Geodesic
Absence of phase transitions in one-dimensional long-range spin systems with random Hamiltonian
Teoretičeskaâ i matematičeskaâ fizika, Tome 43 (1980) no. 2, pp. 253-260 
K. M. Khanin. Absence of phase transitions in one-dimensional long-range spin systems with random Hamiltonian. Teoretičeskaâ i matematičeskaâ fizika, Tome 43 (1980) no. 2, pp. 253-260. http://geodesic.mathdoc.fr/item/TMF_1980_43_2_a7/
@article{TMF_1980_43_2_a7,
     author = {K. M. Khanin},
     title = {Absence of~phase transitions in~one-dimensional long-range spin systems with random {Hamiltonian}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {253--260},
     year = {1980},
     volume = {43},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1980_43_2_a7/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Classical lattice one-dimensional systems with random Hamiltonians $H=\dfrac\beta2\sum\limits_{x_1\ne x_2}\dfrac{\varepsilon(x_1,x_2)\varphi(x_1)\varphi(x_2)}{|x_1-x_2|^\alpha}$ are considered, where $\varepsilon(x_1,x_2)$ are independent random variables for different pairs $(x_1,x_2)$, $E\varepsilon(x_1,x_2)=0$. It is shown that with probability 1 such a system has no phase transition, provided $\alpha>3/2$.

[1] S. F. Edwards, P. W. Anderson, J. Phys. F., 5 (1975), 965 | DOI | MR

[2] D. Sherrington, S. Kirkpatrick, Phys. Rev. Lett., 35 (1975), 1792 | DOI

[3] D. Ruelle, Commun. Math. Phys., 9 (1968), 267 | DOI | MR | Zbl

[4] R. L. Dobrushin, Teoriya veroyatn. i ee primen., 13 (1968), 201

[5] R. L. Dobrushin, Matem. sb., 93 (1974), 29 | MR | Zbl

[6] R. L. Dobrushin, Funkts. analiz, 2 (1968), 31 | Zbl

[7] F. J. Dyson, Commun. Math. Phys., 12 (1969), 91 | DOI | MR | Zbl

[8] K. M. Khanin, Ya. G. Sinai, J. Stat. Phys., 20 (1979), 573 | DOI | MR