Analogously, $v_2$ has external assets of $a^x_{v_2}$ in $\F_{\Delta}$. Hence, every bank has the same external assets in $\F$ and in $\F_{\Delta}$. However, since $a_{v_1} + \delta_{v_1} > a_{v_1}$ or $a_{v_2}+ \delta_{v_2} > a_{v_2}$ the payments in the clearing state in $\F_{\Delta}$ are strictly higher than in $\F$. This contradicts the fact that the clearing state $\vecp$ is the maximal fixed point. With the same argument, $\delta_{\svt} \neq \Delta_{\tvt} + \Delta_{t_{u_2}}$ follows.

Suppose there exists a positive debt swap for a financial network $\F$, in which the total assets of both creditor banks $v_1$ and $v_2$ strictly increase by $\delta_{v_1}>0$ and $\delta_{v_2}>0$. In $\HH$ the debt swap is simulated by increasing the total assets of source nodes $\svo$ and $\svt$ by $\delta_{v_1}$ and $\delta_{v_2}$, respectively. We will show that there exists a suitable set of values for $\delta_{v_1}$ and $\delta_{v_2}$ such that $v_1$ receives $\delta_{v_1}$ additional incoming payments. Intuitively, then, the pairs $u_1$, $v_1$ and $u_2, v_2$ are each contained in the same cycle. Hence, payments could be raised along the cycles, which contradicts maximality of the clearing state.

For a financial network $\F$, suppose the debt swap of $e_1$ and $e_2$ is positive, i.e., $\atvo > \aovo$ and $\atvt > \atvo$. We set $\delta_{v_1}$ and $\delta_{v_2}$ to the increase in total assets of $v_1$ and $v_2$ due to the swap, i.e., $\delta_{v_1} \coloneqq \atvo - \aovo$ and $\delta_{v_2} \coloneqq \atvt - \aovt$. Now consider the financial network $\HH$ corresponding to $\F$ and raise the external assets of $s_{v_1}$ by at most $\delta_{v_1}$ and external assets of $s_{v_2}$ by at most $\delta_{v_2}$. Let $g_{t_{u_1}}: [0, \delta_{v_1}] \times [0, \delta_{v_2}] \to [0, \infty]$ be the increase in total assets at $t_{u_1}$ as a function of the increase in external assets at $\svo$ and $\svt$ in the clearing state of network $\HH$. We define $g_{t_{v_1}}$ similarly. The payment functions of all banks are non-decreasing and continuous, and so are $g_{t_{v_1}}$ and $g_{t_{u_1}}$. In particular, $g_{t_{v_1}} (0,0) = g_{t_{u_1}}(0,0) = 0$.

The sum $g_1 = g_{t_{u_1}} + g_{t_{v_1}}$ is non-decreasing and continuous as well, with $g_1(0, 0) = 0$. By non-expansivity, we have that $g_1(x,y) \le x + y$ for all $(x,y) \in [0, \delta_{v_1}] \times [0, \delta_{v_2}]$. We will now show that whenever the debt swap of $e_1$ and $e_2$ is positive, there exists an $(x,y) \in [0, \delta_{v_1}] \times [0, \delta_{v_2}]$ with $(x,y)>(0,0)$, i.e., $x>0$ and $y>0$, such that $g_1(x,y) = x$. This is a contradiction to Lemma <ref> (2).

We proceed in three steps.

* $g_1(x,0)<x$ for all $x \in (0,\delta_{v_1}]$:

Since $g_1(x, y) \leq x + y$, it follows that $g_1(x, 0) \leq x$, for all $x>0$. Now $g_1(x,0)=x$ for some $x > 0$ would imply $(\tvo, t_{u_1})$-linearity of $s_{v_1}$ (Lemma <ref> (1)) and a contractiction to Lemma <ref> (2).

* $g_1(c,y) > c$ for some $c\in (0, \delta_{v_1}], y \in (0, \delta_{v_2}]$:

Fix a value $y \in (0, \delta_{v_2}]$ with $y \leq \delta_{v_1}$. Then clearly $g_1(0,y)\geq 0$. Moreover, $g_1(0,y)>0$ since otherwise $s_{v_2}$ would be $(\tvt,t_{u_2})$-linear (Lemma <ref> (1)) and we obtain a contradiction to Lemma <ref> (2). Note that $g_1(0,y) \le y \le \delta_{v_1}$. Since $g_1(0,y)>0$ and $g_1$ is monotone, we obtain $g_1\left(g_1(0, y), y\right) \geq g_1(0, y)$.
Using the notation $c \coloneqq g_1(0,y)$, we have $0 < c \leq \delta_{v_1}$ and $g_1(c,y) \leq c$. Since $g_1(c,y) = c$ implies a contradiction to Lemma <ref> (2), we must have $g_1(c,y) > c$.

* $g_1(x,y)=x$ for some $(x,y) \in (0,\delta_{v_1}] \times (0,\delta_{v_2}]$:

(i) and (ii) imply that there exists a pair $(c,y) \in [0,\delta_{v_1}] \times [0,\delta_{v_2}]$ such that $g_1(c,0) < c$ and $g_1(c,y)>c$. By the intermediate value theorem, there exists a $d \in [0,y]$ satisfying $g_1(c,d)=c$. Observe that in particular $y > 0$ holds true.

Theorem <ref> shows that both creditors participating in a debt swap cannot simultaneously strictly increase their total assets. In the next lemmas, we consider the payments on the edges $e_1= (u_1,v_1)$ and $e_2=(u_2,v_2)$ before and the edges $\sig{e}_1 = (u_1,v_2)$ and $\sig{e}_2 = (u_2,v_1)$ after a generic swap $\sigma$. The first lemma shows they cannot both be strictly increase.

For every financial network, there is no debt swap $\sigma$ such that $\sig{p}_{\sig{e}_1} > p_{e_1}$ and $\sig{p}_{\sig{e}_2} > p_{e_2}$.

For a given network $\F$ consider the corresponding network $\G$ after applying source-sink equivalency to the edges $e_1$ and $e_2$. This procedure adds the new banks $t_{u_1},t_{u_2}$ and replaces the edges $(u_1,v_1)$ and $(u_2,v_2)$ by $(u_1,t_{u_1})$ and $(u_2,t_{u_2})$. Additionally, the external assets of $v_1$ and $v_2$ are updated to $a^x_{v_1}+p_{e_1}$ and $a^x_{v_2}+p_{e_2}$, respectively. Analogously, define $\sig{\G}$ for $\sig{\F}$.
Observe that $\G$ and $\sig{\G}$ only differ in the external assets of $v_1$ and $v_2$. Hence, $\G$ is a pre-network of $\sig{\G}$ and, by monotonicity of the allocation rule, both $v_1$ and $v_2$ strictly profit from the edge swap. This is a contradiction to Theorem <ref>.

For every semi-positive debt swap $\sigma$, we have either $\sig{p}_{\sig{e}_1} > p_{e_1}$ and $\sig{p}_{\sig{e}_2} \leq p_{e_2}$, or $\sig{p}_{\sig{e}_2} > p_{e_2}$ and $\sig{p}_{\sig{e}_1} \leq p_{e_1}$.

Proposition <ref> shows that $\sig{p}_{e_1} > p_{e_1}$ and $\sig{p}_{e_2} > p_{e_2}$ can never be satisfied simultaneously. If $\sig{p}_{\sig{e}_1} \leq p_{e_1}$ and $\sig{p}_{\sig{e}_2} \leq p_{e_2}$, then the same source-sink transformation as in the proof of Proposition <ref> shows that the network after swapping $\sig{\G}$ is a pre-network of $\G$. By Corollary <ref> such a swap cannot be semi-positive. Consequently, either $\sig{p}_{\sig{e}_1} > p_{e_1}$ and $\sig{p}_{\sig{e}_2} \leq p_{e_2}$, or $\sig{p}_{\sig{e}_2} > p_{e_2}$ and $\sig{p}_{\sig{e}_1} \leq p_{e_1}$.

A semi-positive swap $\sigma$ Pareto-improves the total assets of $v_1$ and $v_2$. More significantly, every semi-positive debt swap yields a Pareto-improvement in the entire network, i.e., no bank in the financial network is harmed while at least one bank strictly profits. Interestingly, the converse also holds: Every Pareto-improving debt swap is semi-positive.

First, let the debt swap of $e_1$ and $e_2$ be Pareto-improving. For a contradiction, assume it is not semi-positive. Then, w.l.o.g. the following two cases can occur: (1) $\atvo < \aovo$ and $\atvt > \aovt$, or (2) $\atvo \leq \aovo$ and $\atvt \leq \aovt$. Towards (1), it is easy to see that $v_1$ is harmed by the swap. Hence, it is not Pareto-improving. For (2), consider the two networks $\HH$ and $\sig{\HH}$ after applying source-sink equivalency to the swapping edges and $v_1$ and $v_2$ in $\F$ and $\sig{\F}$. Since $\HH$ and $\sig{\HH}$ only differ in the external assets of $v_1$ and $v_2$, $\HH$ is a pre-network of $\sig{\HH}$. Consequently, by Corollary <ref>, the total assets of all banks in $\HH$ are at least as high as those $\sig{\HH}$. Hence, no bank strictly profits by the debt swap, a contradiction.

For the other direction, suppose the debt swap of $e_1$ and $e_2$ is semi-positive and w.l.o.g. $\atvo > \aovo$ and $\atvt = \aovt$. Then, by Lemma <ref>,  $\HH$ is a pre-network of $\sig{\HH}$, and thus, by Corollary <ref>, the total assets of all banks after the swap are at least as high as those before the swap. Thus, the debt swap is a Pareto-improvement.

§ OPTIMIZATION

In this section, we study (global) optimization problems based on sequences of debt swaps. In particular, for a given initial network, we are interested in constructing sequences of debt swaps to a reachable network that maximizes either the resulting total assets of a specific bank or the sum of total assets for all banks. We consider sequences of semi-positive swaps, $v$-improving swaps, arbitrary swaps, and also a restriction where a specific bank $v$ must be involved in every swap.

Before considering sequences of swaps, we prove a slightly more general result. Therefore, consider the  defined in Section <ref> without the neighborhood relation. Instead of local optima we are now interested in global ones.

In the same way, we define a  problem, where instead of the assets of $v$ the objective function is the sum of total assets of all banks.

We again assume to have access to a clearing oracle as discussed in Section <ref>. Similar to the property in the proof of Theorem <ref>, our reductions always ensure that every reachable network has a clearing state that is independent of the allocation rule and can be computed in polynomial time. As such, all results in this section apply beyond edge-ranking rules to all monotone allocation rules.

and  are -hard.

