Uniform Bessel bounds for magnetization crossings in finite Ising chains
We consider a finite Ising chain in zero field and ask when the all-up configuration is as likely as zero magnetization (magnetization one for an odd number of spins). For both free and periodic ends, the crossing coupling of a long chain is half the logarithm of its length minus half the logarithm of that logarithm, up to a term that vanishes as the length grows. We compute further terms with explicit constants and, for free ends, bound the crossing at every length. The second term is missing from an earlier approximation and explains why an earlier numerical fit gives a slope below one half. To locate the crossing, we compare the known magnetization law with its Bessel approximation. The Bessel approximation is above the exact weight, and we bound its relative error for every length, coupling and magnetization. At the crossing this bound tends to zero. The central bin still has about as many domain walls as the logarithm of the length on average, so the crossing has a coupling width of order the inverse logarithm. In a fixed positive dimensionless field the last crossing comes from near the edge of the row, at a coupling of about a quarter of the logarithm of the length. For long chains it is the one-spin crossing when the field exceeds half the logarithm of two. Below that value another bin crosses later.