log-Coulomb gases in the projective line of a $p$-field

Let $X$ be a topological space endowed with a metric $d$ and a finite positive Borel measure $\mu$ satisfying $\mu^{N}\{(x_{1},\dots,x_{N})\in X^{N}:x_{i}=x_{j}\text{ for some $i\neq j$}\}=0$ for every $N\geq 1$. A log-Coulomb gas in $X$ is a statistical model described as follows: Consider $N\geq 1$ particles with fixed charge values $\mathfrak{q}_{1},\dots,\mathfrak{q}_{N}\in\mathbb{R}$ and corresponding variable locations $x_{1},\dots,x_{N}\in X$. Whether the charge values are distinct or not, we assume particles are distinguished by the labels $1,2,\dots,N$, so that unique configurations of the system correspond to unique tuples $(x_{1},\dots,x_{N})\in X^{N}$. Each tuple is called a microstate and has an energy defined by

$$E(x_{1},\dots,x_{N}):=\begin{cases}-\sum_{1\leq i<j\leq N}\mathfrak{q}_{i}\mathfrak{q_{j}}\log d(x_{i},x_{j})&\text{if $x_{i}\neq x_{j}$ for all $i<j$},\\ \infty&\text{otherwise}.\end{cases}$$

Note that $E^{-1}(\infty)$ has measure 0 in $X^{N}$ by our choice of $\mu$, and that $E$ is identically zero if $N=1$. We assume the system is in thermal equilibrium with a heat reservoir at some inverse temperature $\beta>0$, so that its microstates form a canonical ensemble distributed according to the density $e^{-\beta E(x_{1},\dots,x_{N})}$. The canonical partition function $\beta\mapsto Z_{N}(X,\beta)$ is defined as the total mass of this density, namely

$$Z_{N}(X,\beta):=\int_{X^{N}}e^{-\beta E(x_{1},\dots,x_{N})}\,d\mu^{N}=\int_{X^{N}}\prod_{1\leq i<j\leq N}d(x_{i},x_{j})^{\mathfrak{q}_{i}\mathfrak{q}_{j}\beta}\,d\mu^{N}~{}.$$

(1.1.1)

Given $(X,d,\mu)$ and $\mathfrak{q}_{1},\dots,\mathfrak{q}_{N}$, the explicit formula for $Z_{N}(X,\beta)$ is of primary interest, as it yields fundamental relationships between the observable parameters of the system and its temperature. For instance, the system’s dimensionless free energy, mean energy, and energy fluctuation (variance) are respectively given by $-\log Z_{N}(X,\beta)$, $-\partial/\partial\beta\log Z_{N}(X,\beta)$, and $\partial^{2}/\partial\beta^{2}\log Z_{N}(X,\beta)$, all of which are functions of $\beta$ (and hence of temperature). Below are two examples in which the formula for $Z_{N}(X,\beta)$ is known.

Note that $\beta\mapsto Z_{N}(X,\beta)$ always extends to a complex domain containing the line $\operatorname{Re}(\beta)=0$. To simultaneously treat all possible choices of $\mathfrak{q}_{i}$, we extend further to a subset of $\mathbb{C}^{N(N-1)/2}$ as follows:

Once the formula and domain for $\bm{s}\mapsto\mathcal{Z}_{N}(X,\bm{s})$ are known, then for any choice of $\mathfrak{q}_{1},\dots,\mathfrak{q}_{N}\in\mathbb{R}$, the formula and domain for $\beta\mapsto Z_{N}(X,\beta)$ follow by specializing $\mathcal{Z}_{N}(X,\bm{s})$ to ${s_{ij}=\mathfrak{q}_{i}\mathfrak{q}_{j}\beta}$. Thus, given $(X,d,\mu)$, the main problem is to determine the formula and domain of $\bm{s}\mapsto\mathcal{Z}_{N}(X,\bm{s})$.

