# zbMATH — the first resource for mathematics

Critical two-point function for long-range $$O(n)$$ models below the upper critical dimension. (English) Zbl 1387.82012
Summary: We consider the $$n$$-component $$|\phi |^4$$ lattice spin model ($$n \geq 1$$) and the weakly self-avoiding walk ($$n=0$$) on $$\mathbb Z^d$$, in dimensions $$d=1,2,3$$. We study long-range models based on the fractional Laplacian, with spin-spin interactions or walk step probabilities decaying with distance $$r$$ as $$r^{-(d+\alpha )}$$ with $$\alpha \in (0,2)$$. The upper critical dimension is $$d_c=2\alpha$$. For $$\epsilon >0$$, and $$\alpha = \frac{1}{2} (d+\epsilon )$$, the dimension $$d=d_c-\epsilon$$ is below the upper critical dimension. For small $$\epsilon$$, weak coupling, and all integers $$n \geq 0$$, we prove that the two-point function at the critical point decays with distance as $$r^{-(d-\alpha )}$$. This “sticking” of the critical exponent at its mean-field value was first predicted in the physics literature in 1972. Our proof is based on a rigorous renormalisation group method. The treatment of observables differs from that used in recent work on the nearest-neighbour 4-dimensional case, via our use of a cluster expansion.

##### MSC:
 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics 82B28 Renormalization group methods in equilibrium statistical mechanics 82B27 Critical phenomena in equilibrium statistical mechanics
Full Text:
##### References:
 [1] Abdesselam, A, A complete renormalization group trajectory between two fixed points, Commun. Math. Phys., 276, 727-772, (2007) · Zbl 1194.81168 [2] Abdesselam, A., Chandra, A., Guadagni G.: Rigorous quantum field theory functional integrals over the $$p$$-adics I: anomalous dimensions. arXiv:1302.5971 (2013) [3] Aizenman, M; Fernández, R, On the critical behavior of the magnetization in high dimensional Ising models, J. Stat. Phys., 44, 393-454, (1986) · Zbl 0629.60106 [4] Bauerschmidt, R; Brydges, DC; Slade, G, Scaling limits and critical behaviour of the $$4$$-dimensional $$n$$-component $$|φ |^4$$ spin model, J. Stat. Phys, 157, 692-742, (2014) · Zbl 1308.82026 [5] Bauerschmidt, R; Brydges, DC; Slade, G, Critical two-point function of the 4-dimensional weakly self-avoiding walk, Commun. Math. Phys., 338, 169-193, (2015) · Zbl 1320.82031 [6] Bauerschmidt, R; Brydges, DC; Slade, G, Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis, Commun. Math. Phys., 337, 817-877, (2015) · Zbl 1318.60049 [7] Bauerschmidt, R; Brydges, DC; Slade, G, A renormalisation group method. III. perturbative analysis, J. Stat. Phys, 159, 492-529, (2015) · Zbl 1319.82008 [8] Behan, C; Rastelli, L; Rychkov, S; Zan, B, A scaling theory for long-range to short-range crossover and an infrared duality, J. Phys. A, 50, 354002, (2017) · Zbl 1376.82012 [9] Bendikov, A; Cygan, W, $$α$$-stable random walk has massive thorns, Colloq. Math., 138, 105-129, (2015) · Zbl 1329.60114 [10] Bendikov, A., Cygan, W., Trojan B.: Limit theorems for random walks. Stoch. Proc. Appl. 127, 3268-3290 (2017) · Zbl 1395.60050 [11] Brezin, E; Parisi, G; Ricci-Tersenghi, F, The crossover region between long-range and short-range interactions for the critical exponents, J. Stat. Phys., 157, 855-868, (2014) · Zbl 1318.82017 [12] Brydges, DC; Mitter, PK; Scoppola, B, Critical $$({\varPhi }^4)_{3,ϵ }$$, Commun. Math. Phys., 240, 281-327, (2003) · Zbl 1053.81065 [13] Brydges, DC; Slade, G, A renormalisation group method. I. Gaussian integration and normed algebras, J. Stat. Phys., 159, 421-460, (2015) · Zbl 1317.82013 [14] Brydges, DC; Slade, G, A renormalisation group method. II. approximation by local polynomials, J. Stat. Phys., 159, 461-491, (2015) · Zbl 1317.82014 [15] Brydges, DC; Slade, G, A renormalisation group method. IV. stability analysis, J. Stat. Phys., 159, 530-588, (2015) · Zbl 1317.82015 [16] Brydges, DC; Slade, G, A renormalisation group method. V. A single renormalisation group step, J. Stat. Phys., 159, 589-667, (2015) · Zbl 1317.82016 [17] Chen, L-C; Sakai, A, Critical two-point functions for long-range statistical-mechanical models in high dimensions, Ann. Probab., 43, 639-681, (2015) · Zbl 1342.60162 [18] El-Showk, S; Paulos, MF; Poland, D; Rychkov, S; Simmons-Duffin, D; Vichi, A, Solving the 3d Ising model with the conformal bootstrap II. $$c$$-minimization and precise critical exponents, J. Stat. Phys., 157, 869-914, (2014) · Zbl 1310.82013 [19] Fernández, R; Procacci, A, Cluster expansion for abstract polymer models. new bounds from an old approach, Commun. Math. Phys., 274, 123-140, (2007) · Zbl 1206.82148 [20] Fisher, ME; Ma, S; Nickel, BG, Critical exponents for long-range interactions, Phys. Rev. Lett., 29, 917-920, (1972) [21] Friedli, S., Velenik, Y.: Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction. Cambridge University Press, Cambridge (2017) · Zbl 1407.82001 [22] Heydenreich, M, Long-range self-avoiding walk converges to alpha-stable processes, Ann. I. Henri Poincaré Probab. Stat., 47, 20-42, (2011) · Zbl 1210.82055 [23] Heydenreich, M; Hofstad, R; Sakai, A, Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk, J. Stat. Phys., 132, 1001-1049, (2008) · Zbl 1152.82007 [24] Mitter P.: Long range ferromagnets: renormalization group analysis. http://hal.archives-ouvertes.fr/el-01239463 (2013) · Zbl 1318.82017 [25] Mitter, P.K.: On a finite range decomposition of the resolvent of a fractional power of the Laplacian. J. Stat. Phys. 163, 1235-1246 (2016). Erratum. J. Stat. Phys. 166, 453-455 (2017) · Zbl 1342.82040 [26] Mitter, PK, On a finite range decomposition of the resolvent of a fractional power of the Laplacian II. the torus, J. Stat. Phys., 168, 986-999, (2017) · Zbl 1374.82007 [27] Mitter, PK; Scoppola, B, The global renormalization group trajectory in a critical supersymmetric field theory on the lattice $${{\mathbb{Z}}}^3$$, J. Stat. Phys., 133, 921-1011, (2008) · Zbl 1161.82310 [28] Paulos, MF; Rychkov, S; Rees, BC; Zan, B, Conformal invariance in the long-range Ising model, Nucl. Phys. B, 902, 246-291, (2016) · Zbl 1332.82017 [29] Sak, J, Recursion relations and fixed points for ferromagnets with long-range interactions, Phys. Rev. B, 8, 281-285, (1973) [30] Salmhofer, M.: Renormalization: An Introduction. Springer, Berlin (1999) · Zbl 0913.00014 [31] Slade, G.: Critical exponents for long-range $$O(n)$$ models below the upper critical dimension. Commun. Math. Phys. (2016, to appear). arXiv:1611.06169 · Zbl 1391.82022 [32] Slade, G; Tomberg, A, Critical correlation functions for the $$4$$-dimensional weakly self-avoiding walk and $$n$$-component $$|φ |^4$$ model, Commun. Math. Phys., 342, 675-737, (2016) · Zbl 1342.82070 [33] Suzuki, M; Yamazaki, Y; Igarashi, G, Wilson-type expansions of critical exponents for long-range interactions, Phys. Lett., 42A, 313-314, (1972) [34] Ueltschi, D, Cluster expansions and correlation functions, Mosc. Math. J., 4, 511-522, (2004) · Zbl 1070.82002 [35] Wu, TT, Theory of Toeplitz determinants and the spin correlations of the two-dimensional Ising model. I, Phys. Rev., 149, 380-401, (1966)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.