Renormalization: an advanced overview

We present several approaches to renormalization in QFT: the multi-scale analysis in perturbative renormalization, the functional methods \`a la Wetterich equation, and the loop-vertex expansion in non-perturbative renormalization. While each of these is quite well-established, they go beyond standard QFT textbook material, and may be little-known to specialists of each other approach. This review is aimed at bridging this gap.Comment: Review, 130 pages, 33 figures; v2: misprints corrected, refs. added, minor improvements; v3: some changes to sect. 5, refs. adde

Sufficient conditions for log-concave conjecture on all-terminal reliability polynomial of a network

Consider a graph G that is simple, undirected, and connected, and has n vertices and m edges, and let Ni(G) denote the number of connected spanning i-edge-subgraphs in a graph G for an integer i(n－1■i■m). For a graph G and all integers i ’s (n■i■m－1), it is wellknown that the problem of computing all Ni(G) ’s is #P-complete (see e.g., [3, 7, 14, 31]), and that log-concave conjecture (see e.g., [3, 14, 37]), that is, Ni(G)2■Ni－1(G)Ni＋1(G) holds, is still open. In this paper, by introducing new methods of partitioning Ni into a sequence of part integers, and by investigating properties of the sequence, we propose sufficient conditions to ensure the validity of log-concavity of sequence Nn－1(G), Nn(G)…, Nm(G)