Of our two main goals in this paper, the first is to determine the explicit formula and domain of $\bm{s}\mapsto\mathcal{Z}_{N}(\mathbb{P}^{1}(K),\bm{s})$, where $K$ is a $p$-field and its projective line $\mathbb{P}^{1}(K)$ is endowed with a natural metric and measure. To make this precise, we briefly recall well-known properties of $p$-fields (see [Wei95], for instance) and establish some notation. Fix a $p$-field $K$, write $|\cdot|$ for its canonical absolute value, write $d$ for the associated metric (i.e., $d(x,y):=|x-y|$), and define

$$R:=\{x\in K:|x|\leq 1\}\qquad\text{and}\qquad P:=\{x\in K:|x|<1\}~{}.$$

The closed unit ball $R$ is the maximal compact subring of $K$, the open unit ball $P$ is the unique maximal ideal in $R$, and the group of units is $R^{\times}=R\setminus P=\{x\in K:|x|=1\}$. The residue field $R/P$ is isomorphic to $\mathbb{F}_{q}$ for some prime power $q$, and there is a canonical isomorphism of the cyclic group $(R/P)^{\times}$ onto the group of $(q-1)$th roots of unity in $K$. Fixing a primitive such root $\xi\in K$ and sending $P\mapsto 0$ extends the isomorphism $(R/P)^{\times}\to\{1,\xi,\dots,\xi^{q-2}\}$ to a bijection $R/P\to\{0,1,\xi,\dots,\xi^{q-2}\}$ with inverse $y\mapsto y+P$. Therefore $\{0,1,\xi,\dots,\xi^{q-2}\}$ is a complete set of representatives for the cosets of $P\subset R$, i.e.,

$$R=P\sqcup\underbrace{(1+P)\sqcup(\xi+P)\sqcup\dots\sqcup(\xi^{q-2}+P)}_{R^{\times}}~{}.$$

(1.2.1)

Fix a uniformizer $\pi\in K$ (any element satisfying $P=\pi R$) and let $\mu$ be the unique additive Haar measure on $K$ satisfying $\mu(R)=1$. The open balls in $K$ are precisely the sets of the form $y+\pi^{v}R$ with $y\in K$ and $v\in\mathbb{Z}$, and every such ball is compact with measure equal to its radius, i.e., $\mu(y+\pi^{v}R)=|\pi^{v}|=q^{-v}$. In particular, each of the $q$ cosets of $P$ in (1.2.1) is a compact open ball with measure (and radius) $q^{-1}$, and two elements $x,y\in R$ satisfy $|x-y|=1$ if and only if they belong to different cosets. Henceforth, we reserve the symbols $K$, $|\cdot|$, $d$, $R$, $P$, $q$, $\xi$, $\pi$, and $\mu$ for the items above, and we distinguish $|\cdot|$ from the standard absolute value on $\mathbb{C}$ by writing $|\cdot|_{\mathbb{C}}$ for the latter.

We now recall some useful facts from [FP15] in our present notation. The projective line of $K$ is the quotient space $\mathbb{P}^{1}(K)=(K^{2}\setminus\{(0,0)\})/\sim$, where $(x_{0},x_{1})\sim(y_{0},y_{1})$ if and only if $y_{0}=\lambda x_{0}$ and $y_{1}=\lambda x_{1}$ for some $\lambda\in K^{\times}$. Thus we may understand $\mathbb{P}^{1}(K)$ concretely as the set of symbols $[x_{0}:x_{1}]$ with $(x_{0},x_{1})\in K^{2}\setminus\{(0,0)\}$, subject to the relation $[\lambda x_{0}:\lambda x_{1}]=[x_{0}:x_{1}]$ for all $\lambda\in K^{\times}$ and endowed with the topology induced by the quotient $(x_{0},x_{1})\mapsto[x_{0}:x_{1}]$. The projective line is compact and metrizable by the spherical metric $\delta:\mathbb{P}^{1}(K)\times\mathbb{P}^{1}(K)\to\{0\}\cup\{q^{-v}:v\in\mathbb{Z}_{\geq 0}\}$, which is defined via

$$\delta([x_{0}:x_{1}],[y_{0}:y_{1}]):=\frac{|x_{0}y_{1}-x_{1}y_{0}|}{\max\{|x_{0}|,|x_{1}|\}\cdot\max\{|y_{0}|,|y_{1}|\}}~{}.$$

