Stationary Measures for Two Dual Families of Finite and Zero Temperature Models of Directed Polymers on the Square Lattice

Abstract

We study the recently introduced Inverse-Beta (IB) polymer, an exactly solvable, anisotropic finite temperature model of directed polymer on the square lattice, and obtain its stationary measure. In parallel we introduce an anisotropic zero temperature model of directed polymer on the square lattice, the Bernoulli–Geometric polymer, and obtain its stationary measure. This new exactly solvable model is dual to the IB polymer and interpolates between models of first and last passage percolation on the square lattice. Both stationary measures are shown to satisfy detailed balance. We also obtain the asymptotic mean value of (i) the free-energy of the IB polymer; (ii) the optimal energy of the Bernoulli–Geometric polymer. We discuss the convergence of both models to their stationary state. We perform simulations of the Bernoulli–Geometric polymer that confirm our results.

Appendices

Appendix 1: Proof of the Properties of the Finite Temperature Reversibility-Stationarity Map

In this appendix we prove Propositions 3.1, 3.2 being trivial. We thus consider three independent random variables (U, V, W) distributed as in (2.7) and (2.3) and consider the RVs $(U',V',W') = \phi (U,V,W)$ as given in (3.3). The Jacobian of the transformation $(U,V,W) \rightarrow (U',V',W')$ is easily computed as, schematically,

$$\begin{aligned} \mathrm{det}\left( \frac{\partial \phi (U,V,W)}{\partial (U,V,W) } \right) = -\frac{U W+U+V W}{U V} <0. \end{aligned}$$

(7.1)

The PDF of the triplet $(U',V',W')$ is then directly evaluated as

$$\begin{aligned} P(U',V',W')= & {} P_U( \phi ^{(1)}(U',V',W') ) P_V( \phi ^{(1)}(U',V',W')) P_W( \phi ^{(1)}(U',V',W') ) \nonumber \\&\times \frac{U V}{U W+U+V W}. \end{aligned}$$

(7.2)

Where we introduced the PDF of the independent RVs (U, V, W) as noted in (2.7) and (2.3) and used the fact that $\phi$ is an involution. It is then directly checked that

$$\begin{aligned} P(U',V',W') = P_U( U') P_V( V' ) P_W( W' ) , \end{aligned}$$

(7.3)

hence showing that $U'$, $V'$ and $W'$ are independent and distributed as $U' \sim U$, $V' \sim V$ and $W' \sim W$.

Appendix 2: Proof of the Properties of the Zero Temperature Stationarity Map

In this appendix we prove Propositions 4.1 and 4.2. Let us first prove the detailed balance property Proposition 4.2b. We thus consider $\mathsf{U}\perp \mathsf{V}\perp (\mathsf{u}, \mathsf{v})$ distributed as in (2.12) and (2.20). Let us first compute the conditional probability

$$\begin{aligned} \Psi (k_{\mathsf{U}'} , k_{\mathsf{V}'} , k_{\mathsf{U}} , k_{\mathsf{V}}):= Proba\left( ( (\mathsf{U}', \mathsf{V}' )= (k_{\mathsf{U}'} , k_{\mathsf{V}'} ) | (\mathsf{U}, \mathsf{V})= (k_{\mathsf{U}} , k_{\mathsf{V}} ) \right) \end{aligned}$$

(7.4)

where $k_{\mathsf{U}} \in \mathbb {N}$, $k_{\mathsf{V}} \in {\mathbb Z}_-$, $\mathsf{U}' = \mathrm{min}\left( \mathsf{u}, \mathsf{v}+ \mathsf{U}- \mathsf{V}\right)$ and $\mathsf{V}' = \mathrm{min}\left( \mathsf{u}+\mathsf{V}-\mathsf{U}, \mathsf{v}\right) = \mathsf{U}' + \mathsf{V}- \mathsf{U}$. We have

