Positive Harmonic Functions on the Heisenberg group I

TL;DR

This paper classifies positive harmonic functions on the Heisenberg group specifically for the southwest measure, advancing understanding of harmonic analysis in non-commutative geometric settings.

Contribution

It provides a complete classification of positive harmonic functions on the Heisenberg group for a particular measure, which was previously uncharacterized.

Findings

Complete classification of positive harmonic functions for the southwest measure

New insights into harmonic analysis on the Heisenberg group

Foundation for further studies in non-commutative harmonic analysis

Abstract

We present the classification of positive harmonic functions on the Heisenberg group in the case of the southwest measure.

Equations

$$p(x,y,z)$$

$$p(x,y,z)=p(x-1,y,z-y)+p(x,y-1,z).$$

$$(x_0,y_0,z_0)(x,y,z)=(x_0+x,y_0+y,z_0+z+x_{0}y).$$

$$\mu_{0}=\delta_{a^{-1}}+\delta_{b^{-1}}\;\;\mbox{\rm where $a:=(1,0,0)$ and $b=(0,1,0)$.}$$

$$p(g)=p(a^{-1}g)+p(b^{-1}g)+1_{\{e\}}(g).$$

$$f(g)\ge P_{{\mu }_0}f(g)\text{mbox}\mathrm{where}{P}_{{\mu }_0}f(g):=f(a^{-1}g)+f(b^{-1}g).$$

$$p=\sum _{n\ge 0}{P}_{{\mu }_0}^n{1}_{\{e\}}$$

$$g$$

$$h(g)=P_{\mu_{0}}h(g),\;\mbox{\rm for all $g$ in $G$, or equivalently}\;\;$$

$$h(x,y,z)=h(x\!-\!1,y,z\!-\!y)+h(x,y\!-\!1,z),\;\;\mbox{\rm for all $(x,y,z)$ in $\mathbb{Z}^{3}$}.$$

$$h(x,y)=h(x\!-\!1,y)+h(x,y\!-\!1)\;,\;\;\mbox{\rm for all $(x,y)$ in $\mathbb{Z}^{2}$.}\;\;$$

$$\chi(x,y,z)=r^{x}s^{y}\;\;\;\mbox{\rm with $r,s>0$\; and \; $1/r+1/s=1$.}$$

$$h(y,z)=h(y,z\!-\!y)+h(y\!-\!1,z)\;,\;\;\mbox{\rm for all $(y,z)$ in $\mathbb{Z}^{2}$.}\;\;$$

$$p_y(z)$$

$$\sigma (x,y,z)=(y,x,xy-z).$$

$$h_1:=h_0\circ \sigma :(x,y,z)\mapsto p_x(xy-z)$$

$$\widetilde{\mu}_{0}=\tfrac{1}{2}(\delta_{a^{-1}}+\delta_{b^{-1}})\;\;\mbox{\rm where $a:=(1,0,0)$ and $b=(0,1,0)$}$$

$$\tilde{h}=P_{{\tilde{\mu }}_0}h\text{mbox}\mathrm{where}{P}_{{\tilde{\mu }}_0}h(x,y,z)=\frac{1}{2}\left(\tilde{h}(x-1,y,z-y)+\tilde{h}(x,y-1,z)\right),$$

$$h(x,y,z)$$

$$p(x,y,z)=p(y,x,z)=p(x,y,xy-z).$$

$$0\le p(x,y,z)-p(x,y,z-1)\le p(x,y-1,z).$$

$$n_1-n_2=m_1,\dots ,n_{y-1}-n_y=m_{y-1}\mathrm{and}{n}_1+\cdots +n_y=z$$

$$n_y=\frac{1}{y}(z-m_1-2m_2-\cdots -(y-1)m_{y-1})\ge 0\mathrm{and}$$

$$n_1=n_y+m_1+\cdots +m_{y-1}\le \frac{z}{y}+\frac{z}{y}\le x.$$

$$G^{+}:=\{g=(x,y,z)\in G\mid x,y\ge 0\mathrm{and}0\le z\le xy\}$$

$$c=ab{a}^{-1}{b}^{-1}=(0,0,1)$$

$$p(g)=|B_g|\mathrm{where}{B}_g:=\{w\in B\mid g_w=g\}.$$

$$\mathcal{R}$$

$$\pi (w,w^{\prime })=w\mathrm{and}\pi (w,w^{\prime })=w^{\prime }.$$

$$p(g)=|B_g|={\epsilon }_g+\sum _{(w,w^{\prime })\in {\mathcal{R}}_g}\frac{1}{f_w}$$

Full text

1. 1 Introduction

2. 1.1 The partition function $p(x,y,z)$ as a potential

3. 1.2 Construction of positive $\mu_{0}$-harmonic functions

4. 1.3 Classification of positive $\mu_{0}$-harmonic functions

5. 2 The partition function

6. 2.1 The unimodality of the partition functions

7. 2.2 The ratio limit theorem

8. 2.3 When the height of the rectangles is bounded

9. 2.4 Inner and outer corner of a partition

10. 2.5 Partitions with bounded number of corners

11. 2.6 When the height of the rectangles is unbounded

12. 3 Positive harmonic functions

13. 3.1 The function $p_{y}(z)$ as an harmonic function

14. 3.2 Harmonic functions that decay on cosets

15. 3.3 Using the ratio limit theorem

16. 3.4 Extension to finitely supported measures

1 Introduction

In this self-contained paper, we present the classification of the positive harmonic functions on the Heisenberg group $H_{3}(\mathbb{Z})$ in the special case of the southwest measure. This example is striking because the famous partition functions occur as positive harmonic functions. In this case, our main result tells us that roughly all positive harmonic functions are combinations of characters and partition functions (Theorem 1.1).