(1.2.2)

In particular, every open set in $\mathbb{P}^{1}(K)$ is a union of balls of the form

$$B_{v}[x_{0}:x_{1}]:=\{[y_{0}:y_{1}]\in\mathbb{P}^{1}(K):\delta([x_{0}:x_{1}],[y_{0}:y_{1}])\leq q^{-v}\}$$

(1.2.3)

with $[x_{0}:x_{1}]\in\mathbb{P}^{1}(K)$ and $v\in\mathbb{Z}_{\geq 0}$, and every such ball is open and compact. The projective linear group $PGL_{2}(R)$ is the quotient of $GL_{2}(R)=\{A\in M_{2}(R):\det(A)\in R^{\times}\}$ by its center, namely $Z=\{\left(\begin{smallmatrix}\lambda&0\\ 0&\lambda\end{smallmatrix}\right):\lambda\in R^{\times}\}\cong R^{\times}$. It is straightforward to verify that the rule

$$\phi[x_{0}:x_{1}]:=[ax_{0}+bx_{1}:cx_{0}+dx_{1}]~{},$$

where $\phi\in PGL_{2}(R)$ and $\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in GL_{2}(R)$ is any representative of $\phi$, gives a well-defined transitive action of $PGL_{2}(R)$ on $\mathbb{P}^{1}(K)$.

It is routine to verify that $\mu^{N}\{(x_{1},\dots,x_{N})\in(y+\pi^{v}R)^{N}:x_{i}=x_{j}\text{ for some $i\neq j$}\}=0$ and

$$\nu^{N}\{([x_{1,0}:x_{1,1}],\dots,[x_{N,0}:x_{N,1}])\in(\mathbb{P}^{1}(K))^{N}:[x_{i,0}:x_{i,1}]=[x_{j,0}:x_{j,1}]\text{ for some $i\neq j$}\}=0$$

for all $N\geq 1$, so we have suitable metrics and measures to define log-Coulomb gases in $X=y+\pi^{v}R$ and $X=\mathbb{P}^{1}(K)$. Definition 1.3 specializes to these $X$ as follows:

The formulas and domains of absolute convergence for the integrals above can be stated neatly in terms of the following items from [Web21b]:

Proposition 3.15 and Theorem 2.6(c) in [Web21b] imply the following proposition, which shall be generalized slightly in order to prove the main results of this paper:

Our first main result is the following analogue of Proposition 1.7:

The evident similarities between Proposition 1.7 and Theorem 2.1 follow from explicit relationship between the metrics and measures on $K$ and those on $\mathbb{P}^{1}(K)$. These relationships also play a role in the upcoming results, so they are worth recalling now. Note that $[x_{0}:x_{1}]\neq[1:0]$ if and only if $x_{1}\neq 0$, in which case $x=x_{0}/x_{1}$ is the unique element of $K$ satisfying $[x:1]=[x_{0}:x_{1}]$. Therefore the rule $\iota(x):=[x:1]$ defines a bijection $\iota:K\to\mathbb{P}^{1}(K)\setminus\{[1:0]\}$ and relates the metric structures of $K$ and $\mathbb{P}^{1}(K)$ in a simple way: Given $x,y\in K$, (1.2.2) implies $\delta(\iota(x),[1:0])=(\max\{1,|x|\})^{-1}$ and

$$\delta(\iota(x),\iota(y))=\begin{cases}|x-y|&\text{if $x,y\in R$},\\ 1&\text{if $x\in R$ and $y\notin R$},\\ |1/x-1/y|&\text{if }x,y\notin R.\end{cases}$$

(2.1.1)

