PROPUESTA DE PLANIFICACION Y RECOMENDACIONES

Coming back to the percolation model, we introduce boundary sources, i.e. the vertical bonds at the bordersn1 = 0andn2 = 0now have a probabilityα andβ,

respectively, to become active. The vertical bonds in the bulk still have probability

p. The bond at the origin is chosen to be inactive by convention. In the Johansson picture this corresponds to a geometric distribution of the first row and the first line of the integer valued random matrix with strengthαandβ, respectively, the entry in the corner being fixed as zero. By properly choosing α and β one can arrange the height functionh(n1, n2)to have stationary increments in the direction

of increasing coordinates. We are not aware of any publications which investigate this property. Although the stationary state has been determined for the various mappings to vertex and lattice gas models [106,110], the height lines ofh(n1, n2)

provide the simplest way to investigate stationarity, since one does not have to impose periodic boundary conditions in order that the steady state has a simple

4.4. STATIONARITY 71 product form.

The key observation is that it is enough to look at one single square of the shifted lattice(N₀−1/2)2and study the statistics of incoming and outgoing height lines, projected vertically. If there aremlines entering from the bottom andnlines from the left,m_∧n = min_{m, n_}of them annihilate. Letk denote the number of created line pairs in the square, then the number of outgoing lines at the top is

m0 =m−(m∧n)+k, andn0 =n−(m∧n)+kat the right side, see Fig.4.6. Now letk, m, and n be independent and geometrically distributed with parametersp,

α, andβ, respectively. The joint probability,Pm0_,n0, form0 andn0 outgoing lines

is Pm0,n0 = (1−p)(1−α)(1−β) X m,n≥0 pm0∧n0αmβnδm−m0,n−n0 = (1−p)(1₁₋−_αβα)(1−β)pm0∧n0_α(m0−n0)∨0_β(n0−m0)∨0 = 1−p 1₋αβ p αβ m0∧n0 (1₋α)(1₋β)αm0βn0, (4.29) wherem_∨n = max_{m, n_}. Thus if αβ = p the outgoing lines are again dis- tributed independently and geometrically with the same strength as the incoming lines,

Pm0_,n0 = (1−α)αm 0

(1₋β)βn0. (4.30)

For a given p we arrive at a one-parameter family α _∈]p,1[ of stationary solu- tions, which extend to product measures on space-like paths, i.e. chains of sites connected by bonds and being mutually non-time-like,(m1−n1)(m2−n2)≤0,

for each two sites(m1, m2)and(n1, n2)in the path. More precisely, we take two

space-like pathsγ1andγ2, having a time-like separation, i.e. for each site(n1, n2)

inγ2 there is at least one site (m1, m2)in γ1 withm1 ≤ n1 and m2 ≤ n2. The

(possibly infinite) region enclosed by the two paths is bounded by bonds with in- coming lines belonging toγ1, and bonds with outgoing lines belonging toγ2. If the

numbers of incoming lines for each bond are distributed independently geometri- cally with strengthα for horizontal bonds andp/α for vertical bonds, eq. (4.30) immediately tells us by induction that the numbers of outgoing lines for each bond are still independent with the same distribution for horizontal and vertical bonds as for the incoming lines.

For a space-like path approximating the straight line n2 = −1₁₊−b_bn1, b ∈

[₋1,1], one obtains a proof for Proposition 3.1 of Chapter 3 in the PNG limit. We chooseα =β =√p. The geometric distribution of incoming lines at vertical, resp. horizontal bonds becomes independent Poisson point processes along the straight line in the limitp_→0upon rescaling the lattice by√p. The line densities

β β α p/β p/α α (a) /α p p/α α α (Μ,Ν) (0,0) (b)

Figure 4.7: Two examples of propagating the product measure: (a) from a stair- case to the border of a quadrant (the parameters α, β _∈]p,1[can be chosen in- dependently), and (b) from the lower-left sides of a rectangle to the upper-right sides.

for entering horizontal lines can be easily calculated as(1 +b)/√1 +b2_{, likewise}

the density for vertical lines is(1₋b)/√1 +b2_.

Two special cases of interest are sketched in Fig.4.7. Viewed as the discrete PNG model, the anti-diagonal defines initial conditions in form of a fixed macro- scopic slope, being different forx >0andx <0. The macroscopic shapes arising from these initial conditions cover all self-similar macroscopic shapes, possible for this model.

The M _×N rectangle in (Fig. 4.7(b)) can be regarded as a piece of a truely stationary height process defined on the whole plane. This is the discrete analogue of the setting in Section3.3and one expects that one could recover the same scal- ing result. Explicit expressions for the distribution ofh(M, N)are available [11]. Unfortunately, the Riemann-Hilbert techniques used for the proof in the PNG case work only for the square case ofH(M, M)which allows to prove convergence to the scaling functiongdyn₁ (y)only aty= 0, so far. In the next section we illustrate how the stationarity property allows to determine the generator of asymptotics by application of the KPZ scaling theory.

4.5. THE GENERATOR OF ASYMPTOTICS 73