We will also explain with no proof how this result can be extended to finite positive measures on $H_{3}(\mathbb{Z})$ (Theorem 3.9).

1.1 The partition function $p(x,y,z)$ as a potential

We first introduce the “partition function” $p(x,y,z)$ for any integers $x$, $y$, $z$ in $\mathbb{Z}$.

1.1.1 The partition function

This function counts the “number of Young diagrams of area $z$”, also called “partitions of $z$”, included in a rectangle with side lengths $x$ and $y$. More precisely, when $x$, $y$ and $z$ are non-negative, one has

[TABLE]

and $p(x,y,z)=0$ otherwise. The integers $n_{i}$ are the lengths of the rows of the partition. By convention, for $x\geq 0$, one has $p(x,0,z)=0$ when $z\neq 0$, and $p(x,0,0)=1$.

This partition function satisfies the functional equation, for all $g=(x,y,z)$ in $\mathbb{Z}^{3}$, $g\neq(0,0,0)$,

[TABLE]

One checks it by splitting this set of partitions according to the colour of the lower-left case of the rectangle as in Figure 2.

1.1.2 The Heisenberg group

Recall that the Heisenberg group $G:=H_{3}(\mathbb{Z})$ is the set $\mathbb{Z}^{3}$ of triples seen as matrices (x,y,z):=\mbox{\scriptsize\left(!\begin{array}[]{ccc}1&x&z\ 0&1&y\ 0&0&1\end{array}!\right)}. It is endowed with the product

[TABLE]

Let $\mu_{0}$ be the southwest measure on $G$. It is given by

[TABLE]

Let $e:=(0,0,0)$ be the unity of $G$ and ${\bf 1}_{\{e\}}$ be the characteristic function of $\{e\}$. Equation (1.2) can be rewritten as, for all $g=(x,y,z)$ in $G$,

[TABLE]

In particular, the function $f=p$ satisfies

[TABLE]

This inequality (1.6) tells us that the function $f$ is a $\mu_{0}$-superharmonic function on the Heisenberg group $G$.

1.1.3 The potential

More precisely, the partition function $p(g)$ is the potential of $\mu_{0}$ at $e$. This means that one has the equality

[TABLE]

Indeed, as can be seen in Figure 3, for $g$ in $G$,

[TABLE]

A function $h$ on $G$ is said to be $\mu_{0}$-harmonic if it satisfies

[TABLE]

[TABLE]

1.2 Construction of positive $\mu_{0}$-harmonic functions

We want to classify all the positive111A function $f$ on $G$ is said to be positive if $f(g)\geq 0$ for all $g$ in $G$ and $f\not=0$. solutions of (1.6), i.e. all the positive $\mu_{0}$-superharmonic functions $h$ on $G$. We begin with five remarks.

1.2.1 Choquet Theorem

By a theorem of Choquet in [5], every positive superharmonic function $h$ is an average of extremal222A positive (super)harmonic function is said to be extremal if it cannot be written as the sum of two non-proportional positive (super)harmonic functions. positive superharmonic functions $h_{\alpha}$. Moreover when $h$ is harmonic the $h_{\alpha}$ are harmonic. By Riesz decomposition theorem [13, Thm 2.1.4], every positive $\mu_{0}$-superharmonic function can be written in a unique way as the sum of a potential333A potential is a function of the form $f=\sum_{n\geq 0}P_{\mu_{{}_{0}}}^{n}F$ for a positive function $F$ on $G$. and a positive $\mu_{0}$-harmonic function. Therefore it is enough to describe the extremal positive $\mu_{0}$-harmonic functions on $G$.

1.2.2 Choquet-Deny Theorem

If we look for a $\mu_{0}$-harmonic function $h$ which does not depend on $z$, Equation (1.9) becomes

[TABLE]

This equation tells us that the function $h$ is $\mu_{0}$-harmonic on the abelian quotient $\mathbb{Z}^{2}$ of $G$. According to a theorem of Choquet and Deny in [6], since the support of the measure $\mu_{0}$ spans the group $\mathbb{Z}^{2}$, every extremal positive $\mu_{0}$-harmonic function on this abelian group is proportionnal to a character444The proof is very short. One notices that Equality (1.10) gives a decomposition of $h$ as a sum of two positive harmonic functions and hence both of them are proportional to $h$:

[TABLE]

1.2.3 The partition function as a harmonic function

If we look for a $\mu_{0}$-harmonic function $h$ which does not depend on $x$, Equation (1.9) becomes

[TABLE]

A nice example is given by the partition function $(y,z)\mapsto p_{y}(z)$ where

[TABLE]

Hence the function $h_{0}(x,y,z):=p_{y}(z)$ is a $\mu_{0}$-harmonic function on $G$.

1.2.4 Margulis First Theorem

According to the first theorem of Margulis, a theorem he proved in [10] when he was not yet twenty, Choquet-Deny Theorem is still true on a finitely generated nilpotent group $G$ as soon as the support of the measure spans $G$ as a semigroup (See Fact 3.8). This is why it might look surprising at first glance, that there exists a positive $\mu_{0}$-harmonic function $h_{0}$ on $H_{3}(\mathbb{Z})$ which is not invariant by the center. The reason it exists is that the support of $\mu_{0}$ spans $G$ as a group but not as a semigroup. What is more surprising is that this “new” positive harmonic function $h_{0}$ is given by the famous partition function $p_{y}(z)$.