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[circle, draw, inner sep=4] (v) [right=7cm of u2] $v$;
(ancher) [right=of v] ;
(wdots) [above=0.25cm of ancher] $\vdots$;
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[->] (u1) to node[below, midway] $M$ (S1);
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[->] (S1) to node[below, pos=0.2] $M_1$ (x1);
[->] (S1) to node[below=0.1cm, pos=0.2] $M_1$ (x3);
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[->] (S2) to node[below=0.15cm, pos=0.2] $M_2$ (xk);
[->] (Sl) to node[below=0.2cm, pos=0.1] $M_{\ell}$ (x2);
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[->] (x1) to node[below=0.2cm, pos=0.2] $1$ (v);
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[->] (w1) to node[below=0.3cm, pos=0.3] $M$ (v);
[->] (w2) to node[below=0.2cm, pos=0.3] $M$ (v);
[->] (wl) to node[below=0.2cm, pos=0.3] $M$ (v);

Example construction for reduction with  where $\lambda=\ell-c$. Edge labels indicate liabilities.

We first show the statement for . Consider an instance of the  problem with items $\mathcal{X}=\{x_1,x_2,\dots,x_k\}$ and sets $S_1,S_2,\dots,S_{\ell}$ such that $\bigcup_{j \in [\ell]}S_j = \mathcal{X}$, where $[\ell]:= \{1,2,\dots,\ell\}$. We construct an instance of  with banks $v$, $x_i$ for every item and $S_j$ for every set. Moreover, add banks $u_1,u_2,\dots,u_{\ell}$ and $w_1,w_2,\dots,w_{\ell-c}$ for some integer $c$ satisfying $0<c<\ell$.

Similar to the proof of Theorem <ref> we define a value for every set $S_j$ in binary encoding. First, let $d$ be the smallest integer satisfying $2^{d} \geq \max_j |S_j|$ and $d > \log \ell$. Then, set the $d$-th lowest bit to $1$ and encode index $j$ using the lowest $\log \ell$ bits. All other bits are set to $0$. For simplicity, denote the resulting value by $M_j$ for every $j$ and let $M=\max_j M_j$. Observe that all values $M_j$ are pairwise different and lower bounded by $\max_j |S_j|$. The number of bits is polynomial in the input size.

Banks $u_j$ and $w_j$ each have no incoming but one outgoing stub with weight $M$ where the external assets of $u_j$ are set to the same value. If item $x_i$ is element of set $S_j$ in the input instance add an incoming stub to bank $x_i$ and an outgoing stub to bank $S_j$ each with weight $M_j$. Moreover, every bank $S_i$ has one incoming stub with weight $M$. Additionally, include an outgoing stub with weight 1 to each $x_i$ and $k$ incoming stubs with weight $1$ to $v$. Finally, add $\ell-c$ incoming stubs to $v$ with weight $M$.

Banks $S_j$ are the only banks with more than one outgoing stub. We will argue below that every $S_j$ either has no assets or enough to settle all debt. For this reason, the clearing state can be computed efficiently for every monotone allocation rule.

Clearly, the construction can be performed in polynomial time with respect to the input size of the  instance. Figure <ref> illustrates an example construction.

By construction, every consistent set of edges must contain $(S_j,x_i)$ with weight $M_i$ if element $x_i$ is included in set $S_j$. Moreover, since only banks $x_i$ have an outgoing stub with weight 1 and $v$ has $k$ incoming stubs with weight 1, edges $(x_i,v)$ must be contained in every set of edges. Finally, banks $w_i$ and $u_i$ each must have an edge either towards $v$ or a bank $S_j$. If bank $S_j$ has an incoming edge from a bank $w_i$ it has total assets of zero, or otherwise enough assets to pay off all debt if it has an incoming edge from $u_i$.

We claim that there exists a set cover with size $c$ if and only if there exists a consistent set of edges such that $v$ has total assets of $M(\ell-c)+k$ in the corresponding network.

Suppose there exists a set cover $\mathcal{C}\subset [\ell]$ with $|\mathcal{C}|= c$ for the input instance. Then, for every $j \in \mathcal{C}$ include an edge from $u_j$ to $S_j$. As a result, $S_j$ has incoming payments of $M$. For the remaining $\ell-c$ banks $S_j$, choose an incoming edge from a bank $w_i$ each. Hence, no bank $S_j$ with $j \notin \mathcal{C}$ has positive incoming payments. Observe that there are exactly $\ell-c$ many remaining banks $u_i$. For every one of them include the edge $(u_i,v)$ yielding incoming assets of at least $M(\ell-c)$ for $v$ in the resulting network. Observe that the set of banks $S_j$ with incoming assets corresponds to the collection $\mathcal{C}$ covering all elements. Hence, by construction of $M_j$ every bank $x_i$ receives incoming assets of at least 1 and forwards the payments to $v$. In conclusion, $v$ has total assets of $M(\ell-c)+k$.

For the other direction, assume there exists a network with consistent set of edged where $v$ has total assets of $M(\ell-c)+k$. Because the weights of edges $(x_i,v)$ are restricted to 1 and for sufficiently large $M$, bank $v$ can only achieve incoming assets of at least $M(\ell-c)$ if $\ell-c$ banks $u_i$ pay all their assets directly to $v$. The remaining assets of $k$ then must be payments from banks $x_i$. Hence, every $x_i$ receives payments of at least 1. Consequently, there exists a collection of banks $S_j$ with cardinality $c$ covering all $x_i$.

To show the statement for $\MaxAssets$, add $B$ banks $b_1,b_2,\dots,b_B$
where $b_i$ has an incoming stub with weight $B+i$ and an outgoing stub with weight $B+i+1$, for $i<B$. Add an outgoing stub with weight $B+1$ to $v$ and incoming stub with weight $2B$ to $b_B$.
Observe that sufficiently large $B$ there exists exactly two stubs with weight $B+i$. Hence, there exists only one consistent set of edges that joins the stubs to form a path from $v$ to $b_B$. Every bank $b_i$ on the path has the same total assets as $v$. When $B$ is sufficiently large, say, $B = 3(k+\ell)$, then maximizing the sum of total assets in the network reduces to maximizing the total assets of $v$.

The reduction directly implies the following hardness of approximation.

and  are -hard.

Recall that  is -hard in 3-regular graphs [3]. Interpreting a 3-regular graph with $n'$ nodes as an instance of set cover, we obtain $\ell = n'$ sets and $k = 3n'$ elements. Clearly, every feasible set cover in these instances has size $\Theta(n')$. Applying the reduction from Theorem <ref>, we observe that a set cover of size $c$ corresponds to bank $v$ obtaining assets $M(\ell - c) + k$, where $M > \ell$. As such, in these instances the assets are in $M \cdot \Theta(n')$ for every consistent network corresponding to a feasible set cover. As a consequence, the constant-factor approximation gap of  translates to a constant-factor gap for optimizing the assets of $v$. Moreover, since the adaptation to maximizing the sum of all assets requires only $\Theta(k + \ell) = \Theta(|V|)$ additional banks, the result extends to this case as well.

We proceed to show how this result implies hardness results for sequences of swaps. In the instances constructed in the last proof, set $S_j$ is included in the cover $\mathcal{C}$ if and only if bank $S_j$ has positive incoming assets. Every network with consistent set of edges corresponds to some collection of sets $\mathcal{C}$, and vice versa. Suppose we are given such a network for a collection $\mathcal{C}$. We can perform a debt swap where the debtor of one bank $S_j$ is swapped from a bank $w_h$ to a bank $u_i$ or the other way around. In this way, we remove/add $S_j$ from/to the corresponding collection of subsets. Thus, we can reach the network corresponding to the optimal set cover $\mathcal{C}^*$ via a sequence of at most $|(\mathcal{C} \cup \mathcal{C}^*) \setminus (\mathcal{C} \cap \mathcal{C}^*)|$ many (arbitrary, not necessarily $v$-improving) debt swaps.

For a given financial network with monotone allocation rules, an efficient clearing oracle, and a bank $v$, it is -hard to compute a sequence of arbitrary debt swaps such that (1) the total assets of $v$ are maximal or (2) the sum of total assets is maximal.

We strengthen this result as follows. Suppose we start with the collection $[\ell]$, i.e., all sets are in the collection. The optimal set cover $\mathcal{C}^*$ can be obtained by iterative removal of sets from $[\ell]$. Each removal represents a debt swap in the corresponding financial network. It is straightforward to verify that each such debt swap is $v$-improving. Hence, from the initial network corresponding to $[\ell]$, there is a sequence of at most $\ell$ $v$-improving debt swaps to the network corresponding to $\mathcal{C}^*$.

For a given financial network with monotone allocation rules, an efficient clearing oracle, and a bank $v$, it is -hard to compute a sequence of $v$-improving debt swaps such that (1) the total assets of $v$ are maximal or (2) the sum of total assets is maximal.

Semi-Positive Swaps
In contrast to $v$-improving swaps, the reduction in Theorem <ref> does not readily extend to sequences of semi-positive swaps. In the previous section, we proved a number of positive results for semi-positive swaps (and edge-ranking rules).

Here we show hardness of approximation for maximizing assets via choosing an optimal sequence of semi-positive swaps.
Our construction exploits semi-positivity more directly since it restricts the set of reachable networks from a particular initial network – solving only the  problem without this restriction is easy in the instances we consider.

For a given financial network with monotone allocation rules, an efficient clearing oracle and a bank $v$, it is -hard to compute a sequence of semi-positive debt swaps such that (1) the total assets of $v$ are maximal and (2) the sum of total assets is maximal.

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[->] (v1) to[in=140, out=160, looseness=2.2] node[left, midway] $1$ (v);
[->] (v) to[bend left] node[above=0.1cm, midway] $k$ (u);

Example construction for reduction with  where edge labels indicate liabilities.

Towards (1), consider an instance of the  problem with $k$ nodes and $\ell$ edges. We construct a financial network with banks $v_1,v_2,\dots,v_k$ representing the nodes and banks $e_1,e_2,\dots,e_\ell$ representing the edges in the given instance. Additionally, add banks $v,u,x_1,x_2,\dots,x_k$ and $y_1,y_2,\dots,y_k$.