$$\begin{aligned}&\Psi (k_{\mathsf{U}'} , k_{\mathsf{V}'} , k_{\mathsf{U}} , k_{\mathsf{V}})\nonumber \\&\quad = p_{\mathsf{u}\mathsf{v}} \sum _{G_q =0}^{\infty } (1-q) (q)^{G_q} \delta ( k_{\mathsf{U}'} = \mathrm{min}\left( -G_q , -G_q + k_{\mathsf{U}}- k_{\mathsf{V}} \right) ) \delta ( k_{\mathsf{V}'} = k_{\mathsf{U}'}+ k_{\mathsf{V}}- k_{\mathsf{U}} ) \nonumber \\&\qquad +\, (1-p_{\mathsf{u}\mathsf{v}}) \sum _{G_q' =0}^{\infty } (1-q') (q')^{G_q'}\delta ( k_{\mathsf{U}'} = \mathrm{min}\left( 1+ G_q' , k_{\mathsf{U}}- k_{\mathsf{V}} \right) ) \delta ( k_{\mathsf{V}'} = k_{\mathsf{U}'}+ k_{\mathsf{V}}- k_{\mathsf{U}} ) \nonumber \\&\qquad \times \, p_{\mathsf{u}\mathsf{v}} \sum _{G_q =0}^{\infty } (1-q) (q)^{G_q} \delta ( k_{\mathsf{U}'} = -G_q ) \delta (k_{\mathsf{U}}> k_{\mathsf{V}} ) \delta ( k_{\mathsf{V}'} = k_{\mathsf{U}'}+ k_{\mathsf{V}}- k_{\mathsf{U}} ) \nonumber \\&\qquad +\, p_{\mathsf{u}\mathsf{v}} \sum _{G_q =0}^{\infty } (1-q) (q)^{G_q} \delta ( k_{\mathsf{U}'} = -G_q + k_{\mathsf{U}}- k_{\mathsf{V}} ) \delta (k_{\mathsf{U}} \le k_{\mathsf{V}} ) \delta ( k_{\mathsf{V}'} = k_{\mathsf{U}'}+ k_{\mathsf{V}}- k_{\mathsf{U}} ) \nonumber \\&\qquad +\, (1-p_{\mathsf{u}\mathsf{v}}) \sum _{G_q' =0}^{\infty } (1-q') (q')^{G_q'}\delta ( k_{\mathsf{U}'} \!= \! 1\!+\! G_q' ) \delta ( k_{\mathsf{U}}\!-\! k_{\mathsf{V}} \!>\! 1\!+\! G_q' ) \delta ( k_{\mathsf{V}'} = k_{\mathsf{U}'}\!+\! k_{\mathsf{V}}\!-\! k_{\mathsf{U}}) \nonumber \\&\qquad +\, (1\!-\!p_{\mathsf{u}\mathsf{v}}) \sum _{G_q' \!=\!0}^{\infty } (1\!-\!q') (q')^{G_q'}\delta ( k_{\mathsf{U}'} \!=\! k_{\mathsf{U}} \!-\! k_{\mathsf{V}} ) \delta ( k_{\mathsf{U}} \!-\! k_{\mathsf{V}} \le 1\!+\! G_q' ) \delta ( k_{\mathsf{V}'} \!=\! k_{\mathsf{U}'}\!+\! k_{\mathsf{V}}\!-\! k_{\mathsf{U}} ) \nonumber \\&\quad = p_{\mathsf{u}\mathsf{v}} (1-q) (q)^{-k_{\mathsf{U}'}} \delta ( k_{\mathsf{U}'} \le 0 ) \delta (k_{\mathsf{U}}> k_{\mathsf{V}} ) \delta ( k_{\mathsf{V}'} = k_{\mathsf{U}'}+ k_{\mathsf{V}}- k_{\mathsf{U}} ) \nonumber \\&\qquad +\, p_{\mathsf{u}\mathsf{v}} (1-q) (q)^{-k_{\mathsf{U}'} + k_{\mathsf{U}}-k_{\mathsf{V}}} \delta ( -k_{\mathsf{U}'} + k_{\mathsf{U}}-k_{\mathsf{V}} \ge 0 ) \delta (k_{\mathsf{U}} \le k_{\mathsf{V}} ) \delta ( k_{\mathsf{V}'} = k_{\mathsf{U}'}+ k_{\mathsf{V}}- k_{\mathsf{U}}) \nonumber \\&\qquad +\, (1-p_{\mathsf{u}\mathsf{v}}) (1-q') (q')^{ k_{\mathsf{U}'} -1}\delta ( k_{\mathsf{U}'} \ge 1 ) \delta ( k_{\mathsf{U}}- k_{\mathsf{V}} >k_{\mathsf{U}'} ) \delta ( k_{\mathsf{V}'} = k_{\mathsf{U}'}+ k_{\mathsf{V}}- k_{\mathsf{U}} ) \nonumber \\&\qquad +\, (1-p_{\mathsf{u}\mathsf{v}}) (q')^{k_{\mathsf{U}}- k_{\mathsf{V}}-1 }\delta ( k_{\mathsf{U}'} = k_{\mathsf{U}}- k_{\mathsf{V}} ) \delta ( k_{\mathsf{U}'} \ge 1 ) \delta ( k_{\mathsf{V}'} = k_{\mathsf{U}'}+ k_{\mathsf{V}}- k_{\mathsf{U}} ) \nonumber \\&\qquad +\, (1-p_{\mathsf{u}\mathsf{v}}) \delta ( k_{\mathsf{U}'} = k_{\mathsf{U}}- k_{\mathsf{V}} ) \delta ( k_{\mathsf{U}'} \le 0 ) \delta ( k_{\mathsf{V}'} = k_{\mathsf{U}'}+ k_{\mathsf{V}}- k_{\mathsf{U}} ) \end{aligned}$$