1.2.5 Switching and translating harmonic functions

We denote by $\sigma$ the automorphism of $G$ exchanging $a$ and $b$. It is given by

[TABLE]

Since the function $h_{0}$ is $\mu_{0}$-harmonic, the function

[TABLE]

is also $\mu_{0}$-harmonic. For $g_{0}$ in $G$, we denote by $\rho_{g_{{}_{0}}}:g\mapsto gg_{0}$ the right translation by $g_{0}$ on $G$. The translated functions $h_{0}\circ\rho_{g_{{}_{0}}}:g\mapsto h_{0}(gg_{0})$ and $h_{1}\circ\rho_{g_{{}_{0}}}:g\mapsto h_{1}(gg_{0})$ are also $\mu_{0}$-harmonic.

1.3 Classification of positive $\mu_{0}$-harmonic functions

We can now state our main result for the southwest measure $\mu_{0}$ introduced in (1.4).

1.3.1 Main result and strategy

Theorem 1.1**.**

*Let $h$ be an extremal positive $\mu_{0}$-harmonic function on the Heisenberg group $G:=H_{3}(\mathbb{Z})$. Then, up to a multiplicative scalar,

• either $h=\chi$ is a $\mu_{0}$-harmonic character $\chi(x,y,z)=r^{x}s^{y}$ as in (1.11),

• or $h=h_{0}\circ\rho_{g_{{}_{0}}}$ is a translate of the function $h_{0}(x,y,z):=p_{y}(z)$,

• or $h=h_{1}\circ\rho_{g_{{}_{0}}}$ is a translate of the function $h_{1}(x,y,z):=p_{x}(xy-z)$.*

This classification has been annouced on May $28^{th}$ 2019 in a short informal videotaped speech at the Cetraro Conference “Dynamics of group actions”.

As we will see the partition function $p(x,y,z)$ will play a crucial role in the proof of Theorem 1.1. Indeed, in Chapter 2, we will prove a ratio limit theorem for the partition function $p(x,y,z)$. In Chapter 3, we will deduce from this ratio limit theorem the proof of Theorem 1.1.

Notice that the positive $\mu_{0}$-harmonic function $h_{0}$ vanishes. In particular, it does not satisfy the Harnack inequality. This contrasts with the case studied in [10] where the support of $\mu$ spans $G$ as a semigroup.

In the last Section 3.4, we will present the classification of the positive $\mu$-harmonic functions, for all finitely supported measures $\mu$ on $G$.

1.3.2 Dealing with a probability measure

At first glance it might look a little bit weird to deal with $\mu_{0}$-harmonic function for a measure $\mu_{0}$ which is not a probability measure. We could have worked instead with the probability measure

[TABLE]

which is the law for the southwest random walk on $\mathbb{H}_{3}(\mathbb{Z})$. The $\widetilde{\mu}_{0}$-harmonic functions $\widetilde{h}$ on $G$ are the functions satisfying

[TABLE]

is the expected value of the function $h$ after one step of the random walk.

It is easy to see that

[TABLE]

Therefore, classifying positive $\mu_{0}$-harmonic functions is equivalent to classifying positive $\widetilde{\mu}_{0}$-harmonic functions. The main reason we are using $\mu_{0}$ instead of $\widetilde{\mu}_{0}$ is to get rid of all these factors $2^{-x-y}$.

1.3.3 Extremal superharmonic functions

We have seen in (1.5) that the partition function $p$ is $\mu_{0}$-superharmonic and more precisely that it is the potential of $\mu_{0}$ at $e$. For every $g_{0}$ in $G$, the function $p\circ\rho_{g_{{}_{0}}}$ is also a potential of $\mu_{0}$ at $g_{0}^{-1}$. By Riesz decomposition Theorem, those potentials are exactly the extremal positive $\mu_{0}$-superharmonic functions on $G$ which are not harmonic. Therefore,

every extremal positive $\mu_{0}$-superharmonic functions $f$ on $G$ which

is not harmonic is a translate $f=p\circ\rho_{g_{{}_{0}}}$ of the function $p(x,y,z)$.

We would like to end this introduction by pointing out other limit theorems for random walks on the Heisenberg group and other nilpotent groups as [8], [3], [4],[7] eventhough we will not use them here.

2 The partition function

The aim of this chapter is to prove the ratio limit theorem (Proposition 2.2) for the partition function $p(x,y,z)$.

2.1 The unimodality of the partition functions

We recall that, for $x,y,z\geq 0$, the partition function $p(x,y,z)$ counts the number of partitions of $z$ included in a rectangle with side lengths $x$ and $y$. See Definition (1.1.1) and Figure 1.

This function is non-zero for $0\leq z\leq xy$ and satisfies the equalities

[TABLE]

This function is well-studied. For instance one has

Fact 2.1**.**

(Cayley, Sylvester 1850)* The sequence $z\mapsto p(x,y,z)$ is unimodal, i.e. it is increasing for $z\leq xy/2$.*

The proof below relies on the theory of finite dimensional representations of the Lie algebra ${\mathfrak{s}}{\mathfrak{l}}(2,\mathbb{R})$. This proof is due to Hughes in [9]. See [12] for an elementary proof and [14, p. 522] for a survey of various generalizations.

Sketch of proof of Fact 2.1.

