Critical population and error threshold on the sharp peak by Raphael Cerf

By Raphael Cerf

The target of this paintings is to suggest a finite inhabitants counterpart to Eigen's version, which includes stochastic results. the writer considers a Moran version describing the evolution of a inhabitants of measurement m of chromosomes of size over an alphabet of cardinality ?. The mutation chance in step with locus is q. He bargains simply with the pointy height panorama: the replication cost is s>1 for the grasp series and 1 for the opposite sequences. He stories the equilibrium distribution of the method within the regime the place ? 8,m? 8,q?0, q?a?]0, 8[,m?a?[0, eight]

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We suppose that (Ztθ )t≥0 starts from Z0θ = 1. Let τ0 be the hitting time of 0, defined by τ0 = inf n ≥ 0 : Znθ = 0 . The process (Otθ )t≥0 always enters W ∗ at oθenter and it always exits W ∗ \ T θ at oθexit . In particular τ0 coincides with the exit time of W ∗ \ T θ after τ ∗ . From the previous elements, we see that Otθ (0), τ ∗ ≤ t ≤ τ has the same law as Ztθ , 0 ≤ t ≤ τ0 , whence τ τ0 Osθ (0) Zsθ ds O0θ = oθexit = E ds Z0θ = 1 . E f f m m 0 τ∗ Moreover, using the Markov property, we have E τ − τ ∗ O0θ = oθexit = E τ O0θ = oθenter = E τ0 Z0θ = 1 .

0 . Obviously, ∀o ∈ P m+1 π (o) o π1 (o) . , N = o ∈ P m+1 : o(0) = 0 . 1. 5 yields that the map ΦO is below the map ΦO and the map Φ1O is above the map ΦO in the following sense: ∀r ∈ R ∀o ∈ P m+1 ΦO (o, r) ΦO (o, r) Φ1O (o, r) . d. sequence (Rn )n≥1 and the maps ΦO , Φ1O as follows. Let o ∈ P m+1 be the starting point of the process. We set O (0) = O 1 (0) = o and ∀n ≥ 1 1 On1 = Φ1O On−1 , Rn . 1. Suppose that the processes (Ot )t≥0 , (Ot )t≥0 , (Ot1 )t≥0 , start from the same occupancy distribution o.

M } and c ∈ { 0, . . , } such that e = d(j ← c) and d(j) = c. Naturally, d(j ← c) is the vector d in which the j–th component d(j) has been replaced by c: ⎞ d(1) ⎟ ⎜ .. ⎟ ⎜ . ⎟ ⎜ ⎜d(j − 1)⎟ ⎟ ⎜ ⎟ c d(j ← c) = ⎜ ⎟ ⎜ ⎜d(j + 1)⎟ ⎟ ⎜ ⎟ ⎜ ... ⎠ ⎝ ⎛ d(m) We have then p x, x(j ← w) . 1, we have p x, x(j ← w) = w∈A H(w)=c w∈A H(w)=c = 1 m 1 m 1≤i≤m A(x(i))M (x(i), w) A(x(1)) + · · · + A(x(m)) AH H(x(i)) MH (H(x(i)), c) . A H(x(1)) + · · · + AH H(x(m)) 1≤i≤m H ¨ CERF RAPHAEL 28 This sum is a function of H(x) and c only.

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