B Maze Proof and Statistical Mechanics

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Recent research has established a significant connection between maze structures and statistical mechanics, particularly in understanding the behavior of random mazes composed of hexagonal grids. Mathematicians have long explored questions regarding the size of the largest clear paths and the probability of navigating from one edge of the maze to the center and back. The findings indicate that as the grid expands, the critical value for path connectivity increases at a surprisingly slow rate. This suggests a sharp boundary between different connectivity modes within the maze. The implications of this research extend to broader applications in statistical mechanics and complex systems.
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https://www.quantamagazine.org/maze-proof-establishes-a-backbone-for-statistical-mechanics-20240207/

Imagine that a grid of hexagons, honeycomb-like, stretches before you. Some hexagons are empty; others are filled by a 6-foot tall column of solid concrete. The result is a maze of sorts. For over half a century, mathematicians have posed questions about such randomly generated mazes. How big is the largest web of cleared paths? What are the chances that there is a path from one edge to the center of the grid and back out again? How do those chances change as the grid swells in size, adding more and more hexagons to its edges?
 
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Thread 'Erroneously finding discrepancy in transpose rule'

Thread 'Erroneously finding discrepancy in transpose rule'

Obviously, there is something elementary I am missing here. To form the transpose of a matrix, one exchanges rows and columns, so the transpose of a scalar, considered as (or isomorphic to) a one-entry matrix, should stay the same, including if the scalar is a complex number. On the other hand, in the isomorphism between the complex plane and the real plane, a complex number a+bi corresponds to a matrix in the real plane; taking the transpose we get which then corresponds to a-bi...
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