High-Temperature Scaling Limit for Directed Polymers on a Hierarchical Lattice with Bond Disorder

Abstract

Diamond “lattices” are sequences of recursively-defined graphs that provide a network of directed pathways between two fixed root nodes, A and B. The construction recipe for diamond graphs depends on a branching number $b\in {\mathbb {N}}$ and a segmenting number $s\in {\mathbb {N}}$, for which a larger value of the ratio s / b intuitively corresponds to more opportunities for intersections between two randomly chosen paths. By attaching i.i.d. random variables to the bonds of the graphs, we construct a random Gibbs measure on the set of directed paths by assigning each path an “energy” through summing the random variables along the path. For the case $b=s$, we propose a scaling regime in which the temperature grows along with the number of hierarchical layers of the graphs, and the partition function (the normalization factor of the Gibbs measure) appears to converge in law. We prove that all of the positive integer moments of the partition function converge in this limiting regime. The motivation of this work is to prove a functional limit theorem that is analogous to a previous result obtained in the $b<s$ case.

A Proof of Lemma 3.4

Proof of Lemma 3.4

We will apply a strong induction argument in ${\mathbf {m}}=2, 3, 4, \cdots$. Note that the base case $m=2$ holds trivially. Now let us assume for the purpose of induction that the statement of the lemma holds for all $m\in \{2,\dots , {\mathbf {m}}-1\}$, in other terms, there is a $c\equiv c(r,{\mathbf {m}})>0$ such that for any $r\in (-\infty ,\lambda ]$ and $m\in \{2,\ldots ,{\mathbf {m}}\}$ the inequality

$$\begin{aligned} \big |\varrho _ {k}^{(m)}\big (\beta _{n,r}^{(b)}\big )\big |\, \le \, c \Big (\varrho _ {k}^{(2)}\big (\beta _{n,r}^{(b)}\big )\Big )^{\frac{m}{2}} \end{aligned}$$

(A.1)

holds for large enough $n\in {\mathbb {N}}$ and all $1\le k\le n$.

By part (i) of Proposition 3.1, we have the recursive inequality

$$\begin{aligned} \big |\varrho _ {k+1}^{({\mathbf {m}})}\big (\beta _{n,r}^{(b)}\big ) \big |\, < \,&\frac{1}{b^{{\mathbf {m}}-2}}\big |\varrho _ {k}^{({\mathbf {m}})}\big (\beta _{n,r}^{(b)}\big )\big |\,+\, \big |\varrho _ {k}^{({\mathbf {m}})}\big (\beta _{n,r}^{(b)}\big ) \big |\,\Big |U_{\mathbf {m}}\Big ( \varrho _ {k}^{(2)}\big (\beta _{n,r}^{(b)}\big ), \cdots , \varrho _ {k}^{({\mathbf {m}})}\big (\beta _{n,r}^{(b)}\big ) \Big ) \Big | \nonumber \\&\,+\,\Big |V_{\mathbf {m}}\Big ( \varrho _ {k}^{(2)}\big (\beta _{n,r}^{(b)}\big ), \cdots ,\varrho _ {k}^{({\mathbf {m}}-1)}\big (\beta _{n,r}^{(b)}\big ) \Big )\Big | \,. \end{aligned}$$

(A.2)

Since the variable $\omega$ has finite exponential moments, we have that

$$\begin{aligned} \big |\varrho _ {0}^{({\mathbf {m}})}\big (\beta _{n,r}^{(b)}\big )\big | \,=\,\Bigg |{\mathbb {E}}\Bigg [\bigg (\frac{ \exp \big \{\omega \beta _{n,r}^{(b)} \big \} }{{\mathbb {E}}\big [ \exp \big \{\omega \beta _{n,r}^{(b)} \big \} \big ] }-1 \bigg )^{{\mathbf {m}}}\Bigg ]\Bigg | = O \Big (\frac{1}{n^{\frac{{\mathbf {m}}}{2}}} \Big ) \end{aligned}$$

(A.3)

is small with large n. Since $\varrho _ {n}^{(2)}\big (\beta _{n,r}^{(b)}\big )$ converges with large n to $R_b(r)$, and $R_b(r)=-\frac{\kappa _b^2}{r}+ o (r^{-1})$ for $-r\gg 1$ by part (iii) of Lemma 1.1, there is a $-{\widetilde{\lambda }}_1>2\kappa _b^2$ large enough so that when $r\in (-\infty ,{\widetilde{\lambda }}_1]$, then

$$\begin{aligned} \varrho _ {n}^{(2)}\big (\beta _{n,r}^{(b)}\big )\,<\,-\frac{2\kappa _b^2}{r}\,<\,1\, \end{aligned}$$

(A.4)

holds for large enough n.

