Stationary Log-Gamma Polymer in Half-Space

Abstract

We study the half-space log-gamma polymer model with stationary initial conditions. We derive exact formulas for the distribution of the partition function along the diagonal across the entire High density phase and Low density phase. We obtain asymptotics of these distributions under the critical scaling. We also prove the first exponential upper bounds for the upper and lower tail of the scaled free energy for these half-space stationary log-gamma models.

Figures (20)

• Figure 1: This picture describes the phase diagram of the half-space log-gamma polymer model given the boundary parameter $\beta$ and the parameter $t$ that describes the drift. Depending on where $(\beta,t)$ lies in the diagram, the process $Z(\cdot,N) / Z(N,N)$ converges to one of the three spatial processes, $\mathcal{T}_{\beta,t},$$\mathcal{T}_{\beta,0},$ or $\mathcal{T}_{-t,t},$ which is claimed in Conjecture \ref{['conj: stationary measures']}. On the full line $\beta = t$ and the half line $\beta = -t$, $\beta \leq 0$, $t\geq 0$, the process converges to the $\text{Gamma}^{-1}$$(\alpha+t)$ multiplicative random walk. MC, HD, and LD represent Maximal current (green), High density (blue), and Low density (red) phases. We characterize the distribution of $Z(N,N)$ in the High density phase under different conditions of $\beta$ and $t$. When $\beta > t,$ we obtain $Q_{N,\beta >t}^{High}$ and when $\beta < t$, we obtain $Q_{N,\beta <t}^{High}$. Along the Low density boundary, we obtain $Q_{N,t}^{Low}$. In particular, we have ${Q}_{N,t}^{Low}$ on the line $\beta = t$, $t>0$ in High density phase. We obtain these distributions under the condition $t< 0.5$. We do not have the distribution for the Maximal current phase, which is drawn as a dashed line.
• Figure 2: This picture describes the phase diagram of the half-space stationary log-gamma polymer model under critical scaling, which sets $t = \tilde{t}/(\sigma N)^{1/3}$ and $\beta = \tilde{\beta}/(\sigma N)^{1/3}.$ We characterize the distribution of $\lim_{N\rightarrow \infty}\frac{\log Z(N,N) + Nf}{(\sigma N)^{1/3}}$ under three different cases. Under the High density phase, when $\tilde{\beta} > \tilde{t}$, we have $\widetilde{Q}_{\tilde{\beta} > \tilde{t}}^{High}$ and when $-\tilde{t}<\tilde{\beta} < \tilde{t}$, we have $\widetilde{Q}_{\tilde{\beta} < \tilde{t}}^{High}$ as limiting distributions. Under the Low density phase, we obtain $\widetilde{Q}_{\tilde{t}}^{Low}$ as the limiting distribution. In particular, we have $\widetilde{Q}_{\tilde{t}}^{Low}$ on the line $\tilde{\beta} = \tilde{t}$, $\tilde{t}>0$ in the High density regime.
• Figure 3: Both figures are half-space stationary two-parameter log-gamma polymer model with weights described in Definition \ref{['def: two-param stationary model']}. The equality in distribution is justified by Lemma \ref{['lem: symmetry']}.
• Figure 4: Half-space product stationary log-gamma polymer model with weights described in Definition \ref{['def: product stationary model']}.
• Figure 5: This model connects the product stationary case and the two-parameter stationary case. Let $\gamma$ denote the weight at $(2,1)$. If $\gamma = 1,$ i.e., a constant weight $1$, then this model becomes the two-parameter stationary model. If $\gamma = \text{Gamma}^{-1}(\alpha -t)$ and $\beta = \alpha$, then this model becomes the product stationary model.

Theorems & Definitions (125)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Conjecture 1.4
• Definition 1.5
• Theorem 1.6: Low density phase
• Theorem 1.7: Low density phase
• Theorem 1.8: Low density phase
• Definition 1.9
• Theorem 1.10: High density phase with $\beta > t$