Further, if node $v_i$ is incident to edge $e_j$ in the input instance then include $(v_i,e_j)$ with weight 1 in the financial network. Additionally, every bank $v_i$ has an edge $(v_i,v)$ with weight 1. Bank $u$ has incoming edges $(e_j,u)$ with weight $1$ for all $j \in [\ell]$ and $(v,u)$ with weight $k$. Finally, add edges $(y_i,u)$ and $(x_i,u_i)$ for $i \in [k]$ each with weight $d_{v_i}+1$, respectively, where $d_{v_i}$ is the degree of $v_i$ in the input. Set the external assets of every bank $y_i$ to $d_{v_i}+1$. An example of the construction can be seen in Figure <ref>.

It can be easily verified that in all networks constructed via semi-positive swaps, every bank $v_i$ with at least two outgoing edges will either have total assets of 0 or receive sufficient assets to clear all debt. As a consequence, the clearing state of every network can be computed efficiently and is independent of the allocation rule. Thus, the result applies for all monotone allocation rules.

We now show that there exists an independent set with cardinality $c$ if and only if there exists a sequence of semi-positive swaps with resulting total assets of $c$ for $v$.
Suppose there exists an independent set $I$ with cardinality $c$. Swap the edges $(x_i,v_i)$ with $(y_i,u)$, for each $i \in I$. Then, every bank $v_i$ pays assets of 1 to $v$ and distributes the remaining assets of $d_{v_i}$ fully among banks $e_j$. Since $I$ is an independent set, the payments by each $v_i$ get fully forwarded to $u$ via the incident banks $e_j$. As a result, all external assets are still forwarded to $u$ and hence all performed swaps are semi-positive. At the same time, $v$ receives total assets of $c$.

For the other direction assume there exists a sequence of semi-positive debt swaps such that the resulting total assets of $v$ are given by $c$. First, let us observe that no swap of edges with weight 1 can be semi-positive. The only debt swap where all creditors and debtors are pairwise distinct is given by two edges $(v_i,e_j)$ and $(v_{i'},e_{j'})$. However, since such a swap cannot create a new cycle and there exists no path from $e_j$ to $e_{j'}$ and vice versa, no bank can strictly profit while the other bank is not harmed. Hence, it suffices to restrict our attention to swaps between $u$ and a bank $v_i$.

Bank $v$ can only receive payments of 1 from each bank $v_i$. Thus, at least $c$ of the banks $v_i$ must have positive total assets. This implies that $u$ and at least $c$ banks $v_i$ must have executed semi-positive swaps. This only happens if all such banks $v_i$ forward all assets to $u$. As a direct implication, no two distinct banks $v_i$ and $v_{i'}$ pay assets to the same bank $e_j$. In conclusion, the banks represent an independent set of cardinality $c$ in the input instance.

For statement (2) we slightly adapt the construction. Replace the edge $(v,u)$ by a path of $B+1$ edges and $B$ additional banks $w_1,w_2,\dots,w_B$. Assign each edge $(w_i,w_{i+1})$ weight $B+i$, edge $(v,w_1)$ with weight $B$ and $(w_B,u)$ with weight $2B$. Indeed, it suffices to set $B$ to a sufficiently large value of, say, $B = 6k^2$. When executing one additional swap with a node $v_i$ and $u$, the sum of assets of banks $w_i$ increases by at least $6k^2$ whereas the sum of assets in the remaining graph is upper bounded by $5k^2+k$. For this reason, it is always optimal to perform debt swaps to maximize the assets of $v$ and, thus, the cardinality of the corresponding independent set. Maximizing the sum of total assets reduces to maximization of assets of $v$.

The  problem cannot be approximated within $\Omega(k^{1-\epsilon})$, unless =  where $k$ is the number of nodes[16]. In our reduction, $v$ has total assets of $c$ if and only if there exists an independent set of size $c$ in the given  instance. If the input instance of  consists of $k$ nodes, then the number $n$ of banks in the construction is bounded by $O(k^2)$. Therefore, it is straightforward to argue that, for any constant $\varepsilon > 0$, we obtain -hardness of computing a $O(n^{1/2-\varepsilon})$-approximation to the maximal total assets of a given bank $v$ that can be achieved using a sequence of semi-positive debt swaps. The argument applies also when maximizing the sum of all assets, since the total number of banks in the resulting network is still bounded by $O(k^2)$.

For a given financial network with monotone allocation rules, an efficient clearing oracle and a bank $v$, it is -hard to compute an $n^{1/2 - \varepsilon}$-approximation to (1) the maximal total assets of $v$ and (2) the maximal sum of total assets that can be obtained by a sequence of semi-positive debt swaps, for every constant $\varepsilon > 0$.

The next theorem shows that the problem remains -hard even under the restriction that $v$ is involved in every swap and every semi-positive swap is also $v$-improving.

For a given financial network with monotone allocation rules, an efficient clearing oracle, and a bank $v$, it is -hard to compute a sequence of $v$-improving semi-positive debt swaps with creditor $v$ such that (1) the total assets of $v$ are maximal and (2) the sum of total assets is maximal.

For statement (1), consider an instance of  with items $i \in [k]$ and integer values $a_i \in \NN$. Moreover, denote the sum of values by $A$, i.e., $A = \sum_{i \in [k]} a_i$.

For our reduction, we define a financial network with banks $v,u,a_1,a_2,\dots,a_k$ and $k$ auxiliary banks $s_1,s_2,\dots,s_k$. For every $i \in [k]$ there are edges $(a_i,u)$ and $(s_i,v)$ with weights $M>\max_{j \in [k]} a_j$. There is a single edge $(v,u)$ with weight $A/2$. Moreover, bank $a_i$ has external assets $a_i$, for every $i \in [k]$. Clearly, the construction can be computed efficiently in the size of the instance of .

Every bank has at most one outgoing edge and thus pays all assets towards that edge until the debt is settled. The clearing state is the same for every monotone allocation rule and can be computed efficiently.

We show that there is a solution to the instance of  if and only if there is a sequence of $v$-improving semi-positive debt swaps with creditor $v$ such that the total assets of $v$ are $A/2$.

First, assume there exists a partition in two sets $\mathcal{A}_1,\mathcal{A}_2 \subset [k]$ such that $\mathcal{A}_1 \cup \mathcal{A}_2 = [k]$, $\mathcal{A}_1 \cap \mathcal{A}_2 = \emptyset$ and $\sum_{i \in \mathcal{A}_1} a_i = A/2 = \sum_{i \in \mathcal{A}_2} a_i$. Hence, when performing debt swaps of edges $(a_i,u), (s_i,v)$, for $i \in \mathcal{A}_1$, then $v$ has total assets of $A/2$. By construction, all assets of $v$ are payed to $u$. Thus, all swaps are semi-positive.

For the other direction, assume there exists a sequence of $v$-improving semi-positive debt swaps with creditor $v$ such that the total assets of $v$ are given by $A/2$. Then clearly, there exists a set of banks $\mathcal{A} \in [k]$ where total assets add up to $A/2$. There exists a partition $\mathcal{A}, \bar{\mathcal{A}}=\{i \notin \mathcal{A} \mid i \in [k]\}$ for the given input.

Statement (2) follows when including $B$ additional banks $w_b$ on the path from $v$ to $u$ with edge weight $A/2$ for edge $(v,w_1)$ and $B+b$ for every edge $(w_b,\cdot)$ with $b \in [B]$. This construction does not allow any additional debt swaps. Since the total assets of every bank $w_b$ equal those of $v$, maximizing the sum of total assets reduces to maximizing assets of $v$ for sufficiently large $B$.

For financial networks with edge-ranking rules we can show a slightly more elaborate result.

For a given financial network with edge-ranking rules and a bank $v$, it is strongly -hard to compute a sequence of $v$-improving semi-positive debt swaps with creditor $v$ such that (1) the total assets of $v$ are maximal and (2) the sum of total assets is maximal.

[>=stealth', shorten >=1pt, auto,
node distance=1.5cm, scale=1,
transform shape, align=center,
bank/.style=circle, draw, inner sep=1]
[bank] (u1) at (0,0) $u_1$;
[bank] (u2) [right = of u1] $u_2$;
[bank] (u3) [right = of u2] $u_3$;
[bank] (u4) [right = of u3] $u_4$;
(mdots) [right = of u4] $\cdots$;
[bank] (ul) [right = of mdots] $u_l$;

[circle, draw, inner sep=2.5pt] (v) [above =1cm of u4] $v$;
[circle, draw, inner sep=2.5pt] (r) [below =1cm of u4] $r$;

[bank] (s1) [right =0.7cm of u2] $s_2$;
[bank] (s2) [right =0.7cm of u4] $s_4$;
[rectangle, draw=black, inner sep=2.5pt] (xs1) [below=0.05cm of s1] $1$;
[rectangle, draw=black, inner sep=2.5pt] (xs2) [below=0.05cm of s2] $1$;

[bank] (a3) [below=0.5cm of r] $a_3$;
[bank] (a2) [left=1cm of a3] $a_2$;
[bank] (a1) [left=1cm of a2] $a_1$;
(ldots) [right=0.5cm of a3] $\cdots$;
[bank] (ak) [right=0.5cm of ldots] $a_{k}$;
[rectangle, draw=black, inner sep=2.5pt] (xa1) [below=0.05cm of a1] $a_1$;
[rectangle, draw=black, inner sep=2.5pt] (xa2) [below=0.05cm of a2] $a_2$;
[rectangle, draw=black, inner sep=2.5pt] (xa3) [below=0.05cm of a3] $a_3$;
[rectangle, draw=black, inner sep=2.5pt] (xak) [below=0.05cm of ak] $a_k$;

(anchor) [above=0.5cm of v] ;
[bank] (axk) [left=0.5cm of anchor] ;
(adots) [left=0.5cm of axk] $\cdots$;
[bank] (ax2) [left=0.5cm of adots] ;
[bank] (ax1) [left=1cm of ax2] ;
[bank] (sx1) [right=0.5cm of anchor] ;
[bank] (sx2) [right=1cm of sx1] ;
(sdots) [right=0.5cm of sx2] $\cdots$;
[bank] (sxl) [right=0.5cm of sdots] ;

[->] (u1) to node[left=0.4cm, midway] $M$ (r);
[->] (u3) to node[left=0.1cm, midway] $M$ (r);
[->] (ul) to node[left=0.2cm, midway] $M$ (r);
[->] (v) to node[left=0.3cm, pos=0.7] $1$,$T$ (u1);
[->] (v) to node[left=0.2cm, pos=0.7] $2$,$1$ (u2);
[->] (v) to node[left=0.1cm, pos=0.7] $3$,$T$ (u3);
[->] (v) to node[left, pos=0.7] $4$,$1$ (u4);
[->] (v) to node[left=0.3cm, pos=0.7] $l$,$T$ (ul);

[->] (s1) to node[below, pos=0.4] $d$ (u2);
[->] (s2) to node[below, pos=0.4] $d$ (u4);

[->] (a1) to node[left=0.2cm, midway] $c$ (r);
[->] (a2) to node[left=0.1cm, midway] $c$ (r);
[->] (a3) to node[left, midway] $c$ (r);
[->] (ak) to node[left=0.2cm, midway] $c$ (r);

[->] (ax1) to node[left=0.3cm, midway] $c$ (v);
[->] (ax2) to node[right=0.2cm, midway] $c$ (v);
[->] (axk) to node[right, midway] $c$ (v);
[->] (sx1) to node[right, midway] $d$ (v);
[->] (sx2) to node[right=0.1cm, midway] $d$ (v);
[->] (sxl) to node[right=0.2, midway] $d$ (v);

Schematic construction for the reduction from  where blue edge labels indicate edge-ranking rules and black labels indicate edge weights.

