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SUMMARY:The grasshopper problem - Olga Goulko and David Llamas (UMass Bost
 on) 
DTSTART:20230622T131500Z
DTEND:20230622T141500Z
UID:TALK202228@talks.cam.ac.uk
CONTACT:Adrian Kent
DESCRIPTION:A grasshopper lands at a random point on a planar lawn of area
  one. It then makes one jump of fixed distance d in a random direction. Wh
 at shape should the lawn be to maximize the chance that the grasshopper re
 mains on the lawn after jumping? This easily stated yet hard to solve math
 ematical problem has intriguing connections to quantum information and sta
 tistical physics. A generalized version\, where the lawn is placed on a un
 it sphere such that exactly one of every pair of antipodal points belongs 
 to the lawn\, provides insight into a new class of Bell inequalities that 
 are relevant to quantum cryptography. In this setup two parties measure sp
 ins about randomly chosen axes and obtain correlations for pairs of axes s
 eparated by a fixed angle. A discrete version of the planar system can be 
 represented by a generalized Ising model\, where spins do not interact wit
 h their nearest neighbors but rather with spins a fixed distance away\, an
 d this distance d can be large. We show that\, perhaps surprisingly\, in t
 wo dimensions there is no d > 0 for which a disc shaped lawn is optimal. I
 f the jump distance is smaller than the radius of the unit disc\, the opti
 mal lawn resembles a cogwheel\, with transitions to more complex\, disconn
 ected shapes at larger d. A similar picture emerges for the spherical vers
 ion of the problem. In this talk\, we will discuss analytical and numerica
 l results for the grasshopper problem in two and three dimensional flat sp
 ace and on the surface of the sphere\, as well as their connection to Bell
 's inequalities involving random measurement choices. 
LOCATION:MR19
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