Using the definitions (1.2.3) and $v_{K}(x):=-\log_{q}|x|$ for $x\in K^{\times}$, along with (2.1.1) and the strong triangle equality (i.e., $|x+y|=\max\{|x|,|y|\}$ whenever $|x|\neq|y|$), one easily verifies that

$$\iota(y+\pi^{v}R)=\begin{cases}B_{v}(\iota(y))&\text{if }y\in R,\\ B_{v-2v_{K}(y)}(\iota(y))&\text{if }y\notin R,\end{cases}$$

(2.1.2)

whenever $y\in K$ and $v\in\mathbb{Z}_{>0}$. That is, $\iota$ sends the open ball of radius $r\in(0,1)$ centered at $y\in K$ onto the open ball of radius $r/\max\{1,|y|^{2}\}$ centered at $\iota(y)\in\mathbb{P}^{1}(K)\setminus\{[1:0]\}$, so $\iota:K\to\mathbb{P}^{1}(K)\setminus\{[1:0]\}$ is a homeomorphism that restricts to an isometry on $R$ and a contraction on $K\setminus R$.

The map $\iota$ also relates the measures on $K$ and $\mathbb{P}^{1}(K)$ in a simple way: Given $v>0$ and a complete set of representatives $y_{1},\dots,y_{q^{v}}\in R$ for the cosets of $\pi^{v}R\subset R$, applying (2.1.2) to the partition $R=(y_{1}+\pi^{v}R)\sqcup\dots\sqcup(y_{q^{v}}+\pi^{v}R)$ yields

$$\iota(R)=B_{v}[y_{1}:1]\sqcup\cdots\sqcup B_{v}[y_{q^{v}}:1]~{}.$$

Therefore $PGL_{2}(R)$-invariance of $\nu$ (Lemma 1.4) implies $\nu(\iota(R))=q^{v}\cdot\nu(B_{v}[0:1])$. On the other hand,

$$\iota(K\setminus R)=\iota(\{x:|x|\geq q\})=\{\iota(x):\delta(\iota(x),[1:0])\leq q^{-1}\}=B_{1}[1:0]\setminus\{[1:0]\}$$

implies $\iota(R)\sqcup B_{1}[1:0]=\iota(R)\sqcup\iota(K\setminus R)\sqcup\{[1:0]\}=\mathbb{P}^{1}(K)$, which has measure 1. But $\nu(\iota(R))=q\cdot\nu(B_{1}[1:0])$, so the measure of $\iota(R)$ must be $q/(q+1)$, and therefore every ball $B_{v}[x_{0}:x_{1}]\subset\mathbb{P}^{1}(K)$ with $v>0$ has measure $q^{-v}\cdot q/(q+1)$. Combining this with (2.1.2), one concludes that the measure $\nu$ on $\mathbb{P}^{1}(K)\setminus\{[1:0]\}$ pulls back along $\iota$ to an explicit measure on $K$:

$$\nu(\iota(M))=\frac{q}{q+1}\int_{M}\left(\max\{1,|x|^{2}\}\right)^{-1}\,d\mu\qquad\text{for any Borel subset }M\subset K~{}.$$

(2.1.3)

Finally, (1.2.1) and (2.1.2) give a nice refinement of $\mathbb{P}^{1}(K)=\iota(R)\sqcup B_{1}[1:0]$ in terms of the $(q-1)$th roots of unity in $K$, which should be understood as the projective analogue of (1.2.1):

$$\mathbb{P}^{1}(K)=B_{1}[0:1]\sqcup\underbrace{B_{1}[1:1]\sqcup B_{1}[\xi:1]\sqcup\dots\sqcup B_{1}[\xi^{q-2}:1]}_{\iota(R^{\times})}\sqcup B_{1}[1:0]~{}.$$

(2.1.4)

Indeed, all $q+1$ of the parts in the partition are balls with measure $1/(q+1)$ and radius $q^{-1}$, and two points $[x_{0}:x_{1}],[y_{0}:y_{1}]\in\mathbb{P}^{1}(K)$ satisfy $\delta([y_{0}:y_{1}],[y_{0}:y_{1}])=1$ if and only if $[x_{0}:x_{1}]$ and $[y_{0}:y_{1}]$ belong to different parts. Note that $\iota$ sends $R^{\times}$ onto the “equator” $\iota(R^{\times})$, i.e., the set of points in $\mathbb{P}^{1}(K)$ with $\delta$-distance 1 from both the “south pole” $[0:1]$ and the “north pole” $[1:0]$.

