On the stationary solutions
of random polymer models
and their zero-temperature limits

In the recent work [17], a general approach for identifying the spatially independent and identically distributed stationary measures of various classical discrete integrable systems, namely the discrete and ultra-discrete KdV equations, and the discrete and ultra-discrete Toda lattice, was presented. Moreover, parallels were drawn between the argument of [17] and the techniques of [7], which characterised the stationary measures of a class of random polymer models. In particular, in both [17] and [7] appeals were made to particular decompositions of the recursive maps that define the models of interest. After presenting an extended introduction to what is known in the positive-temperature random polymer context, the aim of this article is to highlight the use of similar decompositions in understanding zero-temperature limits of the models in question. Whilst the results we present were for the most part previously known (including all those relevant to the positive-temperature systems), the stationary solutions we derive for the zero-temperature limit of the beta random polymer are apparently new. We further believe that the systematic approach set out here helps clarify why particular stationary measures arise for the different random polymer models considered, and the connections between them. Indeed, in both the positive- and zero-temperature settings, each of the systems is related to one of two basic bijections (which are themselves related), and each of these is naturally connected to certain distributions via an ‘independence preservation’ property. Moreover, the appearance in atoms in the stationary measures of various zero-temperature polymer models is explained in terms of a collapse of part of the domain from the positive- to zero-temperature regime.

Let us start by introducing the central ideas of [17] in the context of two-dimensional stochastic lattice models. In the latter setting we commonly have that the input to a system is provided by an array $(X_{n,m})_{n,m\in{\mathbb{Z}}^{2}}$ of random variables, and the behavior of the partition function $(Z_{n,m})_{n,m\in{\mathbb{Z}}^{2}}$, as determined by $(X_{n,m})_{n,m\in{\mathbb{Z}}^{2}}$, is the main object of interest; see Section 2 for the concrete random polymer examples that will be our primary focus. Writing $(U_{n,m})_{n,m\in{\mathbb{Z}}^{2}}$ and $(V_{n,m})_{n,m\in{\mathbb{Z}}^{2}}$ for the ‘derivative’ (in a suitable sense) of $(Z_{n,m})_{n,m\in{\mathbb{Z}}^{2}}$ in the first- and second-coordinate directions, respectively, one then typically has a recursive relation of the form:

where $R:J_{1}\times I_{1}\times I_{2}\to I_{1}\times I_{2}$ for some subsets $J_{1},I_{1},I_{2}\subseteq\mathbb{R}$ (usually intervals or discrete subsets thereof). In the ‘upper-right corner’-version of such a model, one takes independent random variables $(X_{n,m})_{n,m\in{\mathbb{N}}}$, $(U_{n,0})_{n\in{\mathbb{N}}}$ and $(V_{0,n})_{n\in{\mathbb{N}}}$, and from these inputs determines the remaining variables $(U_{n,m})_{n,m\in{\mathbb{N}}}$ and $(V_{n,m})_{n,m\in{\mathbb{N}}}$ by applying (1.1) repeatedly. Given this structure, if there exists a triplet of probability measures $(\tilde{\mu},\mu,\nu)$ on $J_{1}$, $I_{1}$ and $I_{2}$ such that

where an expression of the form $\mu\times\nu$ means a product of measures, then the stochastic lattice model has a stationary measure; see the discussion in [17], for example. (NB. For a measurable map $f$ and measure $\pi$, we write $f(\pi):=\pi\circ f^{-1}$ for the push-forward of $\pi$ by $f$.) Hence it is important to find such a collection of measures.

