By G. R. Grimmett, D. J. A. Welsh

On 21 March 1990 John Hammersley celebrates his 70th birthday. a couple of his colleagues and acquaintances desire to pay tribute in this social gathering to a mathematician whose remarkable inventiveness has enormously enriched mathematical technological know-how. The breadth and flexibility of Hammersley's pursuits are impressive, doubly so in an age of elevated specialisation. In various hugely person papers on quite a few subject matters, he has theorised, and posed (and solved) difficulties, thereby laying the principles for lots of topics at present less than examine. by means of his obtrusive love for arithmetic and an affinity for the tough challenge, he has been an concept to many. If one needs to unmarried out one specific sector the place Hammersley's contribution has proved specially important, it will most likely be the research of random techniques in area. He used to be a pioneer during this box of regarded significance, a box abounding in it appears uncomplicated questions whose resolutions often require new rules and strategies. This zone is not only a mathematician's playground, yet is of basic value for the certainty of actual phenomena. The primary topic of this quantity displays quite a few features of Hammersley's paintings within the quarter, together with disordered media, subadditivity, numerical equipment, and so on. The authors of those papers subscribe to with these not able to give a contribution in wishing John Hammersley many additional years of fruitful mathematical task.

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**Example text**

Change the colouring on S to χS∗ if q is greater than 1. If q is less than 1, then with probability q change the colouring to χS∗ and with probability 1 − q leave the colouring unchanged. (e) Go to (a). The algorithm is a special case of the general class of algorithms discussed by Hastings (1970). If we let 1A (ω)e−F (ω) , P (A | ) = K(ξ, ) ω∈ΩS (ξ) then the probability distribution for the candidate colouring χS∗ is ∞ e−ρdS (ρdS )n n! n=0 Ln S P (A | ) νSn (d ), 31 Markov Random Fields in Statistics which has density S e−(ρ−λ)dS (ρ/λ)n e−F (χ∗ ) K(ξ, ), with respect to γS (· | ξ).

Condition M (X) is then verified by substitution. 8), condition M (X) is equivalent to R(χ) − R(χX ) = R(χZ−(X+∂X) ) − R(χZ−∂X ), ∀χ ∈ C where R(χ) = log P (χ). In other words, βX R(χ) = R(χ), ∀χ ∈ C. Condition M (X) is therefore equivalent to R ∈ I(βX ). 26 Clifford Theorem 1 then follows immediately since if P is locally Markovian then R ∈ ∩z∈Z I(βz ) = I(β) ⊆ I(βX ) by Lemma 4 and hence P is globally Markovian. Proof of Theorem 2: From Lemma 3, R ∈ I(β) iff R(χ) − R(χZ ) = S(χZ−X ), ∀χ ∈ C X∈Lχ for some S ∈ R.

1957). Percolation processes I. Crystals and mazes. Proceedings of the Cambridge Philosophical Society 53, 629– 641. C. (1971). Abram Samoilovitch Besicovitch. Biographical Memoirs of the Fellows of the Royal Society 17, 1–16. Domb, C. (1947). The problem of random intervals on a line. Proceedings of the Cambridge Philosophical Society 43, 329–341. (1948). Some probability distributions connected with recording apparatus. Proceedings of the Cambridge Philosophical Society 44, 335–341. (1950). Some probability distributions connected with recording apparatus II.