Let $n:=x+y$ and $(Y,H,X)$ be the principal ${\mathfrak{s}}{\mathfrak{l}}_{2}$-triple in the Lie algebra ${\mathfrak{g}}:={\mathfrak{s}}{\mathfrak{l}}(n,\mathbb{R})$ so that $H={\rm diag}(n\!-\!1,n\!-\!3,\ldots,-n\!+\!1)$. This Lie algebra ${\mathfrak{g}}$ has a natural representation in the space $V:=\Lambda^{x}(\mathbb{R}^{n})$. One checks that $p(x,y,z)=\dim V_{xy-2z}$ where $V_{\lambda}$ denotes the eigenspace of $H$ in $V$ for the eigenvalue $\lambda$. The theory of representations of ${\mathfrak{s}}{\mathfrak{l}}(2,\mathbb{R})$ tells us that for $\lambda>0$, one always has $\dim V_{\lambda}\leq\dim V_{\lambda-2}$. ∎

2.2 The ratio limit theorem

Here is the Ratio Limit Theorem for $p(x,y,z)$:

Proposition 2.2**.**

One has $\displaystyle\lim_{\begin{subarray}{c}z\rightarrow\infty\\ xy\!-\!z\rightarrow\infty\end{subarray}}\frac{p(x,y,z-1)}{p(x,y,z)}\;=\;1.$

This limit is taken along sequences of positive triples $(x,y,z)$ such that $z\rightarrow\infty$ and $xy\!-\!z\rightarrow\infty$.

With this generality this theorem seems to be new, eventhough there already exist very precise estimates of $p(x,y,z)$ in certain ranges. For instance, when $x,y\geq z$, the partition function $p(x,y,z)=p(z,z,z)$ depends only on $z$. It is the classical partition function $p(z)$ which admits a famous asymptotic expansion due to Hardy and Ramanujan in 1920 (See [1, Ch. 5]). These estimates have been extended to larger ranges of $(x,y,z)$ as in [15] and [11]. We will not use them.

The proof of Proposition 2.2 is tricky but elementary. The rough idea is to introduce a relation between the set of partitions $w$ of $z$ and the set of partitions $w^{\prime}$ of $z-1$ such that “most of the time” when $w$ and $w^{\prime}$ are related, they are related to approximately the same number of partitions (see Lemma 2.5).

Because of (2.1), we can assume that $y\leq x$ and $z\leq xy/2$.

2.3 When the height of the rectangles is bounded

In this section, we deal with the easy case when the height $y$ remains bounded.

Lemma 2.3**.**

For all $y\geq 1$, one has $\displaystyle\lim_{\begin{subarray}{c}x,z\rightarrow\infty\\ z\,\leq\,xy/2\end{subarray}}\frac{p(x,y,z-1)}{p(x,y,z)}=1.$

Note that in this limit $y$ is fixed, and $x$, $z$ go to $\infty$ with $z\leq xy/2$.

Proof of Lemma 2.3.

This follows from Lemma 2.4 and the inequalities

[TABLE]

The first inequality is the unimodality of the partition function.

For the second inequality, just notice that one can inject the set of partitions of $z$ of height exactly $y$ inside the set of partitions of $z-1$ of height at most $y$ by removing the last square in the bottom row of each partition. ∎

We have used the following Lemma.

Lemma 2.4**.**

$a)$* For all $x,y,z\geq 1$, one has $\;\;p(x,y,z)\leq z^{y-1}$.

$b)$ For all $y\geq 1$, there exists $\alpha_{{y}}>0$ such that, for all $x,z\geq 1$ with $z\leq xy/2$, one has $\;\;p(x,y,z)\geq\alpha_{y}\,z^{y-1}.$*

Proof of Lemma 2.4.

$a)$ The lengths of the last $y\!-\!1$ rows of the partition are bounded by $z-1$ and the first row is deduced from the others.

$b)$ Choose $y\!-\!1$ integers $m_{1},..,m_{y\!-\!1}$ in the interval $[0,\frac{z}{y^{2}}]$. and keep only those for which the system

[TABLE]

has a solution $(n_{1},\ldots,n_{y})$ in $\mathbb{Z}^{y}$. But then one has

[TABLE]

[TABLE]

This gives about $\frac{1}{y}(\frac{z}{y^{2}})^{y-1}$ partitions of $z$ with $x\geq n_{1}\geq\cdots\geq n_{y}\geq 0$. ∎

2.4 Inner and outer corner of a partition

We now introduce notations that will stengthen the connection between partitions and words in the Heisenberg group.

We recall that $a=(1,0,0)$ and $b=(0,1,0)$ are the generators of the Heisenberg group $G=H_{3}(\mathbb{Z})$. Let

[TABLE]

be the semigroup generated by $a$ and $b$ and let

[TABLE]

be the generator of the center $Z$ of $G$.

Let $B_{n}:=\{a,b\}^{n}$ be the set of finite words $w$ in $a$, $b$ of length $\ell_{w}=n$ and let $B:=\cup_{n\geq 0}B_{n}$. Using the product law in $G$, to each word $w\in B$, we can associate an element $g_{w}$ in $G^{+}$. The partition function gives the size of the fibers of this map :

[TABLE]

Indeed, as explained in Figure 3, when $g=(x,y,z)$, each word $w$ in $B_{g}$ corresponds uniquely to a partition of $z$ included in a rectangle with side lengths $x$ and $y$. We introduce now the following relation $\mathcal{R}$ on $B$,

[TABLE]

Let $\pi:\mathcal{R}\rightarrow B$ and $\pi^{\prime}:\mathcal{R}\rightarrow B$ be the two projections

[TABLE]

For $w$, $w^{\prime}$ in $B$, the cardinality of the fiber $f_{w}:=|\pi^{-1}(w)|$ is the number of pairs $ab$ occuring in the word $w$. The size $f_{w}$ is also the number of inner corners of the partition associated to $w$. See Figure 5. Similarly the cardinality of the fiber $f^{\prime}_{w^{\prime}}:=|\pi^{\prime-1}(w^{\prime})|$ is the number of pairs $ba$ occuring in the word $w^{\prime}$. It is equal to the number of outer corners of the partition associated to $w^{\prime}$.

