The Markov Chain and Its Fermionic Representation

Here is a new representation of the Euler ensemble, leading us to the exact analytic solution below.

We start by replacing independent random variables $ \sigma $ with fixed sum by a Markov process, as suggested in ^[1]. We start with $ n $ random values of $ \sigma_i = 1 $ and the remaining $ N-n $ values of $ \sigma_i = -1 $. Instead of averaging over all of these values simultaneously, we follow a Markov process of picking $ \sigma_N,\dots, \sigma_1 $ one after another. At each step there will be $ M = N, \dots, 0 $ remaining $ \sigma $. We get a transition \[ n \Rightarrow n-1 \quad \text{with probability } \frac{n}{M} \] and \[ n \Rightarrow n \quad \text{with the complementary probability.} \]

Multiplying these probabilities and summing all histories of the Markov process is equivalent to the computation of the product of the Markov matrices:

$$\prod_{M=1}^{N} Q(M);$$

$$Q(M)\lvert n\rangle = \frac{M-n}{M}\lvert n\rangle + \frac{n}{M}\lvert n-1\rangle;$$

This Markov process will be random until $ n=0 $. After that, all remaining $ \sigma_k $ will have negative signs and be taken with probability 1, keeping $ n=0 $.

The expectation value of some function $ \hat{X}(\{\sigma\}) $ reduces to the matrix product

$$\mathbb{P}[\hat{X}] = \sum_{n=0}^{N_+} \left\langle n \right\rvert \left(\prod_{M=1}^{N} \hat{Q}(M)\right)\cdot \hat{X} \cdot \left\lvert N_+ \right\rangle;$$

$$\hat{Q}(M)\cdot \hat{X}\lvert n\rangle = \frac{n}{M}\,\hat{X}\Bigl(\sigma_{M}\to 1\Bigr)\lvert n-1\rangle + \frac{M-n}{M}\,\hat{X}\Bigl(\sigma_{M}\to -1\Bigr)\lvert n\rangle.$$

Here $ N_+ = \frac{N + \sum \sigma_l}{2} = \frac{N+qr}{2} $ is the number of positive sigmas. The operator $ \hat{Q}(M) $ sets in $ \hat{X}\lvert n\rangle $ the variable $ \sigma_{M} $ to 1 with probability $ \frac{n}{M} $ and to $-1$ with complementary probability. The generalization of the Markov matrix $ Q(M) $ to the operator $ \hat{Q}(M) $ will be presented shortly.

Once the whole product is applied to $ \hat{X} $, all the sigma variables in all terms will be specified so that the result is a number.

This Markov process is implemented as a computer code in ^[2], leading to a fast simulation with $ O(N^0) $ memory requirement.

Now, we observe that quantum Fermi statistics can represent the Markov chain of Ising variables. Let us construct the operator $ \hat{Q}(M) $ with Fermionic creation and annihilation operators, with occupation numbers \[ \nu_k = \frac{1+\sigma_k}{2} = (0,1). \] These operators obey (anti)commutation relations and create/annihilate the $ \sigma=1 $ state as follows (with Kronecker delta $ \delta[n] \equiv \delta_{n,0} $):

$$\left[a_i, a^\dagger_j\right]_+ = \delta_{ij};$$

$$\left[a_i, a_j\right]_+ = \left[a^\dagger_i, a^\dagger_j\right]_+ = 0;$$

$$a^\dagger_n\lvert\sigma_1,\dots,\sigma_N\rangle = \delta[\sigma_n+1]\,\lvert\sigma_1,\dots,\sigma_n\to 1,\dots,\sigma_N\rangle;$$

$$a_n\lvert\sigma_1,\dots,\sigma_N\rangle = \delta[\sigma_n-1]\,\lvert\sigma_1,\dots,\sigma_n\to -1,\dots,\sigma_N\rangle;$$

$$\hat{\nu}_n = a^\dagger_n a_n;$$

$$\hat{\nu}_n\lvert\sigma_1,\dots,\sigma_N\rangle = \delta[\sigma_n-1]\,\lvert\sigma_1,\dots,\sigma_N\rangle.$$

The number $ n(M) $ of positive sigmas, \[ \sum_{l=1}^M \delta[\sigma_l-1], \] coincides with the occupation number of these Fermi particles:

$$\hat{n}(M) = \sum_{l=1}^M \hat{\nu}_l;$$

This relation leads to the representation

$$\hat{Q}(M) = \hat{\nu}_M\,\frac{\hat{n}(M)}{M} + \Bigl(1-\hat{\nu}_M\Bigr)\frac{M-\hat{n}(M)}{M};$$

The variables $ \sigma_l $ can also be expressed in terms of this operator algebra by using

$$\hat{\sigma}_l = 2\hat{\nu}_l - 1.$$

The Wilson loop in $\eqref{WilsonLoop}$ can now be represented as an average over the small Euler ensemble $ \mathcal{E}(N) $ of a quantum trace expression:

\[ \begin{aligned} \Psi(\gamma, C) &= \frac{\VEV{\hat{W}[C]}_{\hat{\Omega}, \mathcal{E}(N)}} {\VEV{\hat{W}[0]}_{\hat{\Omega}, \mathcal{E}(N)}}, \quad \text{where} \\ \hat{W}[C] &= \mathrm{tr}\Bigl(\hat{Z}(qr)\,\exp\Bigl(\frac{i\gamma\hat{\Gamma}}{\nu}\Bigr) \prod_{M=1}^{N} \hat{Q}(M)\Bigr), \end{aligned} \]

$$\hat{\Gamma} = \sum_l \Delta\vec{C}_l \cdot \hat{\Omega} \cdot \vec{\mathcal P}_l(t);$$