We show statement (1) by reduction from . An instance of  consists of a set of $k$ elements such that $k$ is an integer multiple by $3$. Each item $i \in [k]$ has a value $a_i \in \NN$. We denote the sum of values by $A = \sum_{i \in [k]} a_i$ and let $T =  \frac 3k A$. For each value it holds that $\frac T4 < a_i < \frac T2$. The goal is to decide whether or not the given set of $k$ items can be partitioned into $k$ sets $S_1,\ldots,S_k$ with $\sum_{i \in S_j}a_i = T$, for each $j \in [k]$. Due to the value restriction, each set must have exactly three items.

For an instance of , we construct a financial network as follows. First, add a root $r$ and a bank $a_i$ with external assets $a_i$, for every item $i$. Additionally, every bank $a_i$ has a single outgoing edge $(a_i,r)$ with edge weight $c \geq \frac T2$. Construct bank $v$ and $u_1, u_2, \dots, u_l$, for $l = 2\frac k3-1$, together with edges $(v,u_h)$. For all odd numbers $h$, set the weight of $(v,u_h)$ to $T$ and add the edge $(u_h,r)$ with weight $M > T$. Instead, set the weight of $(v,u_h)$ to $1$, for all even numbers $h$, and include a source $s_h$ with external assets of $1$ and edge $(s_h,u_h)$ and weight $d>1$. Finally, add $k$ auxiliary banks, each with a single outgoing edge to $v$ and weight $c$, and $\frac l2$ auxiliary banks with outgoing edge to $v$ and weight $d>1$. The values $T,M,d$ and $c$ are chosen to be distinct. $v$ is the only bank with more than one outgoing edge, and it distributes all assets according to an edge-ranking rule with $\pi_v(h)=u_h$ for all $h$. That means, first $v$ fully pays off debt to $u_1$, then to $u_2$,..., until $v$ runs out of funds. A schematic construction can be seen in Figure <ref>.

Note that every bank has at most one outgoing edge. As such, the clearing state is independent of the allocation rule and can be determined easily.

We show that there is a solution to the  instance if and only if there is a sequence of $v$-improving semi-positive debt swaps resulting in total assets of at least $A+\frac l2-1$ for $v$.

First, assume there exists a solution to the  instance. Then, swap all edges incident to banks $a_i \in S_1$ with incident edges of $v$. This results in total assets of $T$ for $v$ and thus she can exactly pay off $(v,u_1)$. Clearly, the described swaps are all semi-positive and $v$ is the benefiting creditor. Now, execute a swap with creditors $v$ and $u_2$. Since, this results in additional assets of $1$ in $v$ and the second edge has payments zero and is active, the swap is semi-positive. When repeatedly applying this process for the remaining sets $S_j$, bank $v$ has swapped all incoming edges in the initial network. Consequently, the total assets of $v$ are given by $A+ \frac l2 -1$.

For the other direction, assume there exists a sequence of $v$-improving semi-positive debt swaps resulting in total assets of at least $A+ \frac l2 -1$ for $v$. Because the graph is cycle-free, all assets must originate from external assets. In particular, observe that the sum of external assets in the network is given by $A+ \frac l2 -1$ and hence all banks with external assets must pay all assets to $v$. Clearly, this is only the case if all banks with external assets are swapped to be adjacent to $v$.
In the initial network, due to the allocation rule $v$ can only perform semi-positive swaps with $r$. Moreover, $v$ and $r$ can switch edges as long as the residual capacity of $(v,u_1)$ is sufficiently large. Only when the total assets of $v$ are exactly $T$, can $v$ fully pay off the first edge towards $u_1$ and execute a semi-positive swap with $u_2$. Otherwise, a fraction of the additional assets would either be payed to $u_1$ or $u_3$. These arguments can be repeated for all remaining outgoing edges of $v$. As a result, $v$ and $r$ can only swap all their initial incoming edges if the external assets $a_i$ can be grouped into sets with sum of value exactly $T$. Since $\frac T4 < a_i < \frac T2$, every set has cardinality three.

Statement (2) follows by the same construction as for (1). We claim that there exists a solution to the given instance of  if and only if there exists a sequence of $v$-improving semi-positive debt swaps resulting in at least $4A+3(\frac l2 -1)$ sum of total assets.

If there exists a solution, then proceed as for (1). Clearly, the total assets of all sources add up to $A+\frac l2 -1$ and the total assets of $v$ are given by same amount. Additionally, $v$ pays all assets to banks $u_h$ whereas all payments of $A$ are forwarded to $r$. In total, assets of $4A+3(\frac l2 -1)$ are achieved.

Now, assume there exists a sequence of semi-positive swaps resulting in a sum of total assets of at least $4A+3(\frac l2 -1)$. Clearly, this value can only be achieved when $v$ settles all debt.

By the same argument as before, this directly implies existence of a solution to .

Observe that our constructions in the proofs of Theorems <ref>, <ref>, <ref> and <ref> lead to classes of cycle-free financial networks. Therefore, all -hardness results in this section apply even under the restriction that the underlying graph structure of the financial network is a DAG.


$\inf_{(a,b)\in(0,1]\times(0,1]}\widetilde{D}(a,b;s)=\inf_{x\in(0,1]}\widetilde{D}(x^{1/2},1;s)=\inf_{a\in(0,1]}\widetilde{D}(a,1;s).$
(3.26)

For $(a,u)\in(0,1]\times(0,s]$, we consider
$\widetilde{D}(a,1;u)=p(a^{7}u^{5})+13p(a^{3}u^{3})-46p(a^{4}u)$. For
$y\in(0,s]$, we consider the curve $C^{\prime}_{y}$ defined by $a^{4}u=y$ or
equivalently, by $a=(y/u)^{1/4}$. Note that

$(0,1]\times(0,s]=\bigcup_{y\in(0,s]}\\{(a,u)\in(0,1]\times(0,s]:a^{4}u=y,\
y\leq u\leq s\\}.$

Then, on the curve $C^{\prime}_{y}$, we can write
$\widetilde{D}((y/u)^{1/4},1;u)=p(y^{7/4}u^{13/4})+13p(y^{3/4}u^{9/4})-46p(y)$,
$y\leq u\leq s$. Since $(\partial/\partial
u)\widetilde{D}((y/u)^{1/4},1;u)\leq 0$, we conclude that
$\widetilde{D}((y/u)^{1/4},1;u)$ attains its minimum at $u=s$, and hence, from
(3.26), we have

$m(s)=\inf_{(a,u)\in(0,1]\times(0,s]}\widetilde{D}(a,1;u)=\inf_{y\in(0,s]}\widetilde{D}((y/s)^{1/4},1;s)=\inf_{a\in(0,1]}\widetilde{D}(a,1;s).$
(3.27)

It suffices to show $m(s)\geq 0$ if $s\leq s_{\rm c}$. Since
$xp^{\prime}(x)=q(x):=x-1/x$, we can verify easily that

$\displaystyle a\frac{\partial}{\partial a}\widetilde{D}(a,1;s)$
$\displaystyle=7q(a^{7}s^{5})+13\cdot 3q(a^{3}s^{3})-46\cdot 4q(a^{4}s)$
$\displaystyle=\frac{1}{a^{7}s^{5}}(7a^{14}s^{10}-7+39a^{10}s^{8}-39a^{4}s^{2}-184a^{11}s^{6}+184a^{3}s^{4})=:\frac{1}{a^{7}s^{5}}\delta(a,s).$

For $a,s\in(0,1]$, we see that

$\displaystyle\delta(a,s)$ $\displaystyle\leq
7s^{10}+39s^{8}+184a^{2}s^{4}-39a^{4}s^{2}-7$
$\displaystyle=7s^{10}+39s^{8}-39s^{2}\Big{(}a^{2}-\frac{92}{39}s^{2}\Big{)}^{2}+\frac{92^{2}}{39}s^{6}-7\leq
7s^{10}+39s^{8}+\frac{92^{2}}{39}s^{6}-7.$

Since the last function of $s$ is increasing in $(0,1]$ and it takes a
negative value at $s=11/20$, we have $\delta(a,s)<0$ for
$(a,s)\in(0,1]\times(0,11/20]$. Therefore, $\widetilde{D}(a,1,s)$ is
decreasing in $a$ for $s\in(0,11/20]$, which together with (3.27) implies

$m(s)=\widetilde{D}(1,1,s)=\frac{1+s^{2}}{s^{5}}(s^{8}+12s^{6}-58s^{4}+12s^{2}+1).$

Here we note Remark 12 given in Section 3.4.5. Consequently, $m(s)\geq
m(s_{\rm c})=0$ for $s\in(0,s_{\rm c}]$ as $s_{\rm c}=r_{\rm
c}^{1/2}=0.533\dots\leq 11/20$.

###### Remark 13

We see that

$g_{\mathbb{D}}(-r^{-1/4},r^{-1/4};r)=\frac{6+r+r^{-1}}{4(r^{1/2}+r^{-1/2})}=:\widetilde{g}(r).$

It is readily verified that $\widetilde{g}(1)=1$ and
$d\widetilde{g}(r)/dr=(r-1)^{3}/\\{8r^{3/2}(r+1)^{2}\\}\geq 0$, $r\geq 1$.
Then, $\widetilde{g}(r)>1$ for any $r>1$. Since $1/r_{\rm c}=3.51\cdots>1$,
the PDPP $\mathcal{Z}_{X_{\mathbb{D}}^{r}}$ is still in the partially
attractive phase although $\kappa_{0}(r)$ becomes negative when $r\in(1/r_{\rm
c},\infty)$ due to the symmetry $r\leftrightarrow 1/r$ built in (3.21).

