306.05983]. The subsequent free Askey–Wilson analysis extends explicit generating-function formulas to a broader parameter range and yields a full phase diagram for the stationary slope: interior, low-density, high-density, and a critical line \(c_\ell=c_r\in(1,1/a)\) on which the asymptotic slope is a uniform mixture of the low- and high-density deterministic values [2506.01879]. Cylinder geometries generate a different notion of stationarity. For generalized LPP on the cylinder \(C_L\), the augmented front line \(\tilde F_n\) is a Markov chain, and under the condition \[ \inf_{\Delta\in\mathbb N,\ t\in\mathbb N^*}\frac{\mu_\Delta(t)}{\sum_{s\ge t}\mu_\Delta(s)}\ge \alpha \] it is irreducible, aperiodic, and ergodic, with a unique invariant law \(\tilde\nu_L\) [1912.00645]. In integrable GLPP this invariant law is explicit. In the classical cylinder special cases, geometric weights give \(\nu_L(b)\propto (1-p)^{-k_b}\), where \(k_b\) is the number of local maxima of the bridge \(b\), while exponential weights give \(\nu_L\) uniform on the bridge space \(B_L\) [1912.00645]. This is stationarity of an interface process on a finite periodic geometry rather than of Busemann increments in the quarter-plane. ## 6. Scaling limits, continuum stationarity, and open directions The continuum KPZ analogue of stationary LPP is encoded by the directed landscape \(\mathcal L\), a random continuous function on \(\mathbb R^4_\uparrow\) characterized by three properties: Airy-sheet marginals on fixed-duration slices, independence of increments over disjoint time intervals, and the metric-composition rule \[ \mathcal L(x,r;y,t)=\max_{z\in\mathbb R}\big[\mathcal L(x,r;z,s)+\mathcal L(z,s;y,t)\big] \] for \(r