$$\hat{Z}(s) = \oint \frac{d\omega}{2\pi}\exp\Bigl(i\omega\Bigl(\sum_l \hat{\sigma}_l - s\Bigr)\Bigr);$$

$$\Delta\vec{C}_l = \vec{C}\Bigl(\frac{l+1}{N}\Bigr) - \vec{C}\Bigl(\frac{l}{N}\Bigr),$$

$$\vec{\mathcal P}_l(t)=\sqrt{\frac{\nu}{2(t+t_0)}} \frac{\vec{\mathcal F}_l}{\gamma},\quad \hat{\Omega}\in O(3),$$

$$\vec{\mathcal F}_l = \frac{\Bigl\{\cos\bigl(\hat{\alpha}_l\bigr),\,\sin\bigl(\hat{\alpha}_l\bigr),\,0\Bigr\}} {2\sin\Bigl(\frac{\beta}{2}\Bigr)};$$

\[ \mathcal{E}(N): \quad p,q,r \in \mathbb{Z},\quad -N\le qr\le N,\quad 0

$$\hat{\alpha}_l = \beta\sum_{k=1}^{l-1}(2\hat{\nu}_k-1).$$

The last component of the vector $ \vec{\mathcal F}_l $ is set to 0 as this component does not depend on $ l $ and yields zero in the sum over the loop $ \sum_l \Delta\vec{C}_l = 0 $.

The proof of equivalence to the combinatorial formula with an average over $ \sigma_l=\pm 1 $ can be given using the following Lemma (obvious for a physicist).

$$\textbf{Lemma.} \quad \text{The operators } \hat{\nu}_l \text{ all commute with each other.}$$

\[ \begin{aligned} \hat{\nu}_l\,\hat{\nu}_n &= a^\dagger_l\Bigl(\delta_{ln}-a^\dagger_n a_l\Bigr)a_n\\[1ex] &= a^\dagger_l a_n\,\delta_{ln} - a^\dagger_l\,a^\dagger_n\,a_l\,a_n. \end{aligned} \]

Interchanging indices $ l, n $ shows that both terms are symmetric; hence, \[ \hat{\nu}_l\,\hat{\nu}_n = \hat{\nu}_n\,\hat{\nu}_l. \]

Theorem. The trace formula $\eqref{trace formula}$ equals the expectation value of the momentum loop ansatz $\eqref{LoopSol}$, $\eqref{PFT}$, $\eqref{Fsol}$ in the big Euler ensemble.

Proof. As all the operators $ \hat{\nu}_l $ commute with each other, the operators $ \hat{Q}(M) $ can be applied in arbitrary order to the states $ \Sigma = \lvert \sigma_1,\dots,\sigma_N \rangle $ involved in the trace. The same is true for individual terms in the circulation in the exponential of the Wilson loop. These terms $ \vec{\mathcal F}_l $ involve the operators $ \hat{\alpha}_l $, which commute with each other and with each $ \hat{Q}(M) $. Thus, we can use the ordered product \[ \hat{G}_l = \hat{Q}(l)\exp\Bigl(i\omega\hat{\sigma}_l + \frac{i\gamma}{\nu}\Delta\vec{C}_l\cdot\vec{\mathcal P}_l(t)\Bigr). \] Each operator $ \hat{G}_l $ acting on an arbitrary state $ \Sigma $ will create two terms with $ \delta[\sigma_l \pm 1] $. The product $\hat{Z}_l = \prod_{k=1}^l \hat{G}_k$ then produces $ 2^l $ terms in which each $ \sigma_k $ (for $ k\le l $) is fixed by the Kronecker deltas. Multiplying all the operators $ \hat{G}_M $ yields a superposition $ \hat{\Pi}_N $ of $ 2^N $ terms, each with $ \prod_{M=1}^{N} \delta[\sigma_M-\eta_M] $ for various choices of signs $ \eta_i=\pm 1 $. The product of these deltas projects the total sum over the $ 2^N $ combinations in the trace to a single term corresponding to a particular history $ (\eta_1,\dots,\eta_N) $ of the Markov process. Multiplying by the Markov transition probabilities and by the exponential \[ \exp\Bigl(\frac{i\gamma}{\nu}\sum_l\Delta\vec{C}_l\cdot\hat{\Omega}\cdot\vec{P}_l(t)\Bigr), \] with the operators $ \hat{\sigma} $ replaced by the numbers $ \eta $, reproduces the usual numeric result. Finally, integration over $ \omega $ produces a delta function \[ \delta\Bigl[\sum_l \eta_l - s\Bigr], \] which reduces the trace to the required sum over all histories of the Markov process with fixed $ \sum_l \eta_l $. $\blacksquare$

We have found a third vertex of the triangle of equivalent theories: decaying turbulence in three-dimensional space, the fractal curve in complex space, and Fermi particles on a ring. By degrees of freedom, this corresponds to a one-dimensional Fermi gas in the statistical limit $ N\to\infty $. However, there is no local Hamiltonian in this quantum partition function --- only a trace of certain products of operators in Fock space. Thus, an algebraic (or quantum statistical) problem remains: to find the continuum limit of this theory of the Fermion ring.

References

Alexander Migdal, “To the Theory of Decaying Turbulence”, Fractal and Fractional, vol. 7, no. 10, pp. 754, Oct 2023. DOI: 10.3390/fractalfract7100754 .
Maxim Bulatov and Alexander Migdal, “Numerical Simulation of Euler Ensemble”, Preprints, Feb 2024, unpublished.
James R. Norris, Markov Chains, Cambridge University Press, 2007.