Prove restricted-run tableaux Conjecture 2a

RestrictedRunTableaux/FinitePhaseConeBridge.md

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+# Finite-phase cone bridge theorem for the restricted-run chain
+
+This note supplies the analytic transfer used in Section 7 of `Proof.md`.
+It is written specifically for a bounded Markov-additive process with a finite
+phase space; no claim is made for arbitrary state-dependent Markov chains.
+
+## Theorem
+
+Let `(X_n,J_n)` be a Markov-additive chain on `Z^d x S`, where `S` is finite,
+with kernel
+
+\[
+ p_{st}(z)=\Pr(J_{n+1}=t,\ X_{n+1}-X_n=z\mid J_n=s).
+\]
+
+Assume:
+
+1. the phase chain is irreducible and aperiodic;
+2. the jumps are bounded;
+3. `E[Delta X | J_n=s]=0` for every phase `s`;
+4. the stationary covariance `Sigma` is positive definite;
+5. for every coordinate pair there is a bounded phase function `A^{ij}` with
+   `(I-P)A^{ij}=C^{ij}-Sigma_ij`, where `C_s` is the conditional covariance;
+6. killing is by leaving a Lipschitz star-like cone, possibly with finitely
+   many phase-dependent boundary layers of bounded width;
+7. the killed chain has a positive harmonic function `V_s(x)` satisfying
+   `V_s(x)=u(x)+O(|x|^{p-1})` in every closed subcone, where `u` is the positive
+   Brownian cone harmonic of degree `p>=1`;
+8. the lattice Fourier kernel has no unit-modulus spectrum outside a finite
+   period group.
+
+Then, on each admissible lattice-period class,
+
+\[
+\Pr_{x,s}(X_n=y,J_n=t,\tau>n)
+ \sim c_{\mathcal L}\,V_s(x)V_t^*(y)n^{-p-d/2}, \tag{1}
+\]
+
+for fixed reachable interior `(x,s)` and `(y,t)`. Here `V^*` is the positive
+killed harmonic function of the time-reversed Markov-additive chain, and
+`c_L>0` depends only on the lattice class and covariance normalization.
+
+## Proof
+
+### 1. Whitening and martingale structure
+
+Apply `Sigma^{-1/2}` to the additive coordinate. The process remains a bounded
+martingale additive process and has stationary covariance `I_d`.
+
+For every `i,j`, assumption 5 gives
+
+\[
+M_n^{ij}=X_n^{(i)}X_n^{(j)}-n\delta_{ij}+A^{ij}(J_n).
+\]
+
+Indeed, conditional on `(X_n,J_n)=(x,s)`, the linear cross terms vanish by
+assumption 3 and
+
+\[
+E[M_{n+1}^{ij}-M_n^{ij}\mid J_n=s]
+=C_s^{ij}-\delta_{ij}+(PA^{ij})_s-A_s^{ij}=0.
+\]
+
+Thus the coordinate martingales and all quadratic martingales required in the
+cone estimates are available, with only a bounded terminal correction.
+
+The martingale functional central limit theorem applies because the jumps are
+bounded, the phase chain is ergodic, and the predictable quadratic variation
+satisfies
+
+\[
+n^{-1}\langle X\rangle_n\to I_d
+\]
+
+in probability by the ergodic theorem. Hence
+
+\[
+n^{-1/2}X_{\lfloor nt\rfloor}\Rightarrow B_t.
+\]
+
+### 2. Survival asymptotics
+
+The proof of Theorem 2 of Denisov--Zhang, *Markov Chains in the Domain of
+Attraction of Brownian Motion in Cones*, uses:
+
+- the Markov property;
+- a functional central limit theorem;
+- uniform moment and large-deviation bounds for the increments;
+- a positive killed harmonic function asymptotic to `u`;
+- the martingale identity for `|X_n|^2-dn`.
+
+The first four items hold uniformly over the finite phase space. The fifth is
+replaced by
+
+\[
+|X_n|^2-dn+a(J_n),
+\qquad a=\sum_i A^{ii}, \tag{2}
+\]
+
+which is a martingale by Step 1. Every optional-stopping identity in their
+proof therefore acquires an error bounded by `2||a||_infinity`. Their stopping
+levels are of order `sqrt(n)` and all identities are subsequently divided by
+`n` or by a positive power of `n`; the bounded error is `o(1)` and does not
+alter any estimate or limiting constant. Bounded jumps make their stochastic
+majorization and large-jump remainders immediate. A finite number of
+phase-dependent boundary layers changes the distance to the cone boundary by
+at most a constant, which is likewise absorbed in their uniform estimates.
+
+Consequently, uniformly for fixed phases and `|x|=o(sqrt(n))`,
+
+\[
+\Pr_{x,s}(\tau>n)\sim \varkappa V_s(x)n^{-p/2}, \tag{3}
+\]
+
+and, conditionally on survival,
+
+\[
+X_n/\sqrt n\Rightarrow
+H_0u(z)e^{-|z|^2/2}\,dz. \tag{4}
+\]
+
+The same argument applies to the time-reversed kernel
+
+\[
+p^*_{ts}(-z)=\frac{\pi_s}{\pi_t}p_{st}(z), \tag{5}
+\]
+
+giving `(3)--(4)` with `V^*`.
+
+### 3. Unrestricted finite-phase local limit theorem
+
+For `theta in [-pi,pi]^d`, form the finite Fourier matrix
+
+\[
+F(\theta)_{st}=\sum_z p_{st}(z)e^{i\theta\cdot z}.
