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By Timo Seppäläinen

Summary. This overview article discusses restrict distributions and variance
bounds for particle present in different dynamical stochastic platforms of particles
on the one-dimensional integer lattice: self sustaining debris, independent
particles in a random atmosphere, the random standard process,
the uneven easy exclusion approach, and a category of absolutely asymmetric
zero diversity methods. the 1st 3 versions own linear macroscopic
flux capabilities and lie within the Edwards-Wilkinson universality type with
scaling exponent 1/4 for present fluctuations. For those we end up Gaussian
limits for the present approach. The latter platforms belong to the
Kardar-Parisi-Zhang category. For those we turn out the scaling exponent 0.33 in
the kind of higher and decrease variance bounds.

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Let z(0) = 0, z(t) = z, and introduce the counting variables N I+ (t) = ωnN (t) , n>z(t) N I− (t) = ωnN (t). n≤z(t) Then the current can be expressed as N N N N JzN (t) = I+ (t) − I+ (0) = I− (0) − I− (t), and its variance as N N N N Var JzN (t) = Cov I+ (t) − I+ (0), I− (0) − I− (t) N N N N = Cov I+ (t), I− (0)) + Cov(I+ (0), I− (t)) N N N N − Cov(I+ (0), I− (0)) − Cov(I+ (t), I− (t)). 18) 47 Chapter 5. 19) N Cov[ωm (t), ωkN (0)] = k≤0, m>z + k≤z, m>0 N N N Cov[ωkN (t), ωm (0)] − Cov(I+ (t), I− (t)).

D. with density ρ. 48) below). Label the ζ −η second class particles as {Ym (t) : m ∈ Z} so that initially · · · < Y−1 (0) < Y0 (0) = −n = Q(−n) (0) < 0 < Y1 (0) < Y2 (0) < · · · Let again mQ (t) be the label such that Q(−n) (t) = YmQ (t) (t). Initially mQ (0) = 0. The inclusion Q(−n) (t) ∈ {Ym (t)} persists for all time because 62 Timo Sepp¨al¨ainen the basic coupling preserves the ordering ζ(t) ≥ η + (t). Through the basic coupling mQ jumps to the left with rate q and to the right with rate p, but only when there is a Y -particle adjacent to YmQ .

The gterms above are martingale increments under the distribution P of the environments, relative to the filtration defined by levels of environments: writing ω ¯ m,n = {ωx,s : x ∈ Z, m ≤ s ≤ n}, and with fixed (x, m) and time n = 0, 1, 2 . . , E E ω g(TX x,m ω) ω ¯ m−n+1,m n x,m = y∈Z P ω {X n = (y, m − n)} g(Ty,m−n ω) P(d¯ ωm−n ) = 0. x,m The point above is that the probability P ω {X n = (y, m − n)} is determined by ω ¯ m−n+1,m . It turns out that we can apply a martingale central limit theorem to conclude that, for a fixed t, a vector ¯ n (t, r1 ), H ¯ n (t, r2 ), .

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