By the induction assumption, the term $\big |V_{\mathbf {m}}\big ( \varrho _ {k}^{(2)}\big (\beta _{n,r}^{(b)}\big ), \cdots , \varrho _ {k}^{({\mathbf {m}}-1)}\big (\beta _{n,r}^{(b)}\big ) \big )\big |$ has the bound

$$\begin{aligned} \Big |V_{\mathbf {m}}\Big ( \varrho _ {k}^{(2)}\big (\beta _{n,r}^{(b)}\big ), \cdots , \varrho _ {k}^{({\mathbf {m}}-1)}\big (\beta _{n,r}^{(b)}\big ) \Big )\Big | \,\le&\,V_{\mathbf {m}}\bigg ( \varrho _{k}^{(2)}\big (\beta _{n,r}^{(b)}\big ), \cdots , \Big (\varrho _{k}^{(2)}\big (\beta _{n,r}^{(b)}\big )\Big )^{\frac{\mathbf {m-1}}{2}} \bigg ) \nonumber \\ \,\le&\, {\widehat{c}}\big [\varrho _{k}^{(2)}\big (\beta _{n,r}^{(b)}\big )\big ]^{\frac{{\mathbf {m}}}{2}} \,. \end{aligned}$$

(A.5)

If $r\in (-\infty ,{\widetilde{\lambda }}_1]$, then $\varrho _{k}^{(2)}\big (\beta _{n,r}^{(b)}\big )<1$ for large n and the above is bounded by a constant multiple ${\widehat{c}}$ of $\big [\varrho _{k}^{(2)}\big (\beta _{n,r}^{(b)}\big )\big ]^{{\mathbf {m}}/2}$ as a consequence of part (ii) of Proposition 3.1.

If $\big |\varrho _ {k}^{({\mathbf {m}})}\big (\beta _{n,r}^{(b)}\big ) \big | \le {\mathbf {x}}$ for some ${\mathbf {x}}<1$, the factor $\big |U_{\mathbf {m}}\big ( \varrho _ {k}^{(2)}\big (\beta _{n,r}^{(b)}\big ), \cdots , \varrho _ {k}^{({\mathbf {m}})}\big (\beta _{n,r}^{(b)}\big ) \big )\big |$ in (A.2) has a bound of the form

$$\begin{aligned} \Big |U_{\mathbf {m}}\Big ( \varrho _ {k}^{(2)}\big (\beta _{n,r}^{(b)}\big ), \cdots , \varrho _ {k}^{({\mathbf {m}})}\big (\beta _{n,r}^{(b)}\big ) \Big )\Big |\,\le \, U_{\mathbf {m}}\bigg ( \frac{2\kappa _b^2}{|r|}, \cdots , \Big (\frac{2\kappa _b^{2}}{|r|}\Big )^{\frac{{\mathbf {m}}-1}{2}}, {\mathbf {x}} \bigg ) \,\le \,{\mathbf {c}}{\mathbf {x}} \,+\, \frac{{\mathbf {c}}}{|r|}\,, \end{aligned}$$

(A.6)

where the first inequality holds by (A.4) for sufficiently large $n\in {\mathbb {N}}$ and all $1\le k \le n$ when $r\le {\widetilde{\lambda }}_2$. The second inequality holds for some ${\mathbf {c}}>0$ and all ${\mathbf {x}}<1$ and $r\in (-\infty , {\widetilde{\lambda }}_2]$ since the polynomial $U_{{\mathbf {m}}}$ has no constant term by (i) of Proposition 3.1. Pick some ${\mathbf {x}}$ small enough so that

$$\begin{aligned} \delta \,:=\, {\mathbf {c}}{\mathbf {x}}+\frac{1}{b^{{\mathbf {m}}-2}} \,<\,1 \,. \end{aligned}$$

Let ${\widehat{k}}_n\in {\mathbb {N}}$ be the smallest value such that $\big | \varrho _ {{\widehat{k}}_n}^{({\mathbf {m}})}\big (\beta _{n,r}^{(b)}\big )\big |> {\mathbf {x}}$.