It transpires that for the random polymer models studied in [7] the map $R$ can be usefully decomposed into two maps, each with two inputs and two outputs. More precisely, one can find a bijection $F:I_{1}\times I_{2}\rightarrow J_{1}\times J_{2}$, where $J_{2}\subseteq\mathbb{R}$, such that $R$ can be reconstructed from $F$ and $F^{-1}$. To this end, one first defines an involution $\overline{F}:J_{1}\times I_{1}\times I_{2}\to J_{1}\times I_{1}\times I_{2}$ by setting

where we write $F(b,c)=(F^{(1)}(b,c),F^{(2)}(b,c))$. That is, $(d,e,f):=\overline{F}(a,b,c)$ can be computed by first applying $F$ to $(b,c)$ to give $(d,g)=F(b,c)$, and then applying $F^{-1}$ to $(a,g)$ to give $(e,f)=F^{-1}(a,g)$. If $F$ is chosen appropriately, one then finds that $R=\overline{F}^{(2,3)}$, i.e.

Pictorially, keeping track of the lattice variables, this situation can be represented as follows:

We will recall below explicitly the decomposition of $R$ for the random polymer models of [7]. Moreover, we note that in [17, Section 6], it was observed that if $R$ is the map arising for a certain directed polymer with site weights, then the map $\overline{F}$ is that associated with the discrete Toda lattice, and, similarly, the $R$ of directed last passage percolation corresponds to the $\overline{F}$ associated with the ultra-discrete Toda lattice. (Precisely, up to a simple change of variables, discrete Toda corresponds to the $\overline{F}$ given by $R_{(0,1)}$, and ultra-discrete Toda corresponds to the $\overline{F}$ given by $R^{\mathrm{zero}}_{(0,1)}$, using the notation introduced subsequently at (1.7) and (1.9).) As will become clear in what follows, these connections explain why the systems admit related stationary measures.

The advantage of the above picture is that it gives a route to understanding the stationary measures of $R$, i.e. those satisfying (1.2), via the solutions of a certain ‘detailed balance condition’ for $F$. In particular, we say that a quadruplet $(\mu,\nu,\tilde{\mu},\tilde{\nu})$ on $I_{1}$, $I_{2}$, $J_{1}$ and $J_{2}$ satisfies the detailed balance condition for $F$ if

This condition was called the ‘detailed balance condition for a type II model’ in [17], wherein it was shown to be equivalent to the temporal invariance in distribution of the corresponding bi-infinite dynamical system started from $(\mu\times\tilde{\mu})^{\mathbb{Z}}$. Whilst this usage of the terminology detailed balance is different to the standard one for reversible Markov chains, it similarly provides, in the setting of [17], a local condition for the global invariance of a system. Notice also that the detailed balance condition of (1.6) can be seen as the aforementioned independence preservation property for the bijection $F$. A key result in [17] is the following proposition, which shows that solutions of the detailed balance condition for $F$ precisely correspond to triplets of measures $(\tilde{\mu},\mu,\nu)$ whose product is invariant under $\overline{F}$. Subsequently, applying the link between $R$ and $\overline{F}$ from (1.4), one sees that invariant product measures for $\overline{F}$ yield solutions of (1.2), and vice versa. Thus we have the desired connection between the detailed balance condition for $F$ and stationary measures for $R$.

We now proceed to become more specific in our discussion. For the random polymer models of interest in [7], as we will outline in more detail in Section 2, the maps $R$ that arise take the form

for some ${\alpha},{\beta}\in\mathbb{R}$ satisfying $\max\{{\alpha},{\beta}\}>0$. One can readily check that a corresponding $F$ is given by

(Note that the choice of $F$ is not unique, but the flexibility that it admits is essentially only a change of scale in the second coordinate of its output, and the above form will be convenient for us.) Consequently, if one can characterise detailed balance solutions for the maps $F_{({\alpha},{\beta})}$, then one obtains all the stationary measures of the polymer models. In fact, by scaling and symmetry, there are only essentially four choices of $({\alpha},{\beta})$ that are relevant in the setting of [7]:

• •

  $({\alpha},{\beta})=(0,1)$, which corresponds to the inverse-gamma(/log-gamma) model of [24, 25];

• •

  $({\alpha},{\beta})=(1,0)$, which corresponds to the gamma(/strict-weak) model of [15, 23];

• •

  $({\alpha},{\beta})=(-1,1)$, which corresponds to the inverse-beta model of [27];

• •

  $({\alpha},{\beta})=(1,-1)$, which corresponds to the beta model of [5, 3], see also Remark 2.1 for comments about the connection with the random walk in a beta-distributed random environment (${\beta}$-RWRE) of [5].