The following lemma compares the size of these fibers.

Lemma 2.5**.**

$a)$* For all $(w,w^{\prime})\in\mathcal{R}$, one has $g_{w}=g_{w^{\prime}}c$.

$b)$ For all $(w,w^{\prime})\in\mathcal{R}$, one has $|f_{w}-f^{\prime}_{w^{\prime}}|\leq 2$.

In particular, one also has $f_{w}\leq 3f^{\prime}_{w^{\prime}}$.*

Proof of Lemma 2.5.

$a)$ This follows from the equality $c=aba^{-1}b^{-1}$.

$b)$ Comparing the number of pairs $ab$ and pairs $ba$ occuring in $w$ and in $w^{\prime}$, one gets $|f_{w}-f_{w^{\prime}}|\leq 1\;$ and $\;|f_{w^{\prime}}-f^{\prime}_{w^{\prime}}|\leq 1$. ∎

2.5 Partitions with bounded number of corners

We will need to control the number $p_{{}_{\leq i}}(x,y,z)$ of partitions of $z$ included in a rectangle with side length $x$, $y$ that have at most $i$ inner corner.

The following Lemma 2.6 tells us that $p_{{}_{\leq i}}(x,y,z)$ is negligeable compared to the total number of partitions $p(x,y,z)$.

Lemma 2.6**.**

For all $i\geq 0$, one has $\;\;\displaystyle\lim_{\begin{subarray}{c}x,y,z\rightarrow\infty\\ z\,\leq\,xy/2\end{subarray}}\frac{p_{{}_{\leq i}}(x,y,z)}{p(x,y,z)}=0.$

The limit is taken along sequences where all coordinates $x$, $y$, $z$ go to $\infty$ and $z\leq xy/2$.

Proof of Lemma 2.6.

Use the following slight upgrade of Lemma 2.4. ∎

Lemma 2.7**.**

$a)$* For all $x,y,z,i\geq 1$, one has $\;\;p_{{}_{\leq i}}(x,y,z)\leq(2z)^{2i}$.

$b)$ For all $j>1$, there exists $z_{0}=z_{0}(j)\geq 1$ such that, for all $x,y,z\geq 1$ with $4\,j\leq y\leq x$ and $z_{0}\leq z\leq xy/2$, one has $\;\;p(x,y,z)\geq z^{j}.$*

Proof of Lemma 2.7.

It is similar to Lemma 2.4.

$a)$ We can assume $x=y=z$. We want to choose integers $a_{1},\ldots,a_{i}\geq 1$ and $m_{1},\ldots,m_{i}\geq 0$, bounded by $z$ such that $a_{1}m_{1}+\cdots+a_{i}m_{i}=z$. There are at most $(2z)^{2i}$ possibilities.

$b)$ We give a rough count. Choose $L_{y}\leq y$ as large as possible such that, setting $\ell_{y}=[L_{y}/2]$ and $\ell_{x}=[z/L_{y}]$, one has $\ell_{y}\leq\ell_{x}\leq x/2$. There exists a partition $w_{0}$ of $z$ with $L_{y}$ rows and all of whose rows have length $\ell_{x}$ or $\ell_{x}\!+\!1$. For every sequence $\ell_{x}>m_{1}\geq\cdots\geq m_{\ell_{{}_{y}}}\geq 0$, we can modify this partition $w_{0}$ by adding $m_{j}$ spots to the $j^{\rm th}$ highest row of $w_{0}$ and removing $m_{j}$ spots to the $j^{\rm th}$ lowest row of $w_{0}$, for all $j\leq\ell_{y}$. This gives $N$ different partitions of $z$ where $N:=\binom{\ell_{x}+\ell_{y}-1}{\ell_{y}}\geq\max(2,\ell_{x}/\ell_{y})^{\ell_{y}}$. Hence, one has $p(x,y,z)\geq N$.

First case : when $z\leq y^{2}/2$.   In this case, we have $L_{y}=[\sqrt{2z}]$.

One gets $N\geq 2^{\ell_{y}}\geq 2^{\sqrt{z}/2}\geq z^{j}$.

Second case : when $z\geq y^{2}/2$.   In this case, we have $L_{y}=y$.

If $z\leq y^{4}$, one gets $N\geq 2^{\ell_{y}}\geq 2^{\sqrt[4]{z}/4}\geq z^{j}$.

If $z\geq y^{4}$, one gets $N\geq(\frac{\ell_{x}}{\ell_{y}})^{\ell_{y}}\geq(\frac{z}{y^{2}})^{\ell_{y}}\geq{\sqrt{z}}^{\,\ell_{y}}\geq z^{y/4}\geq z^{j}$. ∎

2.6 When the height of the rectangles is unbounded

We can now explain the proof of the ratio limit theorem.

Proof of Proposition 2.2.

