Bounds on the Ricci curvature and solutions to the Einstein equations for weighted graphs

Abstract

This is a preliminary study of the equation of motion of Euclidean classical gravity on a graph [1], based on the Lin-Lu-Yau Ricci curvature on graphs [2]. We observe that the constant edge weights configuration gives the unique solution on an infinite tree w.r.t. the asymptotically constant boundary condition. We study the minimum and maximum of the action w.r.t. certain boundary conditions, on several types of graphs of interest. We also exhibit a new class of solutions to the equations of motion on the infinite regular tree.

Bending the Bruhat-Tits Tree I:Tensor Network and Emergent Einstein Equations

“…We would like to define graph Ricci curvatures given the set of edge distances we have now defined on the tensor network. Graph curvatures have been considered in the mathematics literature [37][38][39][40]. Inspired by them, we consider the Ricci curvature R(d xy 1 , d xy 2 , .…”

Euclidean (A)dS spaces over $p$-adic numbers

“…This paper is devoted to the investigation of spaces over Q p . The p-adic version of Euclidean AdS 2 is proposed in [10,11], and it is widely used when combining with the anti-de Sitter/conformal field theory correspondence [12][13][14][15][16][17][18][19][20][21]. Considering that de Sitter and anti-de Sitter are two important spaces over R whose Euclidean versions are sphere and hyperboloid in high-dimensional spaces, we study the p-adic version of Euclidean (A)dS spaces(noted as p(A)dS) including the embedding of p(A)dS 2 and the analysis of their subspaces p(A)dS 1 .…”

Effective field theory on a finite boundary of the Bruhat-Tits tree

“…String theories over Q p ( p-adic string) begin with [5,7,8], and the Bruhat-Tits tree( T p ) is regarded as the p-adic string world-sheet in [9]. Spinors, gravity and blackholes on T p are studied in [10][11][12][13][14][15]. Relations between T p and tensor network are studied in [16][17][18].…”