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The lace expansion for the nearest-neighbor models on the BCC lattice Sakai, Akira


The lace expansion was initiated by Brydges and Spencer in 1985. Since then, it has been a powerful tool to rigorously prove mean-field (MF) results for various statistical-mechanical models in high dimensions. For example, Hara and Slade succeeded in showing the MF behavior for nearest-neighbor self-avoiding walk on $\mathbb{Z}^{d \geq 5}$. Recently, van der Hofstad and Fitzner managed to prove the MF results for nearest-neighbor percolation on $\mathbb{Z}^{d \geq 11}$ by using the so-called NoBLE (Non-Backtracking Lace Expansion). For sufficiently spread-out percolation, however, the MF results are known to hold for all $d$ above the percolation upper-critical dimension 6, without using the NoBLE. Our goal is to show the MF behavior for the nearest-neighbor models, for all $d$ above the model-dependent upper-critical dimension, in a simpler and more accessible way. To achieve this goal, we consider the nearest-neighbor models on the $d$-dimensional BCC (Body-Centered Cube) lattice. (This is just like working on the triangular or hexagonal lattice instead of the square lattice in two dimensions.) Because of the nice properties of the BCC lattice, we can simplify the analysis and more easily prove the mean-field results for $d$ close to the corresponding upper-critical dimension, currently $d \geq 6$ for self-avoiding walk and $d \geq 10$ for percolation. This talk is based on joint work with Lung-Chi Chen, Satoshi Handa and Yoshinori Kamijima for self-avoiding walk, and on joint work with the above three colleagues and Markus Heydenreich for percolation.

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