Moreover, after some additional rescaling, one can check that solving the detailed balance equation for the corresponding $F_{({\alpha},{\beta})}$ is equivalent to solving it for one of the following bijections on $(0,\infty)^{2}$:

which has inverse

and the involution

Precisely, we will explain how to relate $R_{(0,1)}$ to $F_{\mathrm{Gam},\mathrm{Be}^{\prime}}$, $R_{(1,0)}$ to $F_{\mathrm{Gam},\mathrm{Be}^{\prime}}^{-1}$, and both $R_{(-1,1)}$ and $R_{(1,-1)}$ to $F_{\mathrm{Be}^{\prime},\mathrm{Be}^{\prime}}$. (See Proposition 3.1 below.) In the course of the study, it will also be convenient to consider $R_{(1,1)}$, which is essentially equivalent to $R_{(-1,1)}$ and also relates to $F_{\mathrm{Be}^{\prime},\mathrm{Be}^{\prime}}$ (see Remark 3.4 for the connection between $R_{(1,1)}$ and $R_{(-1,1)}$, and [7, Lemma 4.1] and Proposition 3.1 again for the link with $F_{\mathrm{Be}^{\prime},\mathrm{Be}^{\prime}}$). Importantly, the solutions of the detailed balance condition for both $F_{\mathrm{Gam},\mathrm{Be}^{\prime}}$ and $F_{\mathrm{Be}^{\prime},\mathrm{Be}^{\prime}}$ are completely described in classical results, see Proposition 4.1 below. As indicated by the notation, the distributions appearing in these statements are either a gamma distribution or a beta prime distribution. (See Appendix A for a list of the various parametric families that appear in this article.) As a result, one is able to characterise the stationary measures for the original polymer models. We summarise the results in Figure 1; for precise claims, see Theorem 4.2 (and also [7]). As further background, we note that [8] contains an application of the stationarity property of the above polymer models to the understanding of their fluctuation exponents.

Towards describing the zero-temperature limits of the above-mentioned polymer models, we define, for each $\varepsilon>0$, a map $S_{\varepsilon}:\mathbb{R}\rightarrow(0,\infty)$ by setting

Moreover, slightly abusing notation, we write $S_{\varepsilon}(x_{1},x_{2},\dots,x_{n}):=(S_{\varepsilon}(x_{1}),S_{\varepsilon}(x_{2}),\dots,S_{\varepsilon}(x_{n}))$. The zero-temperature limit of the map $R_{({\alpha},{\beta})}$ at (1.7) is then given by the piecewise linear function $R^{\mathrm{zero}}_{({\alpha},{\beta})}$, as defined formally by

We highlight that whilst in this article we view the term ‘zero-temperature limit’ purely as a mathematical operation, it is naturally motivated by statistical physics considerations. Indeed, the polymer models described above will typically incorporate a temperature parameter in their definition, and the $\varepsilon\rightarrow 0$ limit of (1.9) represents taking this parameter to 0 in the sense that, just as the map $R_{({\alpha},{\beta})}$ gives a recursion equation for the partition function in the relevant positive-temperature case, the map $R^{\mathrm{zero}}_{({\alpha},{\beta})}$ describes an analogous recursion for the corresponding zero-temperature model. (See Section 2 for some concrete examples of polymer models and their zero-temperature limits.) We further note that the limiting procedure described at (1.9) is commonly referred to as ultra-discretisation in the discrete integrable systems literature.

