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  • 1
    Electronic Resource
    Electronic Resource
    On some limit theorems related to the phase separation line in the two-dimensional ising model (1979)
    Springer
    Probability theory and related fields 50 (1979), S. 287-315 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We prove that at low enough temperatures the phase separation line, when it is suitably normalized, converges almost surely in a suitable probability space to the path of a one-dimensional Brownian bridge. The convergence is in the sense of the distance between compact sets in [0, 1] ×R 1.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00534152
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  • 2
    Electronic Resource
    Electronic Resource
    A percolation problem for {±1}-valued strongly mixing random fields on ℤ d ⋆ (1986)
    Springer
    Probability theory and related fields 73 (1986), S. 597-611 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We present here an upper estimate for the probability that the origin is + connected to the boundary of the cube {|x i |≦k, 1≦i≦d{, under the condition that the expectation E(r(W 0 + )d-1) is finite, where W 0 + is the+cluster of the origin, and r(W 0 + ) is its radius; $$r(W_0^ + ) = \max \left\{ {\left| {x^i } \right|;1 \leqq i \leqq d,x \in W_0^ + } \right\}$$ The upper estimate is given in terms of the mixing coefficient. In particular, if the mixing coefficient decays exponentially, then our upper estimate supply an “almost exponential” decay of the probability in question; it decays faster than exp (-Ck/log k) as k→∞, for some positive constant C. As an example we discuss the two-dimensional Ising model except at the critical point. By using our result, we show the above almost exponential decay for parameters (β, h) satisfying $$h 〈 - 4\beta ^{ - 1} (\beta _c - \beta )v0$$
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00324855
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  • 3
    Electronic Resource
    Electronic Resource
    Coexistence of infinite (*)-clusters II. Ising percolation in two dimensions (1993)
    Springer
    Probability theory and related fields 97 (1993), S. 1-33 
    Details
    ISSN: 1432-2064
    Keywords: 60K35 ; 82B05 ; 82B20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We show a strong type of conditionally mixing property for the Gibbs states ofd-dimensional Ising model when the temperature is above the critical one. By using this property, we show that there is always coexistence of infinite (+ *)-and (−*)-clusters when β is smaller than βc andh=0 in two dimensions. It is also possible to show that this coexistence region extends to some non-zero external field case, i.e., for every β 〈 βc, there exists someh c(β)〉0 such that |h|〈h c(β) implies the coexistence of infinite (*)-clusters with respect to the Gibbs state for (β,h).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01199310
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  • 4
    Electronic Resource
    Electronic Resource
    On the convergence of the kadanoff transformation towards trivial fixed points (1981)
    Springer
    Probability theory and related fields 58 (1981), S. 109-123 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider the Kadanoff transformation T (depending on a positive parameter p) acting on probability measures Μ on the space {+1, −}ℤd. A measure Μ is called a non-trivial fixed point of T, if it is extremal in the set of T-invariant measures but is not a product measure. We describe the set of trivial fixed points and show that non-trivial fixed points exist provided that d≦2 and p large enough. A strong mixing condition on Μ implies convergence of T nΜ towards a trivial fixed point. In particular this applies to the two-dimensional Ising model except at the critical point. What happens at the critical point still remains unknown.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00536199
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  • 5
    Electronic Resource
    Electronic Resource
    Level set representation for the Gibbs states of the ferromagnetic ising model (1991)
    Springer
    Probability theory and related fields 90 (1991), S. 203-221 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We prove the existence of a real valued random field with parameters in thed-dimensional cubic lattice, such that the distribution of the level set of this random field is a Gibbs state for the nearest neighbour ferromagnetic Ising model. Using this, we prove the continuity of the percolation probability with respect to the parameter (β,h) in the uniqueness region except on the critical curve Γ c ={(β,h c (β))}, whereh c(β) is the critical level of the external field above which percolation takes place.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01192162
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  • 6
    Electronic Resource
    Electronic Resource
    A sharp transition for the two-dimensional Ising percolation (1993)
    Springer
    Probability theory and related fields 97 (1993), S. 489-514 
    Details
    ISSN: 1432-2064
    Keywords: 82B20 ; 60K35 ; 82B43
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We show that the percolation transition for the two-dimensional Ising model is sharp. Namely, we show that for every reciprocal temperature β〉0, there exists a critical valueh c (β) of external magnetic fieldh such that the following two statements hold. (i) Ifh〉h c (β), then the percolation probability (i.e., the probability that the origin is in the infinite cluster of + spins) with respect to the Gibbs state μβ,h for the parameter (β,h) is positive. (ii) Ifh〈h c (β), then the connectivity function τ β,h + (0,x) (the probability that the origin is connected by + spins tox with respect to μβ,h ) decays exponentially as |x|→∞. We also shows that the percolation probability is continuous in (β,h) except on the half line {(β, 0); β≧β c }.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01192961
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  • 7
    Electronic Resource
    Electronic Resource
    Coexistence of the infinite (*) clusters: — A remark on the square lattice site percolation (1982)
    Springer
    Probability theory and related fields 61 (1982), S. 75-81 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We show that the critical probability p c is strictly greater than 1/2 for the square lattice site percolation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00537226
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  • 8
    Electronic Resource
    Electronic Resource
    Ising models on the lattice Sierpinski gasket (1996)
    Springer
    Journal of statistical physics 84 (1996), S. 295-307 
    Details
    ISSN: 1572-9613
    Keywords: Ising model ; lattice Sierpinski gasket ; Dobrushin-Shlosmann mixing condition
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract Ferromagnetic Ising models on the lattice Sierpinski gasket are considered. We prove the Dobrushin-Shlosmann mixing condition and discuss corresponding properties of the stochastic Ising models.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02179588
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