The Balancing Theorem for Spanning Trees of Rectangular Grid Graphs

Abstract

We compare spanning-tree counts of rectangular grid graphs at fixed area. The main result is a balancing theorem: if $AB=ab$ and $A\le a\le b\le B$, then the more balanced rectangle has larger spanning-tree count, and in fact \[ \logτ(a,b)-\logτ(A,B) \ge \operatorname{arsinh}(1)\bigl((A+B)-(a+b)\bigr). \] The universal constant $\operatorname{arsinh}(1)=\log(1+\sqrt{2})\approx0.88137$ is optimal. Thus the log-ratio between equal-area rectangles grows at least linearly in the reduction of the sum of side lengths. The proof starts from the Laplacian product formula, passes to hyperbolic coordinates, and gives an exact finite-size identity for the log-gain of a balancing move. A monotone trapezoidal estimate gives the optimal main term, while a positive monotone residual gives strictness. The same decomposition identifies the closest-to-square divisor rectangle as the unique fixed-area maximizer and recovers the bounded-aspect rectangular asymptotic through the logarithmic corner term, including the square-grid case.