Define $\lambda = \min ( {\widetilde{\lambda }}_1, {\widetilde{\lambda }}_2 )$. By the bounds (A.5) and (A.6), there is a $\delta \in (0,1)$ and a $C'>0$ such that for all $r\in (-\infty , \lambda )$ there is an $n\in {\mathbb {N}}$ large enough so that for all $k<\min (n,{\widehat{k}}_n)$

$$\begin{aligned} \big | \varrho _ {k+1}^{({\mathbf {m}})}\big (\beta _{n,r}^{(b)}\big )\big | \, \le \, \delta \big |\varrho _ {k}^{({\mathbf {m}})}\big (\beta _{n,r}^{(b)}\big )\big |\,+\,C'\Big (\varrho _{k}^{(2)}\big (\beta _{n,r}^{(b)}\big )\Big )^{\frac{{\mathbf {m}}}{2}}\,. \end{aligned}$$

(A.7)

Using (A.7) recursively, it follows that for $k\le \min (n,{\widehat{k}}_n)$

$$\begin{aligned} \big |\varrho _ {k}^{({\mathbf {m}})}\big (\beta _{n,r}^{(b)}\big )\big | \, \le&\, \delta ^{k}\big |\varrho _ {0}^{({\mathbf {m}})}\big (\beta _{n,r}^{(b)}\big )\big |\,+\,C' \sum _{j=0}^{k-1}\delta ^{k-1-j}\Big (\varrho _{j}^{(2)}\big (\beta _{n,r}^{(b)}\big )\Big )^{\frac{{\mathbf {m}}}{2}}\nonumber \\ \,\le&\, O \Big (\frac{1}{n^{\frac{{\mathbf {m}}}{2}}} \Big ) \,+\,\frac{C'}{1-\delta }\left( \varrho _{k}^{(2)}\big (\beta _{n,r}^{(b)}\big )\right) ^{\frac{{\mathbf {m}}}{2}}\nonumber \\ \,\le&\, C\Big (\varrho _{k}^{(2)}\big (\beta _{n,r}^{(b)}\big )\Big )^{\frac{{\mathbf {m}}}{2}}\,. \end{aligned}$$

(A.8)

The second inequality holds by (A.3) for the first term and since $\varrho _{k}^{(2)}\big (\beta _{n,r}^{(b)}\big )$ is increasing with $k\in {\mathbb {N}}$ for the second term. The third inequality holds for some $C>0$ since a multiple of $\big (\varrho _{k}^{(2)}\big (\beta _{n,r}^{(b)}\big )\big )^{\frac{{\mathbf {m}}}{2}}$ can be used to cover the error $O \big (n^{-\frac{{\mathbf {m}}}{2}} \big )$. To understand this, recall that $\varrho _{0}^{(2)}\big (\beta _{n,r}^{(b)}\big )=\kappa _b^2/n+ o (1/n)$ for $n\gg 1$ and, again, invoke that $\varrho _{k}^{(2)}\big (\beta _{n,r}^{(b)}\big )$ is increasing with k. If $-\lambda >0$ is large enough so that

$$\begin{aligned} \delta \,:=\, {\mathbf {c}}C\big (R_b(\lambda )\big )^{\frac{{\mathbf {m}}}{2}}+\frac{1}{b^{{\mathbf {m}}-2}} \,<\,1 \,, \end{aligned}$$

(A.9)

then for all $r\in (-\infty ,\lambda )$ the inequality ${\widehat{k}}_n>n$ will hold for sufficiently large n since $\varrho _{k}^{(2)}\big (\beta _{n,r}^{(b)}\big )$ converges to $R_b(\lambda )$ as $n\rightarrow \infty$. Hence, (A.8) will hold for all $k\le n$ when n is large enough.

The above argument proves the statement of the lemma is true for $\lambda \in {{\mathbb {R}}}$ sufficiently far in the negative direction so that (A.9) holds. For arbitrary $\lambda '\in {{\mathbb {R}}}$, pick $N\in {\mathbb {N}}$ large enough so that $\lambda =\lambda '-N$ satisfies (A.9). Then the reasoning above applies in the same way to show that ${\widehat{k}}_n> n-N$ for large n, and thus (A.8) will hold for all $1\le k\le n-N$. The inequality can then be extended to $n-N< k\le n$ through (A.2). $\square$