##  4 Concluding Remarks

Peres and Virág proved a relationship between the Szegő Kernel
$S_{\mathbb{D}}$ and the Bergman kernel $K_{\mathbb{D}}$ in the context of
probability theory: A GAF is defined so that its covariance kernel is given by
$S_{\mathbb{D}}$. Then the zero point process $\mathcal{Z}_{X_{\mathbb{D}}}$
is proved to be a DPP for which the correlation kernel is given by
$K_{\mathbb{D}}$. The background of their work is explained in the monograph
[35], in which we find that the Edelman–Kostlan formula [24] gives the density
of $\mathcal{Z}_{X_{\mathbb{D}}}$ with respect to $m/\pi$ as

$\rho^{1}_{\mathbb{D},\mathrm{PV}}(z)=\frac{1}{4}\Delta\log
S_{\mathbb{D}}(z,z),\quad z\in\mathbb{D},$

where $\Delta:=4\partial_{z}\partial_{\overline{z}}$. Moreover, we have the
equality

$K_{\mathbb{D}}(z,w)=\partial_{z}\partial_{\overline{w}}\log
S_{\mathbb{D}}(z,w)=S_{\mathbb{D}}(z,w)^{2},\quad z,w\in\mathbb{D}.$ (4.1)

On the other hand, as explained above (2.9), for the kernels on simply
connected domain $D\subsetneq\mathbb{C}$, the equality

$S_{D}(z,w)^{2}=K_{D}(z,w),\quad z,w\in D,$ (4.2)

is established.

In the present paper, we have reported our work to generalize the above to a
family of GAFs and their zero point processes on the annulus $\mathbb{A}_{q}$.
By comparing the expression (1.19) for the density obtained from Theorem 1.3
with (C.4) in Proposition C.1 in Appendix C.2 given below, we can recover the
Edelman–Kostlan formula as follows,

$\rho^{1}_{\mathbb{A}_{q}}(z;r)=\frac{\theta(-r)}{\theta(-r|z|^{4})}S_{\mathbb{A}_{q}}(z,z;r|z|^{2})^{2}=\frac{1}{4}\Delta\log
S_{\mathbb{A}_{q}}(z,z;r),\quad z\in\mathbb{A}_{q}.$

However, (4.1) does not hold for the weighted Szegő kernel for
$H^{2}_{r}(\mathbb{A}_{q})$. As shown by (C.3), the second log-derivative of
$S_{\mathbb{A}_{q}}(z,w;r)$ cannot be expressed by $S_{\mathbb{A}_{q}}(z,w;r)$
itself but a new function $S_{\mathbb{A}_{q}}(z,w;rz\overline{w})$ should be
introduced. In addition the proportionality between the square of the Szegő
kernel and the Bergman kernel (4.2) is no longer valid for the point processes
on $\mathbb{A}_{q}$ as shown in Proposition C.2 in Appendix C.3.

The Borchardt identity plays an essential role in the proof of Peres and
Virág, which is written as

$\mathop{\mathrm{perdet}}_{1\leq i,j\leq
n}\Big{[}S_{\mathbb{D}}(z_{i},z_{j})\Big{]}=\det_{1\leq i,j\leq
n}\Big{[}S_{\mathbb{D}}(z_{i},z_{j})^{2}\Big{]},\quad n\in\mathbb{N},\quad
z_{1},\dots,z_{n}\in\mathbb{D}.$

Since the $n$-point correlation function
$\rho^{n}_{\mathbb{D},\mathrm{PV}}(z_{1},\dots,z_{n})$ of
$\mathcal{Z}_{X_{\mathbb{D}}}$ is given by the left-hand side, $\forall
n\in\mathbb{N}$, this equality proves that $\mathcal{Z}_{X_{\mathbb{D}}}$ is a
DPP. For $S_{\mathbb{A}_{q}}$ the corresponding equality does not hold. We
have proved, however, that all correlation functions of our two-parameter
family of zero point processes
$\\{\mathcal{Z}_{X_{\mathbb{A}_{q}}^{r}}:q\in(0,1),r>0\\}$ on $\mathbb{A}_{q}$
can be expressed using $\mathop{\mathrm{perdet}}$ defined by (1.15) and we
stated that they are permanental-determinantal point processes (PDPPs).

We would like to place an emphasis on the fact that the present paper is not
an incomplete work nor just replacing determinants by
$\mathop{\mathrm{perdet}}$’s. The essentially new points, which are not found
in the previous works [64, 35], are the following:

(i) Even if we start from the GAF whose covariance kernel is given by the
original Szegő kernel
$S_{\mathbb{A}_{q}}(\cdot,\cdot)=S_{\mathbb{A}_{q}}(\cdot,\cdot;q)$ on
$\mathbb{A}_{q}$, the full description of conditioning with zeros needs a
series of new covariance kernels.

(ii) The covariance kernels of the induced GAFs generated by conditioning of
zeros are identified with the weighted Szegő kernel
$S_{\mathbb{A}_{q}}(\cdot,\cdot;\alpha)$ studied by Mccullough and Shen [56].
In the present study, the weight parameter $\alpha$ plays an essential role,
since it is determined by $\alpha=r\prod_{\ell=1}^{n}|z_{\ell}|^{2}$ and
represents a geometrical information of the zeros in $\mathbb{A}_{q}$
$\\{z_{1},\dots,z_{n}\\},n\in\mathbb{N}$ put in the conditioning.

(iii) Corresponding to such an inductive structure of conditional GAFs, the
correlation kernel of our PDPP of $\mathcal{Z}_{\mathbb{A}_{q}^{r}},r>0$ is
given by $S_{\mathbb{A}_{q}}(\cdot,\cdot;\alpha)$ with
$\alpha=r\prod_{\ell=1}^{n}|z_{\ell}|^{2}$ in order to give the correlation
function for the points $\\{z_{1},\dots,z_{n}\\}$;
$\rho^{n}_{\mathbb{A}_{q}}(z_{1},\dots,z_{n};r)$. In addition, the $n$-product
measure of the Lebesgue measure on $\mathbb{C}$ divided by $\pi$,
$(m/\pi)^{\otimes n}$, should be weighted by
$\theta(-r)/\theta(-r\prod_{k=1}^{n}|z_{k}|^{4})$ to properly provide
$\rho^{n}_{\mathbb{A}_{q}}(\cdot;r)$.

(iv) The parameter $r$ also plays an important role to describe the symmetry
of the GAF and its zero point process under the transformation which we call
the $(q,r)$-inversion,

$(z,r)\,\longleftrightarrow\,\left(\frac{q}{z},\frac{q^{2}}{r}\right)\,\in\mathbb{A}_{q}\times(0,\infty).$
(4.3)

And if we adjust $r=q$ the GAF and its zero point process become invariant
under conformal transformations which preserve $\mathbb{A}_{q}$.

Inapplicability of the Borchardt identity to our zero poin processes
$\mathcal{Z}_{X_{\mathbb{A}_{q}}^{r}}$ causes interesting behaviors of them as
interacting particle systems. We have proved that the short-range interaction
between points is repulsive with index $\beta=2$ in a similar way to the usual
DPP, but attractive interaction is also observed in
$\mathcal{Z}_{X_{\mathbb{A}_{q}}^{r}}$. The index for decay of correlations is
given by $\eta=4$. We found that there is a special value $r=r_{0}(q)\in(q,1)$
for each $q\in(0,1)$ at which the coefficient of the power-law decay of
correlations changes its sign. We have studied the zero point process obtained
in the limit $q\to 0$, which has a parameter $r\in[0,\infty)$. In this PDPP
$\mathcal{Z}_{X_{\mathbb{D}}^{r}}$, $r_{\rm c}:=r_{0}(0)$ can be regarded as
the critical value separating two phases in the sense that if $r\in[0,r_{\rm
c})$ the zero point process is completely repulsive, while if $r\in(r_{\rm
c},\infty)$ attractive interaction emerges depending on distances between
points.

There are many open problems, since such PDPPs have not been studied so far.
Here we list out some of them.

(1) We prove that the GAF $X_{\mathbb{A}_{q}}^{r}$ and its zero point process
$\mathcal{Z}_{X_{\mathbb{A}_{q}}^{r}}$ have the rotational invariance and the
$(q,r)$-inversion symmetry, and when $r=q$, they are invariant under conformal
transformations which preserve $\mathbb{A}_{q}$ (Proposition 1.1). We claimed
in Remark 1 that $X_{\mathbb{A}_{q}}^{r}$ can be extended to a one-parameter
family of GAFs $\\{X_{\mathbb{A}_{q}}^{r,(L)}:L\in\mathbb{N}\\}$ having the
rotational invariance and the $(q,r)$-inversion symmetry and this family is an
extension of $\\{X_{\mathbb{D}}^{(L)}:L\in\mathbb{N}\\}$ studied in [35,
Sections 2.3 and 5.4] in the sense that $\lim_{q\to
0}X_{\mathbb{A}_{q}}^{r,(L)}|_{r=q}\stackrel{{\scriptstyle\rm
d}}{{=}}X_{\mathbb{D}}^{(L)}$, $L\in\mathbb{N}$. In [35, Section 2.5], it is
argued that $\\{X_{\mathbb{D}}^{(L)}:L>0\\}$ is the only GAFs, up to
multiplication by deterministic non-vanishing analytic functions, whose zeros
are isometry-invariant under the conformal transformations preserving
$\mathbb{D}$. This assertion is proved by the fact that the zero point process
of the GAF is completely determined by its first correlation function.
Therefore, the “canonicality” of the GAFs $\\{X_{\mathbb{D}}^{(L)}:L>0\\}$ is
guaranteed by the uniqueness, up to multiplicative constant, of the density
function with respect to $m(dz)/\pi$,
$\rho_{\mathbb{D},\mathrm{PV}}^{1}(z)=1/(1-|z|^{2})^{2}$, which is invariant
under the Möbius transformations preserving $\mathbb{D}$. We have found,
however, that the density function with parameter $r>0$,
$\varrho(z;r),z\in\mathbb{A}_{q}$ is not uniquely determined to be
$\rho_{\mathbb{A}_{q}}^{1}(z;r)$ as (1.19) by the requirement that it is
rotationally invariant and having the $(q,r)$-inversion symmetry. For example,
we have the three-parameter ($\alpha_{1}>1-\alpha_{2}$, $\alpha_{2}>0$,
$\alpha_{3}\in\mathbb{R}$) family of density functions,

$\varrho(z;r;\alpha_{1},\alpha_{2},\alpha_{3})=\frac{\theta(-r)^{\alpha_{3}}}{\theta(-|z|^{2\alpha_{1}}r^{\alpha_{2}})}f^{\rm
JK}(|z|^{2},-|z|^{2\beta_{1}}r^{\beta_{2}})^{2}$

with $\beta_{1}=(\alpha_{1}-\alpha_{2})(\alpha_{1}+\alpha_{2}-1)/4+1/2$,
$\beta_{2}=\alpha_{2}(\alpha_{1}+\alpha_{2}-1)/2$, which satisfy the above
requirement of symmetry. We see that
$\varrho(z;r;2,1,1)=\rho_{\mathbb{A}_{q}}^{1}(z;r)$ and $\lim_{q\to
0}\varrho(z;q;\alpha_{1},\alpha_{2},\alpha_{3})=\rho_{\mathbb{D},\mathrm{PV}}^{1}(z)$.
The present study of the GAFs on $\mathbb{A}_{q}$ and their zero point
processes will be generalized in the future.

