A solution space for a system of null-state partial differential equations. I. (English) Zbl 1314.35188
It is the first article from the series of works considering the partition function \(F\) for the Schramm-Löwner evolution (SLE) describing a variety of problems of mathematical statistical physics. This range of problems includes bond percolation, Ising and Potts models, etc. The main approach operates with the function \(F\) as a solution of \(2N+3\) partial differential equations (PDE) of \(2N\) variables: \(2N\) elliptic null-state PDE with independent variables, which are the results of the lattice vertices mapping into real numbers, and three conformal Ward identities. The number \(2N\) corresponds to number of polygon’s edges, where the process occurs. This part of the series presents a general introduction into the problem and the proof of the fact that the dimension of PDE solutions is less or equal to the \(N\)th Catalan number.
For Part II, III and IV see [the authors, ibid. 333, No. 1, 435–481 (2015; Zbl 1314.35189); ibid. 333, No. 2, 597–667 (2015; Zbl 1311.35314); ibid. 333, No. 2, 669–715 (2015; Zbl 1314.35190)].
For Part II, III and IV see [the authors, ibid. 333, No. 1, 435–481 (2015; Zbl 1314.35189); ibid. 333, No. 2, 597–667 (2015; Zbl 1311.35314); ibid. 333, No. 2, 669–715 (2015; Zbl 1314.35190)].
Reviewer: Eugene Postnikov (Kursk)
MSC:
| 35Q82 | PDEs in connection with statistical mechanics |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 60J67 | Stochastic (Schramm-)Loewner evolution (SLE) |
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 82B27 | Critical phenomena in equilibrium statistical mechanics |
| 82B43 | Percolation |
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