By (2.1) and Lemma 2.3, we can assume that $x,y,z$ are going to $\infty$ with $y\leq x$ and $z\leq xy/2$. For $g=(x,y,z)$ in $G^{+}$, one sets $\mathcal{R}_{g}:=\{(w,w^{\prime})\in\mathcal{R}\mid g_{w}=g\},$ and one computes

[TABLE]

where $\varepsilon_{g}=1$ if $\mathcal{R}_{g}=\emptyset$ and $\varepsilon_{g}=0$ otherwise. Similarly, by Lemma 2.5.$a$, one has

[TABLE]

where $\varepsilon^{\prime}_{g}=0$ or $1$. Combining (2.4), (2.5) and Lemma 2.5.$b$, one gets

[TABLE]

We recall that $p_{{}_{\leq i}}(g)$ is the number of $w$ with $f_{w}\leq i$. Therefore, one has

[TABLE]

We let $x,y,z$ go to infinity with $z\leq xy/2$. According to Lemma 2.6, for all $i\geq 1$, the ratios $p_{{}_{\leq i}}(g)/p(g)$ converge to [math]. Therefore

[TABLE]

and therefore the sequence $\displaystyle\frac{p(gc^{-1})}{p(g)}$ converges to $1$ as required. ∎

3 Positive harmonic functions

We now start the classification of extremal positive $\mu_{0}$-harmonic functions $h$. In Section 3.1, we deal with the case where $h$ has a non-zero limit along an orbit of $a^{-1}$ or $b^{-1}$. In Sections 3.2 and 3.3, we deal with the case where $h$ goes to zero along all orbits of $a^{-1}$ and $b^{-1}$. In Section 3.4 we present the generalization of this classification to any finitely supported measure $\mu$ on $G$.

3.1 The function $p_{y}(z)$ as an harmonic function

In this section we characterize the functions $h_{0}\circ\rho_{g_{{}_{0}}}$ and $h_{1}\circ\rho_{g_{{}_{0}}}$ among extremal positive $\mu_{0}$-harmonic functions by their behavior along the orbits $a^{-\mathbb{N}}g_{0}$ and $b^{-\mathbb{N}}g_{0}$ of $G$.

We recall that $a=(1,0,0)$ and $b=(0,1,0)$ are the generators of the Heisenberg group $G=H_{3}(\mathbb{Z})$, that $\mu_{0}=\delta_{a^{-1}}+\delta_{b^{-1}}$, and that $h_{0}$ and $h_{1}$ are the $\mu_{0}$-harmonic functions $h_{0}(x,y,z)=p_{y}(z)$ and $h_{1}(x,y,z)=p_{x}(xy-z)$.

We first begin by an alternative construction of the function $h_{0}$. Let $H_{0}$ be the abelian subgroup of $G$ generated by $a$ and let $\psi_{0}:={\bf 1}_{H_{0}}$ be the characteristic function of $H_{0}$. One has

[TABLE]

Lemma 3.1**.**

One has the equality $\displaystyle\;\;h_{0}\;=\;\lim_{n\rightarrow\infty}P^{n}_{\mu_{0}}\psi_{0}.$

Remark 3.2*.*

Since the function $\psi_{0}$ is $\mu_{0}$-subharmonic, i.e. $\;\psi_{0}\leq P_{\mu_{0}}\psi_{0}\,,$ the sequence $n\mapsto P_{\mu_{0}}^{n}\psi_{0}$ is increasing.

Proof of Lemma 3.1.

One can compute explicitely this function $P_{\mu_{0}}^{n}\psi_{0}$. It does not depend on $x$. Indeed $P_{\mu_{0}}^{n}\psi_{0}(x,y,z)$ is the number of ways of writing the element $(n-y,y,z)$ as a word $w$ of length $n$ in $a$ and $b$. This proves the equality, involving the partition function,

[TABLE]

Letting $n$ go to $\infty$, we conclude using (1.2.3). ∎

Lemma 3.3**.**

Let $g_{0}\in G$ and $h$ be an extremal positive $\mu_{0}$-harmonic function on $G$ such that $\displaystyle\limsup_{n\rightarrow\infty}h(a^{-n}g_{0})>0$. Then one has $h=\lambda\,h_{0}\circ\rho_{g_{{}_{0}}}$ with $\lambda>0$.

In particular, the positive $\mu_{0}$-harmonic function $h_{0}\circ\rho_{g_{{}_{0}}}$ is extremal.

Proof of Lemma 3.3.

We can assume that $g_{0}=e$. Since the function $h$ is positive and $\mu_{0}$-harmonic, the sequence $n\mapsto h(a^{-n})$ is decreasing. Hence it has a limit $\lambda$. By assumption, this limit $\lambda$ is positive. By construction, one has the equality $h\geq\lambda\psi_{0}$. Since $h$ is $\mu_{0}$-harmonic, one also has the inequality $h\geq\lambda P^{n}_{\mu_{0}}\psi_{0}$ for all $n\geq 0$. Therefore, by Lemma 3.1, one gets $h\geq\lambda h_{0}$. Since $h$ is extremal, it has to be proportional to $h_{0}$ and therefore one has $h=\lambda h_{0}$.

It remains to check that $h_{0}$ is extremal. If one can write $h_{0}=h^{\prime}_{0}+h^{\prime\prime}_{0}$ with both $h^{\prime}_{0}$ and $h^{\prime\prime}_{0}$ positive $\mu_{0}$-harmonic, for at least one of them, say $h^{\prime}_{0}$, the sequence $h^{\prime}_{0}(a^{-n})$ does not converge to [math] for $n\rightarrow\infty$. Hence, by the previous discussion, $h^{\prime}_{0}$ is proportional to $h_{0}$. This proves that $h_{0}$ is extremal. ∎

Exchanging the role of $a$ and $b$ we get

Corollary 3.4**.**

Let $h$ be an extremal positive $\mu_{0}$-harmonic function on $G$ such that $\displaystyle\limsup_{n\rightarrow\infty}h(b^{-n}g_{0})>0$. Then one has $h=\lambda\,h_{1}\circ\rho_{g_{{}_{0}}}$ for some $\lambda>0$.