(2) As shown by (1.20), the asymptotics of the density of zeros
$\rho_{\mathbb{A}_{q}}^{1}(z)\sim(1-|z|^{2})^{-2}$ with respect to $m(dz)/\pi$
in the vicinity of the outer boundary of $\mathbb{A}_{q}$ can be identified
with the metric in the hyperbolic plane called the Poincaré disk model (see,
for instance, [34, 18]). The zero point process $\mathcal{Z}_{X_{\mathbb{D}}}$
of Peres and Virág can be regarded as a uniform DPP on the Poincaré disk model
[64, 22, 13]. Is there any meaningful geometrical space in which the present
zero point process $\mathcal{Z}_{X_{\mathbb{A}_{q}}^{r}}$ seems to be uniform?
As mentioned in Remark 3, conditioning with zeros does not induce any new GAF
on $\mathbb{D}$ [64], but it does on $\mathbb{A}_{q}$. Is it possible to give
some geometrical explanation for such a new phenomenon appearing in replacing
$\mathbb{D}$ by $\mathbb{A}_{q}$ reported in the present paper?

(3) As mentioned above and in Theorem 1.6 (i), we have found power-law decays
of unfolded 2-correlation functions to the unity with an index $\eta=4$.
Although the coefficient of this power-law changes depending on $q$ and $r$,
the index $\eta=4$ seems to be universal in the PDPPs
$\mathcal{Z}_{X_{\mathbb{A}_{q}}^{r}}$, $\mathcal{Z}_{X_{\mathbb{D}}^{r}}$ and
the DPP of Peres and Virág $\mathcal{Z}_{X_{\mathbb{D}}}$ (except the PDPPs at
$r=r_{0}(q)\in(q,1),q\in[0,1)$). The present proof of Theorem 1.6 (i) relied
on brute force calculations showing vanishing of derivatives up to the third
order. Simpler proof is required. In the metric of a proper hyperbolic space,
the decay of correlation will be exponential. In such a representation, what
is the meaning of the ‘universal value’ of $\eta$?

(4) As mentioned in Remark 9, the simplified PDPPs
$\\{\mathcal{Z}_{X_{\mathbb{D}}^{r}}:r\in(0,\infty)\\}$ can be regarded as an
interpolation between the DPP of Peres and Virág
$\mathcal{Z}_{X_{\mathbb{D}}}$ and its deterministic perturbation at the
origin $\mathcal{Z}_{X_{\mathbb{D}}}+\delta_{0}$. The first approximation of
the perturbation of the deterministic zero near $r=\infty$ is given by
$\frac{-1}{\sqrt{1+r}}\zeta_{0}/\zeta_{1}$ by solving the approximated linear
equation $\frac{\zeta_{0}}{\sqrt{1+r}}+\zeta_{1}z=0$. Here the ratio
$\zeta_{0}/\zeta_{1}$ is distributed according to the push-forward of the
uniform distribution on the unit sphere by the stereographic projection (see
Krishnapur [46] for the matrix generalization). Can we trace such a flow of
zeros in $\\{\mathcal{Z}_{X_{\mathbb{D}}^{r}}:r\in(0,\infty)\\}$ more
precisely?

(5) The simplified PDPP $\mathcal{Z}_{X_{\mathbb{D}}^{r}}$ was introduced as a
$q\to 0$ limit of the PDPP $\mathcal{Z}_{\mathbb{X}_{\mathbb{A}_{q}}^{r}}$ in
this paper. On the other hand, as shown by (1.23) the GAF $X_{\mathbb{D}}^{r}$
can be obtained from the GAF $X_{\mathbb{D}}$ of Peres and Virág by adding a
one-parameter ($r>0$) perturbation on a single term. Can we explain the
hierarchical structures of these GAF and PDPP on $\mathbb{D}$ and the
existence of the critical value $r_{\rm c}$ for correlations of
$\mathcal{Z}_{X_{\mathbb{D}}^{r}}$ apart from all gadgets related to elliptic
functions? Can we expect any interesting phenomenon at $r=r_{\rm c}$?

(6) As mentioned at the end of Section 1.3, the zero point process of the GAF
$X_{\mathbb{D}}$ studied by Peres and Virág [64] is the DPP
$\mathcal{Z}_{X_{\mathbb{D}}}$, whose correlation kernel is given by
$K_{\mathbb{D}}(z,w)=S_{\mathbb{D}}(z,w)^{2}=1/(1-z\overline{w})^{2}$,
$z,w\in\mathbb{D}$ with respect to the reference measure $m/\pi$. Krishnapur
[46] introduced a one-parameter ($\ell\in\mathbb{N}$) extension of DPPs
$\\{\mathcal{Z}_{X_{\mathbb{D}}}^{(\ell)}:\ell\in\mathbb{N}\\}$, whose
correlation kernels are given by
$K_{\mathbb{D}}^{(\ell)}(z,w)=1/(1-z\overline{w})^{\ell+1}$ with respect to
the reference measure $\ell(1-|z|^{2})^{\ell-1}m/\pi$ on $\mathbb{D}$. He
proved that $\mathcal{Z}_{\mathbb{D}}^{(\ell)}$ is realized as the zeros of
$\det[\sum_{n\in\mathbb{N}}\mathsf{G}_{n}z^{n}]$, where
$\\{\mathsf{G}_{n}\\}_{n\in\mathbb{N}}$ are i.i.d. complex Ginibre random
matrices of size $\ell\in\mathbb{N}$. A similar extension of the present PDPP
$\mathcal{Z}_{\mathbb{A}_{q}^{r}}$ on $\mathbb{A}_{q}$ will be challenging.

(7) For $0<t\leq 1$, let $\mathbb{D}_{t}:=\\{z\in\mathbb{C}:|z|<t\\}$. The CLT
for the number of points $\mathcal{Z}_{X_{\mathbb{D}}}(\mathbb{D}_{t})$ as
$t\to 1$ can be easily shown, since $\mathcal{Z}_{X_{\mathbb{D}}}$ is a DPP
and then $\mathcal{Z}_{X_{\mathbb{D}}}(\mathbb{D}_{t})$ can be expressed as a
sum of independent Bernoulli random variables [64, Corollary 3 (iii)] [69].
For $0<q\leq s<t\leq 1$, let $\mathbb{A}_{s,t}:=\\{z\in\mathbb{C}:s<|z|<t\\}$.
It would also be expected that the CLT holds for
$\big{(}\mathcal{Z}_{\mathbb{A}_{q}}(\mathbb{A}_{s,\sqrt{q}}),\mathcal{Z}_{\mathbb{A}_{q}}(\mathbb{A}_{\sqrt{q},t})\big{)}$
as $s\to q$ and $t\to 1$ simultaneously in some sense. Is there a useful
expression for those random variables as above and can we prove the CLTs for
them?

(8) In the present paper, we have tried to characterize the density functions
and the unfolded 2-correlation functions of the PDPPs. As demonstrated by
Fig.2, change of global structure is observed at $r=r_{\rm c}$ for the
unfolded 2-correlation function. Precise description of such a topological
change is required. More detailed quantitative study would be also
interesting. For example, we can show that $G_{\mathbb{A}_{q}}^{\vee}(x;r)$
plotted in Fig.2 attains its maximum at $x=\sqrt{q}$ when $r=1$ and the value
is given by
$G_{\mathbb{A}_{q}}^{\vee}(\sqrt{q};1)=(q_{2}(q)^{8}+q_{3}(q)^{8})/(16qq_{1}(q)^{8})>1$,
where $q_{1}(q):=\prod_{n=1}^{\infty}(1+q^{2n})$,
$q_{2}(q):=\prod_{n=1}^{\infty}(1+q^{2n-1})$, and
$q_{3}(q):=\prod_{n=1}^{\infty}(1-q^{2n-1})$. How about the local minima?
Moreover, systematic study on three-point and higher-order correlations will
be needed to obtain a better understanding of differences between PDPPs and
DPPs.

(9) Matsumoto and one of the present authors [54] studied the real GAF on a
plane and proved that the zeros of the real GAF provide a Pfaffian point
process (PfPP). There a Pfaffian–Hafnian analogue of Borchardt’s identity was
used [36]. Is it meaningful to consider the Pfaffian–Hafnian analogues of
PDPPs? Systematic study on the comparison among DPPs, PfPPs, permanental PPs,
Hafnian PPs, PDPPs, and Hafnian–Pfaffian PPs will be a challenging future
problem.

(10) The symmetry of the present GAF and its zero point process under the
$(q,r)$-inversion (4.3) mentioned above and the pairing of uncorrelated points
in the GAF $X_{\mathbb{A}_{q}}$ shown by Proposition 2.4 suggest that the
inner boundary $\gamma_{q}$ plays essentially the same role as the outer
boundary $\gamma_{1}$. As an extension of the Riemann mapping function for a
simply connected domain $D\subsetneq\mathbb{C}$, a function mapping a multiply
connected domain to the unit disk is called the Ahlfors map [79, 7]. (See also
Remark 7 again.) Could we use such Ahlfors maps to construct and analyze GAFs
and their zero point processes on general multiply connected domains?

Acknowledgements The present study was stimulated by the useful discussion
with Nizar Demni when the authors attended the workshop ‘Interactions between
commutative and non-commutative probability’ held in Kyoto University, Aug
19–23, 2019. The present authors would like to thank Michael Schlosser for
useful discussion on the elliptic extensions of determinantal and permanental
formulas. The present work has been supported by the Grant-in-Aid for
Scientific Research (C) (No.19K03674), (B) (No.18H01124), and (S)
(No.16H06338) of Japan Society for the Promotion of Science (JSPS).

