scholarly journals Geometric combinatorics in discrete settings

2011 ◽  
Author(s):  
David Covert
Keyword(s):  
Geometric Combinatorics

scholarly journals Enumeration of Hybrid Domino-Lozenge Tilings II: Quasi-Octagonal Regions

10.37236/4669 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Tri Lai
Keyword(s):  
Product Formula ◽  
Cambridge University ◽  
Replacement Method ◽  
Lozenge Tilings ◽  
Simple Product ◽  
Geometric Combinatorics

We use the subgraph replacement method to prove a simple product formula for the tilings of an  8-vertex counterpart of Propp's quasi-hexagons (Problem 16 in New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999), called quasi-octagon.

Geometric combinatorics of Kalman algebras

2006 ◽  
Vol 42 (11) ◽  
pp. 1604-1611
Author(s):  
N. I. Osetinskii ◽  
O. O. Vasil’ev ◽  
F. S. Vainstein
Keyword(s):  
Geometric Combinatorics

scholarly journals The grasshopper problem

2017 ◽  
Vol 473 (2207) ◽  
pp. 20170494 ◽  
Author(s):  
Olga Goulko ◽  
Adrian Kent
Keyword(s):  
Simulated Annealing ◽  
Ground State ◽  
Bell Inequalities ◽  
Spin Model ◽  
Random Point ◽  
Parallel Tempering ◽  
Discrete Version ◽  
Fixed Distance ◽  
Random Direction ◽  
Geometric Combinatorics

We introduce and physically motivate the following problem in geometric combinatorics, originally inspired by analysing Bell inequalities. A grasshopper lands at a random point on a planar lawn of area 1. It then jumps once, a fixed distance d , in a random direction. What shape should the lawn be to maximize the chance that the grasshopper remains on the lawn after jumping? We show that, perhaps surprisingly, a disc-shaped lawn is not optimal for any d >0. We investigate further by introducing a spin model whose ground state corresponds to the solution of a discrete version of the grasshopper problem. Simulated annealing and parallel tempering searches are consistent with the hypothesis that, for d < π −1/2 , the optimal lawn resembles a cogwheel with n cogs, where the integer n is close to π ( arcsin ⁡ ( π d / 2 ) ) − 1 . We find transitions to other shapes for d ≳ π − 1 / 2 .

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