(7.5)

Using this last expression and the expression of $\textit{Proba}((\mathsf{U}, \mathsf{V})= (k_{\mathsf{U}} , k_{\mathsf{V}} ) )$ given in (2.22), we obtain

$$\begin{aligned} \tilde{\Psi }(k_{\mathsf{U}'} , k_{\mathsf{V}'} , k_{\mathsf{U}} , k_{\mathsf{V}}):= & {} Proba\left( ( (\mathsf{U}', \mathsf{V}' )= (k_{\mathsf{U}'} , k_{\mathsf{V}'} ) , (\mathsf{U}, \mathsf{V})= (k_{\mathsf{U}} , k_{\mathsf{V}} ) \right) \nonumber \\= & {} \Psi (k_{\mathsf{U}'} , k_{\mathsf{V}'} , k_{\mathsf{U}} , k_{\mathsf{V}}) \textit{Proba}((\mathsf{U}, \mathsf{V})= (k_{\mathsf{U}} , k_{\mathsf{V}} ) ) \nonumber \\= & {} \Psi (k_{\mathsf{U}'} , k_{\mathsf{V}'} , k_{\mathsf{U}} , k_{\mathsf{V}}) \times \nonumber \\&\times \left( p_{\mathsf{U}} \delta (k_{\mathsf{U}} \le 0 ) (1-q/q_b)(q/q_b)^{-k_{\mathsf{U}}} + (1 - p_{\mathsf{U}} ) \delta (k_{\mathsf{U}} \ge 1 )\right. \nonumber \\&\times \left. (1-q_b q')(q_bq')^{k_{\mathsf{U}} -1} \right) \nonumber \\&\times \left( p_{\mathsf{V}} \delta (k_{\mathsf{V}} \le 0 )(1-q_b)(q_b)^{-k_{\mathsf{V}}} + (1 - p_{\mathsf{V}} ) \delta (k_{\mathsf{V}} = 0 ) \right) \end{aligned}$$

(7.6)

and it is then straightforward (although technically complicated due to the large number of terms) to check the detailed balance property Proposition 4.2. Namely one shows that the equality

$$\begin{aligned} \tilde{\Psi }(k_{\mathsf{U}'} , k_{\mathsf{V}'} , k_{\mathsf{U}} , k_{\mathsf{V}}) = \tilde{\Psi }(k_{\mathsf{U}} , k_{\mathsf{V}} , k_{\mathsf{U}'} , k_{\mathsf{V}'}) \end{aligned}$$

(7.7)

holds. Let us emphasize here that this property is rather special: the fact that (7.7) works requires a large number of cancellation between terms that are made possible by the choice of only three parameters $p_{\mathsf{U}} =\frac{1-q_b q'}{1-qq'}$, $p_{\mathsf{V}}= \frac{1-q'}{1-q_b q'}$ and $p_{\mathsf{u}\mathsf{v}}= \frac{1-q'}{1-qq'}$, a characteristic sign of the existence of exact solvability properties for the model. Finally, summing (7.7) on $k_{\mathsf{U}'}$ and $k_{\mathsf{V}'}$ gives the stationarity property Proposition 4.1:

$$\begin{aligned} \textit{Proba}((\mathsf{U}, \mathsf{V}) = ( k_{\mathsf{U}} , k_{\mathsf{V}}) ) = \textit{Proba}((\mathsf{U}', \mathsf{V}') = (k_{\mathsf{U}}, k_{\mathsf{V}})). \end{aligned}$$

(7.8)