##  Appendix A A Hyperdeterminantal Point Processes

Recall that determinant and permanent are defined for an $n\times n$ matrix (a
2nd order tensor on an $n$-dimensional space) $M=(m_{i_{1}i_{2}})_{1\leq
i_{1},i_{2}\leq n}$ as

$\displaystyle\det M$ $\displaystyle=\det_{1\leq i_{1},i_{2}\leq
n}[m_{i_{1}i_{2}}]:=\sum_{\sigma\in\mathfrak{S}_{n}}{\rm
sgn}(\sigma)\prod_{\ell=1}^{n}m_{\ell\sigma(\ell)}=\frac{1}{n!}\sum_{(\sigma_{1},\sigma_{2})\in\mathfrak{S}_{n}^{2}}{\rm
sgn}(\sigma_{1}){\rm
sgn}(\sigma_{2})\prod_{\ell=1}^{n}m_{\sigma_{1}(\ell)\sigma_{2}(\ell)},$
$\displaystyle\mathop{\mathrm{per}}M$
$\displaystyle=\mathop{\mathrm{per}}_{1\leq i_{1},i_{2}\leq
n}[m_{i_{1}i_{2}}]:=\sum_{\sigma\in\mathfrak{S}_{n}}\prod_{\ell=1}^{n}m_{\ell\sigma(\ell)}=\frac{1}{n!}\sum_{(\sigma_{1},\sigma_{2})\in\mathfrak{S}_{n}^{2}}\prod_{\ell=1}^{n}m_{\sigma_{1}(\ell)\sigma_{2}(\ell)},$
(A.1)

where $\mathfrak{S}_{n}$ denotes the symmetric group of order $n$. The notion
of determinant has been extended as follows. Cayley’s first hyperdeterminant
is defined for a $k$-th order tensor (hypermatrix) on an $n$-dimensional space
$M=(m_{i_{1}\dots i_{k}})_{1\leq i_{1},\dots,i_{k}\leq n}$ as

$\mathop{\mathrm{Det}}M=\mathop{\mathrm{Det}}_{1\leq i_{1},\dots,i_{k}\leq
n}[m_{i_{1}\dots
i_{k}}]:=\frac{1}{n!}\sum_{(\sigma_{1},\dots,\sigma_{k})\in\mathfrak{S}_{n}^{k}}\prod_{i=1}^{k}{\rm
sgn}(\sigma_{i})\prod_{\ell=1}^{n}m_{\sigma_{1}(\ell)\dots\sigma_{k}(\ell)}.$
(A.2)

It is straightforward to see that $\mathop{\mathrm{Det}}M=0$ if $k$ is odd.
Gegenbauer generalized (A.2) to the case where some of the indices are non-
alternated. If $\mathcal{I}$ denotes a subset of $\\{1,\dots,k\\}$, one has

${\mathop{\mathrm{Det}_{\mathcal{I}}}}M={\mathop{\mathrm{Det}_{\mathcal{I}}}_{1\leq
i_{1},\dots,i_{k}\leq n}}[m_{i_{1}\dots
i_{k}}]:=\frac{1}{n!}\sum_{(\sigma_{1},\dots,\sigma_{k})\in\mathfrak{S}_{n}^{k}}\prod_{i\in\mathcal{I}}{\rm
sgn}(\sigma_{i})\prod_{\ell=1}^{n}m_{\sigma_{1}(\ell)\dots\sigma_{k}(\ell)}.$
(A.3)

These extensions of the determinant are called hyperdeterminants. See [53, 25,
51] and references therein.

###### Lemma A.1

Let $A=(a_{i_{1}i_{2}})$ and $B=(b_{i_{1}i_{2}})$ be $n\times n$ matrices.
Then $\mathop{\mathrm{per}}A\det B=\mathop{\mathrm{Det}_{\\{2,3\\}}}C$, where
$C=(c_{i_{1}i_{2}i_{3}})$ is the $n\times n\times n$ hypermatrix with the
entries

$c_{i_{1}i_{2}i_{3}}=a_{i_{2}i_{1}}b_{i_{2}i_{3}},\quad
i_{1},i_{2},i_{3}\in\\{1,\dots,n\\}.$ (A.4)

In particular,
$\mathop{\mathrm{perdet}}M=\mathop{\mathrm{Det}_{\\{2,3\\}}}[m_{i_{2}i_{1}}m_{i_{2}i_{3}}]$,
where $\mathop{\mathrm{perdet}}M$ is defined by (1.15).

  * Proof.

By the definition (A.1),

$\displaystyle\mathop{\mathrm{per}}A\det B$
$\displaystyle=\sum_{\tau_{1}\in\mathfrak{S}_{n}}\prod_{i=1}^{n}a_{i\tau_{1}(i)}\sum_{\tau_{2}\in\mathfrak{S}_{n}}{\rm
sgn}(\tau_{2})\prod_{j=1}^{n}b_{j\tau_{2}(j)}=\sum_{\tau_{1}\in\mathfrak{S}_{n}}\sum_{\tau_{2}\in\mathfrak{S}_{n}}{\rm
sgn}(\tau_{2})\prod_{i=1}^{n}a_{i\tau_{1}(i)}b_{i\tau_{2}(i)}$
$\displaystyle=\frac{1}{n!}\sum_{\sigma_{1}\in\mathfrak{S}_{n}}\sum_{\sigma_{2}\in\mathfrak{S}_{n}}\sum_{\sigma_{3}\in\mathfrak{S}_{n}}{\rm
sgn}(\sigma_{1}^{-1}\circ\sigma_{3})\prod_{i=1}^{n}a_{i\,\sigma_{1}^{-1}\circ\sigma_{2}(i)}b_{i\,\sigma_{1}^{-1}\circ\sigma_{3}(i)}$
$\displaystyle=\frac{1}{n!}\sum_{\sigma_{1}\in\mathfrak{S}_{n}}\sum_{\sigma_{2}\in\mathfrak{S}_{n}}\sum_{\sigma_{3}\in\mathfrak{S}_{n}}{\rm
sgn}(\sigma_{1}){\rm
sgn}(\sigma_{3})\prod_{i=1}^{n}a_{\sigma_{1}(i)\sigma_{2}(i)}b_{\sigma_{1}(i)\sigma_{3}(i)}.$

We change the symbols of permutations as
$\sigma_{1}\to\rho_{2},\sigma_{2}\to\rho_{1},\sigma_{3}\to\rho_{3}$. Then the
above is written as
$(1/n!)\sum_{\rho_{1}\in\mathfrak{S}_{n}}\sum_{\rho_{2}\in\mathfrak{S}_{n}}\sum_{\rho_{3}\in\mathfrak{S}_{n}}{\rm
sgn}(\rho_{2}){\rm
sgn}(\rho_{3})\prod_{i=1}^{n}a_{\rho_{2}(i)\rho_{1}(i)}b_{\rho_{2}(i)\rho_{3}(i)}$.
Hence if we assume (A.4), then this is written as
$(1/n!)\sum_{(\sigma_{1},\sigma_{2},\sigma_{3})\in\mathfrak{S}_{n}^{3}}\prod_{i\in\\{2,3\\}}{\rm
sgn}(\sigma_{i})\prod_{j=1}^{n}c_{\sigma_{1}(j)\sigma_{2}(j)\sigma_{3}(j)}.$
By the definition (A.3), the proof is complete.

Theorem 1.3 of the present paper can be written in the following way.

###### Theorem A.2

$\mathcal{Z}_{X_{\mathbb{A}_{q}}^{r}}$ is a hyperdeterminantal point process
(hDPP) in the sense that it has correlation functions expressed by
hyperdeterminants as

$\displaystyle\rho^{n}_{\mathbb{A}_{q}}(z_{1},\dots,z_{n};r)=\frac{\theta(-r)}{\theta(-r\prod_{k=1}^{n}|z_{k}|^{4})}$
$\displaystyle\qquad\qquad\times\mathop{\mathrm{Det}_{\\{2,3\\}}}_{1\leq
i_{1},i_{2},i_{3}\leq
n}\Big{[}S_{\mathbb{A}_{q}}\Big{(}z_{i_{2}},z_{i_{1}};r\prod_{\ell=1}^{n}|z_{\ell}|^{2}\Big{)}S_{\mathbb{A}_{q}}\Big{(}z_{i_{2}},z_{i_{3}};r\prod_{\ell=1}^{n}|z_{\ell}|^{2}\Big{)}\Big{]},$

for every $n\in\mathbb{N}$ and $z_{1},\dots,z_{n}\in\mathbb{A}_{q}$ with
respect to $m/\pi$.

##  Appendix B B Conformal Map from $\mathbb{A}_{q}$ to $D(s)$

A general Schwarz–Christoffel formula for conformal maps from $\mathbb{A}_{q}$
to a doubly connected domain is given as Eq.(1) in [21] and on page 68 in
[23]. We can read that a conformal map from $\mathbb{A}_{q}$ to a chordal
standard domain $D(s)$, $s>0$ is given in the form

$f(z)=C\int_{-1}^{z}\frac{\theta(-\sqrt{-1}qu,\sqrt{-1}qu)}{\theta(u)}du,$

where $C$ is a parameter. We can show that the integral is transformed into an
integral of the Weierstrass $\wp$-function and hence the map is expressed by
the $\zeta$-function. A result is given by (1.38) in Remark 10. We note that
the obtained function $H_{q}$ is related to the Villat kernel $\mathcal{K}$
(see, for instance, [28]),

$\mathcal{K}(z)=\mathcal{K}(z;q):=\sum_{n\in\mathbb{Z}}\frac{1+q^{2n}z}{1-q^{2n}z}=\frac{1+z}{1-z}+2\sum_{n=1}^{\infty}\left(\frac{q^{2n}}{q^{2n}-z}+\frac{q^{2n}z}{1-q^{2n}z}\right),\quad
z\in\mathbb{A}_{q},$

by a simple relation $H_{q}(z)=\sqrt{-1}\mathcal{K}(z),z\in\mathbb{A}_{q}$.
Moreover, we can verify the equality $\mathcal{K}(z)=2\rho_{1}(z)$,
$z\in\mathbb{A}_{q}$.