The domain upon which the limit at (1.9) is defined will vary depending on the particular choice of $({\alpha},{\beta})$, and, in the $(1,-1)$ case, we will make a change of scale before taking the zero-temperature limit (see Subsection 2.2 for details). For the special cases of polymer models listed above, these limits again correspond to models that have been studied hitherto. Specifically:

• •

  $({\alpha},{\beta})=(0,1)$ corresponds to directed last passage percolation (LPP) with exponential/geometric waiting times, which essentially appeared in [20];

• •

  $({\alpha},{\beta})=(1,0)$ corresponds to directed first passage percolation (FPP) with exponential/geometric waiting times, as studied in the language of queuing systems in [18];

• •

  $({\alpha},{\beta})=(-1,1)$ or $({\alpha},{\beta})=(1,1)$ corresponds to the Bernoulli-exponential/geometric polymer of [28];

• •

  $({\alpha},{\beta})=(1,-1)$ corresponds to Bernoulli-exponential/geometric FPP, which was introduced as a zero-temperature limit of the ${\beta}$-RWRE in [5], and has also been called the river delta model (see [6]).

(See [27, Sections V.B.1, V.B.2 and V.B.3] for further background on the first three models, respectively.) Unlike in the positive-temperature case, the maps $R^{\mathrm{zero}}_{({\alpha},{\beta})}$ in these classes can not always be decomposed as per the picture at (1.5) directly, even by taking the map $F^{\mathrm{zero}}_{({\alpha},{\beta})}$ to be the zero-temperature limit of $F_{({\alpha},{\beta})}$. (See Remark 2.2 and the preceding discussion below.) However, by making certain changes of scale corresponding to those of the positive-temperature case, one can nonetheless identify a family of both continuous and discrete stationary measures for each $R^{\mathrm{zero}}_{({\alpha},{\beta})}$ by considering the detailed balance condition for one of the following bijections on $\mathbb{R}^{2}$:

which has inverse

and the involution

where we use the notation $x\wedge y:=\min\{x,y\}$ (and later also $x\vee y:=\max\{x,y\}$). Note that $F_{\mathrm{Gam},\mathrm{Be}^{\prime}}^{\mathrm{zero}}=F_{\mathrm{E},\mathrm{AL}}$ and $F_{\mathrm{Be}^{\prime},\mathrm{Be}^{\prime}}^{\mathrm{zero}}=F_{\mathrm{AL},\mathrm{AL}}$, with these zero-temperature limits being defined as at (1.9). See Proposition 3.5 for the links between maps, and Theorem 4.7 for our main result on the stationary measures of zero-temperature random polymer models, which we believe is new for the $({\alpha},{\beta})=(1,-1)$ case in particular¹¹1In [6, Section 1.3], it was written that the stationary measure of Bernoulli-exponential FPP, which is the zero-temperature limit of the beta polymer, was computed in [27]. However, this seems to stem from a conflation of Bernoulli-exponential FPP and the Bernoulli-exponential polymer, which was studied in [27]. We highlight that these models are not the same; whereas the first is a zero-temperature limit of the beta polymer, the second is a zero-temperature limit of the inverse-beta polymer. Moreover, in [27], it is stressed that the beta polymer is not present in the framework of that article, and ‘somehow lives in a different class’.. (See Figure 1 again.) Similarly to the positive-temperature case, as indicated by the notation, the distributions appearing in the latter result are either an exponential(/geometric) distribution or an asymmetric Laplace(/discrete asymmetric Laplace) distribution.

Two outstanding issues with the argument of the previous paragraph are as follows. Firstly, whilst we believe the detailed balance solutions that we present for $F_{\mathrm{AL},\mathrm{AL}}$ are the only ones, we do not yet have a proof of this. (For $F_{\mathrm{E},\mathrm{AL}}$, characterisation of solutions of the detailed balance equation is classical.) Secondly, since the links between the zero-temperature polymer setting and the basic bijections $F_{\mathrm{E},\mathrm{AL}}$ and $F_{\mathrm{AL},\mathrm{AL}}$ are in general not bijective, it is not possible to conclude that the stationary measures we identify for the polymer ones provide a complete characterisation of such. See Section 6 for more detailed comments in this direction.