In particular, the positive $\mu_{0}$-harmonic function $h_{1}\circ\rho_{g_{{}_{0}}}$ is extremal.

3.2 Harmonic functions that decay on cosets

We now discuss positive harmonic functions on $G$ that decay to [math] along the orbits $a^{-\mathbb{N}}g_{0}$ and $b^{-\mathbb{N}}g_{0}$.

Let $G_{n}$ be the subset of $G$ consisting of elements of “degree” $n$,

[TABLE]

By definition and by (1.7), a positive $\mu_{0}$-harmonic function $h$ on $G$ satisfies the equality, for all $n\geq 1$,

[TABLE]

For an integer $A>0$, we set

[TABLE]

The following lemma tells us when the contributions of $G_{n,A}$ and $G^{\sigma}_{n,A}$ in Formula (3.1) is negligeable.

Lemma 3.5**.**

Let $h$ be a positive $\mu_{0}$-harmonic function on $G$ such that,

[TABLE]

Then, for all $A>0$ and $g_{0}$ in $G$, one has

[TABLE]

Proof of Lemma 3.5.

It is enough to prove (3.4) with $g_{0}=e$. Moreover, since $G^{\sigma}_{n,A}$ is the image of $G_{n,A}$ by the involution $\sigma$ which exchanges $a$ and $b$, it is enough to prove (3.4) with $g\in G_{n,A}$. Equivalently, it is enough to prove

[TABLE]

Note that, when $n>A$, every word $w\in B_{n,A}$ can be written as

[TABLE]

with $s\in B_{A+1}$ a word of length $A\!+\!1$. See Figure 6.

One splits the set $B_{n,A}$ according to $m\geq A$ or $m<A$. Therefore, for $n\geq 2A$, one has the inclusion

[TABLE]

Therefore, using (3.1), one gets the inequalities

[TABLE]

For all $\varepsilon>0$, we choose $A$ large enough so that, by the second assumption (3.3), one has $h(b^{-A})\leq\varepsilon$. Then the last sum is a sum over the fixed finite set $B_{2A}$, and, by the first assumption (3.3), this last sum converges to [math] when $n$ goes to infinity. This proves (3.5) as required. ∎

3.3 Using the ratio limit theorem

Combining Lemma 3.5 with the ratio limit theorem we can finish the last case of the proof of Theorem 1.1.

Lemma 3.6**.**

Let $h$ be a positive $\mu_{0}$-harmonic function on $G$ such that, for all $g_{0}$ in $G$, $\displaystyle\lim_{n\rightarrow\infty}h(a^{-n}g_{0})=\lim_{n\rightarrow\infty}h(b^{-n}g_{0})=0$. Then $h$ is invariant by the center $Z=c^{\mathbb{Z}}$ of $G$.

Proof of Lemma 3.6.

Using (3.1) with $g_{0}$ and $g_{0}c$, we compute,

[TABLE]

We fix $\varepsilon>0$. According to the ratio limit theorem (Proposition 2.2), there exists an integer $A>0$ such that, for all $g=(x,y,z)$ in $G^{+}$ with $z\geq A$ and $xy-z\geq A$, one has

[TABLE]

Therefore, using (3.6), (3.7) and Definition (3.2), one gets

[TABLE]

By (3.1), the first term is equal to $\varepsilon h(g_{0})$. Therefore using twice Lemma 3.5 and letting $n$ go to infinity, one gets $|h(g_{0})-h(g_{0}c)|\leq\varepsilon\,h(g_{0})$. Since $\varepsilon$ is arbitrary small, this proves that $h(g_{0})=h(g_{0}c)$ as required. ∎

Corollary 3.7**.**

Let $h$ be an extremal positive $\mu_{0}$-harmonic function on $G$ such that, for all $g_{0}$ in $G$, $\displaystyle\lim_{n\rightarrow\infty}h(a^{-n}g_{0})=\lim_{n\rightarrow\infty}h(b^{-n}g_{0})=0$. Then $h$ is a character of $G$.

In particular, every $\mu_{0}$-harmonic character of $G$ is an extremal positive $\mu_{0}$-harmonic function.

Proof of Corollary 3.7.

By Lemma 3.6, the function $h$ is $\mu_{0}$-harmonic on the abelian group $G/Z$. By Choquet-Deny Theorem, it is a character.

It remains to check that a $\mu_{0}$-harmonic character $\chi$ is extremal. Assume that $\chi=h^{\prime}+h^{\prime\prime}$ with both $h^{\prime}$ and $h^{\prime\prime}$ positive $\mu_{0}$-harmonic. For all $g_{0}$ in $G$, the sequences $h^{\prime}(a^{-n}g_{0})$ and $h^{\prime}(b^{-n}g_{0})$ converge to [math] for $n\rightarrow\infty$. Hence, by the previous discussion and by Choquet Theorem, the function $h^{\prime}$ is an integral $h^{\prime}=\int_{C}\chi^{\prime}{\rm d}\sigma(\chi^{\prime})$ where $\sigma$ is a finite positive measure on the set $C$ of (harmonic) character $\chi^{\prime}$ of $G$. Since $h^{\prime}\leq\chi$, the measure $\sigma$ must be supported by $\chi$. This proves that $\chi$ is extremal. ∎

This ends the proof of Theorem 1.1.