##  Appendix C C Bergman Kernel and Szegő Kernel of an Annulus

###  C.1 $K_{\mathbb{A}_{q}}$ expressed by Weierstrass $\wp$-function

A CONS for the Bergman space on $\mathbb{A}_{q}$ is given by
$\\{\widetilde{e}^{(q)}_{n}(z)\\}_{n\in\mathbb{Z}}$ where we set

$\widetilde{e}^{(q)}_{n}(z)=\begin{cases}\displaystyle{\sqrt{\frac{n+1}{1-q^{2(n+1)}}}z^{n},}&n\in\mathbb{Z}\setminus\\{-1\\},\cr\displaystyle{\sqrt{\frac{1}{-2\log
q}}z^{-1}},&n=-1.\end{cases}$

The Bergman kernel of $\mathbb{A}_{q}$ is then given by

$\displaystyle K_{\mathbb{A}_{q}}(z,w)$
$\displaystyle:=k_{L^{2}_{\textrm{B}}(\mathbb{A}_{q})}(z,w)=\sum_{n\in\mathbb{Z}}\widetilde{e}^{(q)}_{n}(z)\overline{\widetilde{e}^{(q)}_{n}(w)}$
$\displaystyle=-\frac{1}{2\log
q}\frac{1}{z\overline{w}}+\frac{1}{z\overline{w}}\sum_{n\in\mathbb{Z}\setminus\\{0\\}}\frac{n}{1-q^{2n}}(z\overline{w})^{n},\quad
z,w\in\mathbb{A}_{q}.$ (C.1)

Using (2.39) and the notation (2.40), we can verify that this kernel is
expressed using the Weierstrass $\wp$-function (2.39) as [8]

$K_{\mathbb{A}_{q}}(z,w)=-\frac{1}{2\log
q}\frac{1}{z\overline{w}}-\frac{1}{z\overline{w}}\Big{(}\wp(\phi_{z\overline{w}})+\frac{P}{12}\Big{)},\quad
z,w\in\mathbb{A}_{q}.$ (C.2)

###  C.2 Second log-derivatives of $S_{\mathbb{A}_{q}}$

Here we prove the following.

###### Proposition C.1

For $r>0$, the following equality holds,

$\partial_{z}\partial_{\overline{w}}\log
S_{\mathbb{A}_{q}}(z,w;r)=\frac{\theta(-r)}{\theta(-r(z\overline{w})^{2})}S_{\mathbb{A}_{q}}(z,w;rz\overline{w})^{2},\quad
z,w\in\mathbb{A}_{q}.$ (C.3)

In particular,

$\Delta\log
S_{\mathbb{A}_{q}}(z,z;r)=4\frac{\theta(-r)}{\theta(-r|z|^{4})}S_{\mathbb{A}_{q}}(z,z;r|z|^{2})^{2},\quad
z\in\mathbb{A}_{q}.$ (C.4)

  * Proof.

Let
$\vartheta_{1}(\xi):=\sqrt{-1}q^{1/4}q_{0}e^{-\sqrt{-1}\xi}\theta(e^{2\sqrt{-1}\xi})$
[29, (11.2.2)]. This is one of the well-known four kinds of Jacobi theta
functions $\vartheta_{i}(\xi),i=0,1,2,3$. (See [29, Section 1.6] and [62,
Section 20.5].) Using $\vartheta_{1}$, (2.34) in Proposition 2.2 is written as

$S_{\mathbb{A}_{q}}(z,w;r)=\frac{\sqrt{-1}\vartheta_{1}^{\prime}(0)\vartheta_{1}(\phi_{-rz\overline{w}}/2)}{2\vartheta_{1}(\phi_{-r}/2)\vartheta_{1}(\phi_{z\overline{w}}/2)},$

where the notation (2.40) has been used. This gives

$\partial_{z}\partial_{\overline{w}}\log
S_{\mathbb{A}_{q}}(z,w;r)=-\Big{(}\partial_{\xi}^{2}\log\vartheta_{1}(\xi)\Big{|}_{\xi=\phi_{-rz\overline{w}}/2}-\partial_{\xi}^{2}\log\vartheta_{1}(\xi)\Big{|}_{\xi=\phi_{z\overline{w}}/2}\Big{)}/(4z\overline{w}).$

We use the equality
$\wp(2\omega_{1}z/\pi)=(\pi/(2\omega_{1}))\\{\vartheta_{1}^{\prime\prime\prime}(0)/(3\vartheta_{1}^{\prime}(0))-\partial_{z}^{2}\log\vartheta_{1}(z)\\}$
(see Eq. (23.6.14) in [62]). In the setting (2.38) we have

$\partial_{z}\partial_{\overline{w}}\log
S_{\mathbb{A}_{q}}(z,w;r)=\big{(}\wp(\phi_{-rz\overline{w}})-\wp(\phi_{z\overline{w}})\big{)}/(z\overline{w}).$
(C.5)

Now we use (2.55) in Lemma 2.5 given in Section 2.4 [19]. Combining with
(2.29) in Proposition 2.1, (C.5) gives

$\displaystyle\partial_{z}\partial_{\overline{w}}\log
S_{\mathbb{A}_{q}}(z,w;r)$ $\displaystyle=f^{\rm
JK}(z\overline{w},-rz\overline{w})f^{\rm
JK}(z\overline{w},-(rz\overline{w})^{-1})/(z\overline{w})$
$\displaystyle=S_{\mathbb{A}_{q}}(z,w;rz\overline{w})S_{\mathbb{A}_{q}}(z,w;(rz\overline{w})^{-1})/(z\overline{w}).$
(C.6)

The expression (2.29) of $S_{\mathbb{A}_{q}}(\cdot,\cdot;r)$ in Proposition
2.1 gives

$\displaystyle S_{\mathbb{A}_{q}}(z,w;(rz\overline{w})^{-1})$
$\displaystyle=f^{\rm JK}(z\overline{w},-(rz\overline{w})^{-1})=-f^{\rm
JK}((z\overline{w})^{-1},-rz\overline{w})$
$\displaystyle=-S_{\mathbb{A}_{q}}(z^{-1},w^{-1};rz\overline{w}),$ (C.7)

where (2.32) is used. On the other hand, the expression (2.34) of
$S_{\mathbb{A}_{q}}(\cdot,\cdot;r)$ in Proposition 2.2 gives
$S_{\mathbb{A}_{q}}(z,w;rz\overline{w})=q_{0}^{2}\theta(-r(z\overline{w})^{2})/\theta(-rz\overline{w},z\overline{w})$,
and

$S_{\mathbb{A}_{q}}(z^{-1},w^{-1};rz\overline{w})=\frac{q_{0}^{2}\theta(-rz\overline{w}(z\overline{w})^{-1})}{\theta(-rz\overline{w},(z\overline{w})^{-1})}=\frac{q_{0}^{2}\theta(-r)}{\theta(-rz\overline{w},(z\overline{w})^{-1})}=-z\overline{w}\frac{q_{0}^{2}\theta(-r)}{\theta(-z\overline{w}r,z\overline{w})},$

where (2.13) was used. Hence,
$S_{\mathbb{A}_{q}}(z^{-1},w^{-1};rz\overline{w})=-z\overline{w}\\{\theta(-r)/\theta(-r(z\overline{w})^{2})\\}S_{\mathbb{A}_{q}}(z,w;rz\overline{w})$
and (C.7) gives
$S_{\mathbb{A}_{q}}(z,w;(rz\overline{w})^{-1})=z\overline{w}\\{\theta(-r)/\theta(-r(z\overline{w})^{2})\\}S_{\mathbb{A}_{q}}(z,w;rz\overline{w})$.
Then (C.6) proves the proposition.

###  C.3 Relation between $K_{\mathbb{A}_{q}}$ and $S_{\mathbb{A}_{q}}$

We prove the following relation between the Bergman kernel
$K_{\mathbb{A}_{q}}$ and the Szegő kernel $S_{\mathbb{A}_{q}}$ of an annulus.

###### Proposition C.2

The equality

$S_{\mathbb{A}_{q}}(z,w)^{2}=K_{\mathbb{A}_{q}}(z,w)+\frac{a}{z\overline{w}},\quad
z,w\in\mathbb{A}_{q},$ (C.8)

holds, where

$a=a(q)=e_{2}+\frac{P}{12}+\frac{1}{2\log
q}=-2\sum_{n\in\mathbb{N}}\frac{(-1)^{n}nq^{n}}{1-q^{2n}}+\frac{1}{2\log q}.$
(C.9)

  * Proof.

By Proposition 2.1, $S_{\mathbb{A}_{q}}(z,w)^{2}=f^{\rm
JK}_{q}(z\overline{w},-q)^{2}$. Since $f^{\rm JK}(z,a)=f^{\rm
JK}(z,a/q^{2})/z$ is given by (2.33), we have $f^{\rm
JK}(z\overline{w},-q)=f^{\rm JK}(z\overline{w},-q^{-1})/(z\overline{w})$, and
hence

$S_{\mathbb{A}_{q}}(z,w)^{2}=\frac{1}{z\overline{w}}f^{\rm
JK}(z\overline{w},-q)f^{\rm JK}(z\overline{w},-q^{-1}).$ (C.10)

Here we use (2.55) in Lemma 2.5 given in Section 2.4 [19]. Then

$f^{\rm JK}(z\overline{w},-q)f^{\rm
JK}(z\overline{w},-q^{-1})=\wp(\pi+\pi\tau_{q})-\wp(\phi_{z\overline{w}})=e_{2}-\wp(\phi_{z\overline{w}}),$

where we have used the setting (2.38), the notation (2.40) and the evenness of
$\wp(z)$. The equality (C.10) is thus written as
$S_{\mathbb{A}_{q}}(z,w)^{2}=-\wp(\phi_{z\overline{w}})/(z\overline{w})+e/(z\overline{w})$.
Now we use (C.2). Then (C.8) is obtained with $a$ given by the first
expression in (C.9). If we set $z=-q/\overline{w}$ in (C.8), then Lemma 2.3
gives an equality, $0=K_{\mathbb{A}_{q}}(-q/\overline{w},w)-a/q$. By (C.1)
with a short calculation, the second expression for $a$ in (C.9) is obtained.

###### Remark 14

The relationship (C.8) between $S_{\mathbb{A}_{q}}$ and $K_{\mathbb{A}_{q}}$
with an additional term $a$ is concluded from a general theory (see, for
instance, Exercise 3 in Section 6, Chapter VII of [60], and Section 25 of
[7]). It was shown in [12] that $a$ is readily determined by Lemma 2.3 as
shown above, if the equality (C.8) is established. Here we showed direct proof
of (C.8) using the equality (2.55) between $f^{\rm JK}$ and $\wp$ [19]. By the
explicit formulas (C.9) for $a$, we see that $\lim_{q\to 0}a(q)=0$. Therefore,
the relation (C.8) is reduced in the limit $q\to 0$ to
$S_{\mathbb{D}}(z,w)^{2}=K_{\mathbb{D}}(z,w),z,w\in\mathbb{D}$, which is a
special case of (2.9), as expected.

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