bibkey: karlinmcgregor1959coincidence authors: Samuel Karlin and James L. McGregor year: 1959 title: Coincidence probabilities doi: 10.2140/pjm.1959.9.1141 url: https://msp.org/pjm/1959/9-4/pjm-v9-n4-p13-s.pdf claim: The source relates ordered transition determinants to noncrossing paths and gives a local-character consequence of order-two positivity. The original even theta jump form fails that all-times ordered determinant hypothesis; no sharp spectral gap follows from this route. strata_touched: [] license: bibliographic-reference-only triage: anchor

Ordered semigroups and the original theta jumps

Published determinant interface

The journal reference is Pacific Journal of Mathematics 9 (1959), 1141–1164. The inspected publisher PDF has SHA-256 e198ad4388e3744ff670ab58472d192feabee7538edb53a41ed47ac1ae91bf4a. Only bibliographic information and the following source applications are retained; no third-party implementation or PDF is copied into the project.

Introduction (C), printed pp.1142–1143, relates ordered transition probability determinants to noncoincidence for a one-dimensional strong Markov process with continuous paths. Continuity prevents two ordered particles from exchanging positions without meeting. A jump process is not assigned this hypothesis merely because its state space is a line.

Section8, Theorem5, printed pp.1157–1158, assumes stationary transition probabilities on the real line, convergence to1 on a neighborhood of the starting point as time tends to zero, order-two positivity of the ordered transition determinants, and an exponent $a>0$ such that

$$P(t,x,\{y:|y-x|\ge \delta \})=o(t^a)(t\downarrow 0)$$

for every $x$ and $\delta >0$. Its conclusion is a bound $M{t}^{\beta }$, uniform on each fixed compact set of starting points, for every fixed $\beta ,\delta >0$. The first inequality in its proof places a middle neighborhood between the starting point and the distant destination: order-two positivity bounds a direct crossing by the product of the two intermediate transitions.

Section9, Theorem6, printed pp.1159–1160, classifies homogeneous processes with totally positive transition functions as deterministic drift or Wiener processes, with possible exponential killing. The original theta process is not homogeneous. That classification is not used as a theorem about it. Nor are the pointwise transition hypotheses of Theorem5 asserted without verification; the model check below uses its ordered-determinant mechanism directly on the known closed form.

Test the unchanged even form before borrowing oscillation theory

Retain the original minimal realization, $D=E_{\Gamma }+E_{\mathrm{prime}}$ in $L_{\mathrm{even}}^2(\nu )$, with $d\nu =\rho dx=2\Phi \cosh (x/2)dx$. Let $A\ge 0$ be its energy operator and $P_t=e^{-tA}$. Reflection reduces both. On the positive half-line the even space has radial measure $d{\nu }_{\mathrm{rad}}(r)=2\rho (r)dr$. The full symmetric conductance is $J=J_{\Gamma }+J_p$, where

$$J_{\Gamma }(dx,dy)=\Phi (x)\Phi (y)\psi (|x-y|)dxdy,\psi (u)=\frac{e^{-u/2}}{1-e^{-2u}}>0(u>0).$$

Every prime-power graph remains in the nonnegative $J_p$. In the nonnegative-generator convention the real polarized form is

$$D(f,g)=\frac{1}{2}\iint (f(y)-f(x))(g(y)-g(x))J(dx,dy).$$

Choose three nonzero nonnegative even compact smooth functions $f_L,f_M,f_R$, supported in the reflections of strictly separated positive intervals $I_L<I_M<I_R$. They belong to the original even core and have pairwise zero $L^2(\nu )$ overlap. Symmetry and disjointness give, for distinct indices,

$$D(f_i,f_j)=-\iint f_i(x)f_j(y)J(dx,dy)<0.\text{\hspace{1em}(O1)}$$

The conductance integral is finite by the form bound. Its Gamma part is strictly positive before the minus sign; prime terms cannot cancel it. Thus this test uses every actual long edge and no numerical cutoff.

The existing closed-form/semigroup correspondence gives, for $f,g\in D[D]$,

$$⟨f,P_{t}g{⟩}_{\nu }=⟨f,g{⟩}_{\nu }-tD(f,g)+o(t).\text{\hspace{1em}(O2)}$$

Operator-domain membership is unnecessary. In the spectral calculus, write $f=(A+1{)}^{-1/2}F$ and $g=(A+1{)}^{-1/2}G$. The multipliers $(1-e^{-t\lambda })/[t(1+\lambda )]$ are bounded by1 and converge to $\lambda /(1+\lambda )$, which verifies the form limit in (O2).

Put $a_{ij}(t)=⟨f_i,P_t{f}_j{⟩}_{\nu }$, $m=\Vert f_M{\Vert }_{\nu }^2>0$ and $c_{LR}=-D(f_L,f_R)>0$. The ordered rows $(L,M)$ and columns $(M,R)$ have determinant

$$\begin{array}{rl}\Delta (t)& =a_{LM}(t)a_{MR}(t)-a_{LR}(t)a_{MM}(t)\\ & =-m{c}_{LR}t+o(t)<0for\; all\; sufficiently\; smallt>0.\end{array}\text{\hspace{1em}(O3)}$$

Both factors in the first product are $O(t)$; the second product has $a_{LR}=t{c}_{LR}+o(t)$ and $a_{MM}=m+O(t)$. An order-two positive radial transition kernel would give a nonnegative determinant after integration against these ordered nonnegative source and destination weights. Equivalently that integrated condition can itself be stated without assuming a transition density. Equation(O3) violates it.

Consequently the original even theta semigroup cannot supply the all-times order-two positivity required by this proposed oscillation route. Positivity preservation of the Dirichlet semigroup remains true; it does not impose signs on these ordered two-by-two minors. The continuous Gamma conductance already causes the obstruction, so deleting prime atoms would not restore this particular condition.

What is still needed at the critical threshold

The known critical family satisfies $A{h}_k=h_k/2$, but that equation alone does not locate every other nonzero eigenvalue above it. A nodal-domain upper bound cannot be reversed into such an ordering. This note neither counts the nodal domains of $h_1$ nor identifies a subthreshold eigenvalue.

The original remainder still requires $D(r)\ge \Vert r{\Vert }_{\nu }^2/2$. Equations(O1)–(O3) exclude the stated all-times ordered-semigroup hypothesis. They do not exclude a weaker resolvent or fixed-time oscillation theorem, a different justified comparison, or the sharp Poincare bound itself. A FIB boundary description must preserve the long jump pairings before borrowing a theorem based on ordered paths. Branch order alone does not supply that theorem’s conditions.

The short-time determinant mechanism is reused from the inspected classical source. Its application to the original minimal even form is paper mathematics, without a new general-method, priority, numerical certificate or Lean claim. Weaker spectral-ordering hypotheses remain unverified for this operator; no exhaustive absence claim is made. RH and the full Robin inequality remain unresolved.