3.4 Extension to finitely supported measures

In this section we give the classification of the positive $\mu$-harmonic functions on the Heisenberg group for all finitely supported measure $\mu$.

Let $G=H_{3}(\mathbb{Z})$ be the Heisenberg group and $S$ be a finite subset of $G$. We denote by $G_{S}$ the subgroup of $G$ generated by $S$. Let $\mu=\sum_{s\in S}\mu_{s}\delta_{s}$ be a positive measure on $G$ with support $S$.

We recall that a function $h$ on $G$ is said to be $\mu$-harmonic if

[TABLE]

We want to describe the cone $\mathcal{H}^{+}$ of positive $\mu$-harmonic functions $h$ on $G$. By Choquet Theorem, it is enough to describe the extremal rays of this cone $\mathcal{H}^{+}$.

There are two constructions of extremal positive $\mu$-harmonic functions.

3.4.1 The harmonic characters $\chi$

The $\mu$-harmonic characters are the characters $\chi:G\rightarrow\mathbb{R}_{>0}$ of $G$ such that $\sum_{s\in S}\mu_{s}\,\chi(s)=1$. Such a function $h=\chi$ is an extremal positive $\mu$-harmonic function on $G$ which is invariant by the center $Z$ of $G$.

We now recall Margulis Theorem which tells us that this first construction is the only possible when $G_{\mu}^{+}=G$.

Fact 3.8**.**

(Margulis)* Let $\mu$ be a finite positive measure on a finitely generated nilpotent group $G$. If the semigroup $G_{\mu}^{+}$ generated by the support of $\mu$ is equal to $G$, then every extremal positive $\mu$-harmonic function $h$ on $G$ is a character.*

Sketch of proof of Fact 3.8 for $G=H_{3}(\mathbb{Z})$.

Because of the assumption $G_{\mu}^{+}=G$, we can assume that $\mu_{c}>0$ and $\mu_{a}>0$. The first part of the argument is as in the abelian case : since $\;h(x,y,z)\geq\mu_{c}\,h(x,y,z+1)$, these two $\mu$-harmonic functions are proportional and we get that, for some $t>0$, one has $h(x,y,z)=h(x,y,0)t^{z}$. We now want to prove that $t=1$.

Let $K_{t}$ be the set of positive harmonic functions $h_{0}(x,y,z)=\psi_{0}(x,y)t^{z}$ with $h_{0}(e)=1$. Since $G_{\mu}^{+}=G$, the convex set $K_{t}$ is compact for the pointwise convergence. The element $a\in G$ acts continuously by “right-translation and renormalization” on $K_{t}$. By Schauder fixed point theorem, this action has a fixed point $h_{0}$ in $K_{t}$. It can be written as $h_{0}(x,y,z)=r^{x}\varphi_{0}(y)t^{z}$ with $r>0$. But then one writes $h_{0}(g)\geq\mu_{a}h_{0}(ag)$ for all $g$ in $G$, or equivalently $\varphi_{0}(y)\geq\mu_{a}r\varphi_{0}(y)t^{y}$ for all $y\in\mathbb{Z}$. This proves that $t=1$. ∎

When $G_{\mu}^{+}\neq G$, a second construction is possible.

3.4.2 The functions $h_{S_{{}_{0}},\chi_{{}_{0}}}$

induced from a harmonic character

Let $S_{0}\subset S$ be an abelian subset. Denote by $\mu_{S_{{}_{0}}}:=\sum_{s\in S_{{}_{0}}}\mu_{s}\,\delta_{s}$ the measure restriction of $\mu$ to $S_{0}$. Let $\chi_{0}$ be a $\mu_{S_{{}_{0}}}$-harmonic character of $G_{S_{{}_{0}}}$. We extend $\chi_{0}$ as a function

[TABLE]

on $G$ which is [math] outside $G_{S_{{}_{0}}}$. This function $\psi_{0}$ is $\mu$-subharmonic, so that the sequence $P_{\mu}^{n}\psi_{0}$ is increasing. We set

[TABLE]

We can tell exactly for which pairs $(S_{0},\chi_{0})$ the function $h_{S_{{}_{0}},\chi_{{}_{0}}}$ is finite. In this case the function $h_{S_{{}_{0}},\chi_{{}_{0}}}$ is an extremal positive $\mu$-harmonic function on $G$.

We can now state the extension of Theorem 1.1 to a more general finitely supported measure $\mu$ on $G$.

Theorem 3.9**.**

Let $G=H_{3}(\mathbb{Z})$ and $\mu$ be a positive measure on $G$ whose finite support $S$ generates the group $G$. Then every extremal positive $\mu$-harmonic function $h$ on $G$ is proportional either to a character $\chi$ of $G$ or to a translate $h_{S_{{}_{0}},\chi_{{}_{0}}}\circ\rho_{g_{{}_{0}}}$ of a function induced from a harmonic character.

Corollary 3.10**.**

*Let $G=H_{3}(\mathbb{Z})$, $Z$ its center and $\mu$ a probability measure on $G$ whose finite support $S$ generates the group $G$. The following are equivalent:

$(i)$ Every positive $\mu$-harmonic function on $G$ is $Z$-invariant.

$(ii)$ $G_{\mu}^{+}$ contains two non-central elements whose product is in $Z\smallsetminus\{0\}$.*

Theorem 3.9 and Corollary 3.10 will be proven in the sequel paper [2].

We will also see that on the nilpotent group of rank 4 with cyclic center, there exist extremal positive harmonic functions which are neither an harmonic character nor a function induced from a harmonic character.