Gumbel statistics for the longest interval of identical spins in a one-dimensional Gibbs measure |
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Authors: | F Redig |
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Institution: | (1) FaculteitWiskunde en Informatica, Technische Universiteit Eindhoven, Postbus 513, 5600 MB Eindhoven, THE NETHERLANDS, NL |
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Abstract: | We consider one-dimensional Gibbs measures on spin configurations σ ∈ {–1,+1}ℤ. For N ∈ ℕ let l
N
denote the length of the longest interval of consecutive spins of the same kind in the interval 0,N]. We show that the distribution of a suitable continuous modification l
c
(N) of l
N
converges to the Gumbel distribution, i.e., for some α, β ∈ (0, ∞) and γ ∈ ℝ,
lim
N
→∞ ℙ(l
c
(N) ≤ α log N + βx + γ) = e
–e
–x
.
Received: 2 September 2002 |
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Keywords: | : Gibbs measures extreme values Gumbel distribution |
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