As well as identifying the stationary measures of random polymer models, we investigate how these are related. To this end, we start by considering the connections between our basic maps, showing how each of $F_{\mathrm{Gam},\mathrm{Be}^{\prime}}$ (and its inverse), $F_{\mathrm{AL},\mathrm{AL}}$ and $F_{\mathrm{E},\mathrm{AL}}$ (and its inverse) can be obtained through certain limiting operations from $F_{\mathrm{Be}^{\prime},\mathrm{Be}^{\prime}}$, see Figure 2. (Cf. The centrality of the inverse-beta model in [27, Figure 1].) Furthermore, applying the same changes of variables, we can relate (some of the continuous) solutions of the detailed balance condition for these maps. Such observations naturally translate to the maps underlying the positive- and zero-temperature random polymer models considered in this article, and in some instances transfer to convergence of the partition function of the models in question. The latter links are shown by arrows on Figure 1, and discussed in more detail in Remarks 5.6, 5.8 and 5.13. Whilst we do give a new connection between the bijection $F_{({\alpha},{\beta})}$ and detailed balance solutions in the beta random polymer setting and those in the inverse-gamma setting (see Propositions 5.3(d) and 5.5(d)), we highlight that there remains a question concerning the relation of the partition function of these two models (and similarly in the zero-temperature setting).

The remainder of the article is organised as follows. Section 2 contains precise definitions of the random polymer models studied in this article. In Section 3, we explain how the underlying recursion maps relate to one of the bijections $F_{\mathrm{Gam},\mathrm{Be}^{\prime}}$, $F_{\mathrm{Be}^{\prime},\mathrm{Be}^{\prime}}$, $F_{\mathrm{E},\mathrm{AL}}$ and $F_{\mathrm{AL},\mathrm{AL}}$. In Section 4, we then apply the preceding results to obtain descriptions of stationary measures for the random polymer models. Following this, Section 5 features a discussion of various relations between the detailed balance solutions identified for the basic bijections, and between the stationary measures for the random polymer maps $R_{({\alpha},{\beta})}$, as well as some remarks on connections between the partition functions of the corresponding polymer models. We conclude in Section 6 with some conjectures and problems left open by this work. As noted above, we also include an appendix in which definitions of the various parametric families appearing in the discussion are collated.

As we commented in the introduction, there are four basic bijections that appear in this study: $F_{\mathrm{Gam},\mathrm{Be}^{\prime}}$, $F_{\mathrm{Be}^{\prime},\mathrm{Be}^{\prime}}$, $F_{\mathrm{E},\mathrm{AL}}$ and $F_{\mathrm{AL},\mathrm{AL}}$. The aim of this section is to explain how the bijections $F_{(0,1)}$, $F_{(1,0)}$, $F_{(-1,1)}$ (or $F_{(1,1)}$) and $F_{(1,-1)}$ arising in the positive-temperature polymer models can each be transformed to one of $F_{\mathrm{Gam},\mathrm{Be}^{\prime}}$, $F_{\mathrm{Gam},\mathrm{Be}^{\prime}}^{-1}$ or $F_{\mathrm{Be}^{\prime},\mathrm{Be}^{\prime}}$ by a bijective transformation of the state space. We will further present corresponding links between the zero-temperature maps $R^{\mathrm{zero}}_{(0,1)}$, $R^{\mathrm{zero}}_{(1,0)}$, $R^{\mathrm{zero}}_{(1,1)}$ and $\tilde{R}^{\mathrm{zero}}_{(1,-1)}$ and the bijections $F_{\mathrm{E},\mathrm{AL}}$, $F_{\mathrm{E},\mathrm{AL}}^{-1}$ or $F_{\mathrm{AL},\mathrm{AL}}$.