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  • 1
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract It is shown that aD-component Euclidean quantum field, ϕ=(ϕ1,...,ϕD), with λ|ϕ|4+β|ϕ2| interaction, can be obtained as a limit of (ferromagnetic) classical rotator models; this extends a result of Simon and Griffiths from the caseD=1. For these Euclidean field models, it is then shown that a Lee-Yang theorem applies forD=2 or 3 and that Griffiths' second inequality is valid forD=2; a complete proof is included of a Lee-Yang theorem for plane rotator and classical Heisenberg models. As an application of Griffiths' second inequality forD=2, an interesting relation between the “parallel” and “transverse” two-point correlations is obtained.
    Type of Medium: Electronic Resource
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  • 2
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract A series of inequalities for partition, correlation, and Ursell functions are derived as consequences of the Lee-Yang Theorem. In particular, then-point Schwinger functions ofeven φ4 models are bounded in terms of the 2-point function as strongly as is the case for Gaussian fields; this strengthens recent results of Glimm and Jaffe and shows that renormalizability of the 2-point function by fourth degree counter-terms implies existence of a φ4 field theory with a moment generating function which is entire of exponential order at most two. It is also noted that ifany (even) truncated Schwinger function vanishes identically, the resulting field theory is a generalized free field.
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Communications in mathematical physics 66 (1979), S. 181-196 
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract A refined and extended version of the Buckingham-Gunton inequality relating various pairs of critical exponents is shown to be valid for a large class of statistical mechanical models. If this inequality is an equality (in the refined sense) and one of the critical exponents has a non-Gaussian value, then any scaling limit must be non-Gaussian. This result clarifies the relationship between the nontriviality or triviality of the scaling limit for ordinary critical points in four dimensions (or tricritical points in three dimensions) and the existence of logarithmic factors in the asymptotics which define the two critical exponents.
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  • 4
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Simple exact expressions are derived for all the Lyapunov exponents of certainN-dimensional stochastic linear dynamical systems. In the case of the product of independent random matrices, each of which has independent Gaussian entries with mean zero and variance 1/N, the exponents have an exponential distribution asN→∞. In the case of the time-ordered product integral of exp[N −1/2 dW], where the entries of theN×N matrixW(t) are independent standard Wiener processes, the exponents are equally spaced for fixedN and thus have a uniform distribution as N→∞.
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract A central limit theorem is given which is applicable to (not necessarily monotonic) functions of random variables satisfying the FKG inequalities. One consequence is convergence of the block spin scaling limit for the magnetization and energy densities (jointly) to the infinite temperature fixed point of independent Gaussian blocks for a large class of Ising ferromagnets whenever the susceptibility is finite. Another consequence is a central limit theorem for the density of thesurface of the infinite cluster in percolation models.
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  • 6
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We prove the GHS inequality for families of random variables which arise in certain ferromagnetic models of statistical mechanics and quantum field theory. These include spin −1/2 Ising models, ϕ4 field theories, and other continuous spin models. The proofs are based on the properties of a classG − of probability measures which contains all measures of the form const exp(−V(x))dx, whereV is even and continuously differentiable anddV/dx is convex on [0, ∞). A new proof of the GKS inequalities using similar ideas is also given.
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    Communications in mathematical physics 26 (1972), S. 169-204 
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The paper considers the possibility of constructing ultralocal theories, whose Hamiltonians contain no gradient terms and are therefore diagonal in position space, entirely in terms of currents with an equal time current algebra replacing the canonical commutation relations. It is shown that the free current theory can be defined in terms of a certain representation of the current algebra related to the group,S L(2,R). This representation is then constructed by using certain results of Araki and in the process a new infinitely divisible state on the universal covering group ofSL(2,R) is displayed. An ultralocal free theory can also be constructed for the canonical fields, and its relation to the free current theory is shown to involve a certain renormalization procedure reminiscent of the thermodynamic limit.
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    Constructive approximation 7 (1991), S. 389-399 
    ISSN: 1432-0940
    Keywords: Primary 11M26 ; Secondary 60K35 ; 82A25 ; Riemann Hypothesis ; GHS inequality ; Ising model ; Lee-Yang theorem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract LetV(t) be the even function on (−∞, ∞) which is related to the Riemann xi-function by Ξ(x/2)=4∫ −∞ ∞ exp(ixt−V(t))dt. In a proof of certain moment inequalities which are necessary for the validity of the Riemann Hypothesis, it was previously shown thatV'(t)/t is increasing on (0, ∞). We prove a stronger property which is related to the GHS inequality of statistical mechanics, namely thatV' is convex on [0, ∞). The possible relevance of the convexity ofV' to the Riemann Hypothesis is discussed.
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  • 9
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract For products,A(t)·A(t−1)...A(1), of i.i.d.N×N random matrices, with i.i.d. entries, a triangle law governs theN→∞ distribution of Lyapunov exponents, much like Wigner's quarter-circle law governs the singular values ofA(1). Our proof requires finite fourth moments and a bounded density; the result was previously derived only in the Gaussian case.
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  • 10
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We study the asymptotic behavior of partial sums S nfor certain triangular arrays of dependent, identically distributed random variables which arise naturally in statistical mechanics. A typical result is that under appropriate assumptions there exist a real number m, a positive real number λ, and a positive integer k so that (S n−nm)/n1−1/2k converges weakly to a random variable with density proportional to exp(−λ¦s¦ 2k/(2k)!). We explain the relation of these results to topics in Gaussian quadrature, to the theory of moment spaces, and to critical phenomena in physical systems.
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