+\]
+
+Near every point of the finite period group, analytic perturbation of the
+simple Perron eigenvalue gives
+
+\[
+\lambda(\theta)=1-\tfrac12\theta^T\theta+O(|\theta|^3), \tag{6}
+\]
+
+after whitening. On the complement of small neighborhoods of the period
+group, the spectral radius is strictly less than one. Fourier inversion and
+Laplace's method therefore give the lattice local central limit theorem,
+uniformly in the finitely many initial and terminal phases:
+
+\[
+\Pr_{s}(X_m=v,J_m=t)
+= c_{\mathcal L}\pi_t(2\pi m)^{-d/2}
+ e^{-|v|^2/(2m)}+o(m^{-d/2}) \tag{7}
+\]
+
+on an admissible lattice class, and zero on the other classes. This is the
+finite-matrix specialization of the multidimensional Nagaev--Guivarc'h local
+limit theorem.
+
+### 4. Conditioned local limit at fluctuation scale
+
+Repeat the proof of Denisov--Wachtel Theorem 5, summing also over the
+intermediate phase. Choose `m=floor(epsilon^3 n)` and split at time `n-m`:
+
+\[
+\Pr_{x,s}(X_n=y,J_n=t,\tau>n)
+=\sum_{z,r}K_{n-m}((x,s),(z,r))K_m((z,r),(y,t)). \tag{8}
+\]
+
+For `|z-y|>=epsilon sqrt(n)`, bounded-increment exponential estimates make the
+sum negligible compared with `n^{-p/2-d/2}`. If `z` is farther than
+`epsilon sqrt(n)` from the cone boundary, replacing the killed second kernel
+by the unrestricted kernel costs the same negligible amount. Insert (7) in
+the remaining sum, divide the first kernel by (3), and use the conditioned
+integral limit (4). Exactly the same Gaussian convolution as in
+Denisov--Wachtel yields, uniformly for `y=O(sqrt(n))` away from a vanishing
+boundary layer,
+
+\[
+\Pr_{x,s}(X_n=y,J_n=t,\tau>n)
+\sim c_{\mathcal L}\varkappa V_s(x)n^{-p/2-d/2}
+H_0u(y/\sqrt n)e^{-|y|^2/(2n)}. \tag{9}
+\]
+
+Their boundary and large-`y` estimates use only (3), bounded jumps, and the
+Brownian boundary estimate, so they are unchanged and extend (9) uniformly to
+all `y` in the cone.
+
+### 5. Fixed endpoint
+
+Fix `0<a<1`, put `m=floor((1-a)n)`, and split the killed bridge at `n-m`.
+By (5),
+
+\[
+K_m((z,r),(y,t))
+=\frac{\pi_t}{\pi_r}K_m^*((y,t),(z,r)). \tag{10}
+\]
+
+Insert the fluctuation-scale local limit (9) for the forward and reversed
+kernels. Terms with `|z|>A sqrt(n)` are negligible by (3)--(4). On
+`|z|<=A sqrt(n)`, the sum is a Riemann sum and converges to
+
+\[
+\int_K u(w)^2e^{-|w|^2/(2a(1-a))}\,dw,
+\]
+
+which is finite and strictly positive. The powers contributed by the two
+local kernels and the `n^{d/2}` lattice points in the central region combine
+to
+
+\[
+n^{-(p/2+d/2)}n^{-(p/2+d/2)}n^{d/2}
+=n^{-p-d/2}.
+\]
+
+This proves (1). Positivity follows from positivity of `V`, `V^*`, the lattice
+constant on a reachable class, and the Gaussian integral. QED.
+
+## Specialization to restricted-run tableaux
+
+For the six-state chain in `Proof.md`:
+
+- the jumps are bounded and conditionally centered;
+- the phase chain is irreducible and aperiodic because each `B_i` has a
+  self-loop;
+- the exact covariance is
+  `Sigma=(5/9)*[[2,-1],[-1,2]]`;
+- the Poisson correctors are rational and checked by `check_model.py`;
+- Sections 4--5 of `Proof.md` construct `V` exactly and prove `V~xy(x+y)`;
+- whitening gives a wedge angle `pi/3`, hence `p=3`;
+- the Fourier period group has three points, so the bridge is evaluated on its
+  admissible class `N=3n`.
+
+Equation (1) therefore gives
+
+\[
+\Pr((A_1,(3,1))\to(B_3,(1,1)),\tau>3n)
+\sim c n^{-4},\qquad c>0.
+\]
+
+Using the exact equal-weight identity
+
+\[
+\Pr(\text{bridge})=4G(n)/(3\,8^n)
+\]
+
+gives
+
+\[
+G(n)\sim C_1 8^n/n^4,
+\qquad C_1=3c/4>0.
+\]
+
+## Sources used in the transfer
+
+- Denisov--Wachtel, *Random Walks in Cones*, especially Theorems 5--6 and
+  their proofs by short-time convolution and time reversal.
+- Denisov--Zhang, *Markov Chains in the Domain of Attraction of Brownian
+  Motion in Cones*, especially Theorem 2; the discussion following Assumption
+  (M2) explicitly notes that the exact conditional-covariance assumption can
+  be relaxed at the cost of more complicated calculations.
+- Herve--Pene, *The Nagaev--Guivarc'h Method via the Keller--Liverani Theorem*,
+  for multidimensional local limits of additive functionals of strongly
+  ergodic Markov chains.