So far we have only considered log-Coulomb gases with $N$ labeled (and hence distinguishable) particles. Our second main result concerns the situation in which all particles are identical with charge $\mathfrak{q}_{i}=1$ for all $i$, in which case a microstate $(x_{1},\dots,x_{N})\in X^{N}$ is regarded as unique only up to permutations of its entries. Since the energy $E(x_{1},\dots,x_{N})$ and measure on $X^{N}$ are invariant under such permutations, each unlabeled microstate makes the contribution $e^{-\beta E(x_{1},\dots,x_{N})}d\mu^{N}$ to the integral $Z_{N}(X,\beta)$ in (1.1.1) precisely $N!$ times. Therefore the canonical partition function for the unlabeled microstates is given by $Z_{N}(X,\beta)/N!$. We further assume that the system exchanges particles with the heat reservoir with chemical potential $\eta$ and define the fugacity parameter $f=e^{\eta\beta}$. In this situation the particle number $N\geq 0$ is treated as a random variable and the canonical partition function is replaced by the grand canonical partition function

$$Z(f,X,\beta):=\sum_{N=0}^{\infty}Z_{N}(X,\beta)\frac{f^{N}}{N!}$$

(2.2.1)

with the familiar convention $Z_{0}(X,\beta)=1$. Many properties of the system can be deduced from the grand canonical partition function. For instance, if $\beta>0$ is fixed and $Z_{N}(X,\beta)$ is sub-exponential in $N$, then $Z(f,X,\beta)$ is analytic in $f$ and the expected number of particles in the system is given by $f\frac{\partial}{\partial f}\log(Z(f,X,\beta))$. The canonical partition function for each $N\geq 0$ can also be recovered by evaluating the $N$th derivative of $Z(f,X,\beta)$ with respect to $f$ at $f=0$.

We are interested in the examples $Z(f,R,\beta)$, $Z(f,P,\beta)$, and $Z(f,\mathbb{P}^{1}(K),\beta)$, which turn out to share several common properties and simple relationships. By setting $s_{ij}=\beta$ in Definition 1.5, one sees that $|Z_{N}(R,\beta)|_{\mathbb{C}}$, $|Z_{N}(P,\beta)|_{\mathbb{C}}$, and $|Z_{N}(\mathbb{P}^{1}(K),\beta)|_{\mathbb{C}}$ are bounded above by 1 for all $N\geq 0$ and all $\beta>0$, and hence $Z(f,R,\beta)$, $Z(f,P,\beta)$, and $Z(f,\mathbb{P}^{1}(K),\beta)$ are analytic in $f$ when $\beta>0$. Sinclair recently found an elegant relationship between the first two, which is closely related to the partition of $R$ in (1.2.1):

Roughly speaking, the $q$th Power Law states that a log-Coulomb gas in $R$ exchanging energy and particles with a heat reservoir “factors” into $q$ identical sub-gases (one in each coset of $P$) that exchange energy and particles with the reservoir. For $\beta>0$, note that the series equation $Z(f,R,\beta)=(Z(f,P,\beta))^{q}$ is equivalent to the coefficient identities

$$\frac{Z_{N}(R,\beta)}{N!}=\sum_{\begin{subarray}{c}N_{0}+\dots+N_{q-1}=N\\ N_{0},\dots,N_{q-1}\geq 0\end{subarray}}\prod_{k=0}^{q-1}\frac{Z_{N_{k}}(P,\beta)}{N_{k}!}\qquad\text{for all $N\geq 0$.}$$

(2.2.2)

The $\beta=1$ case of (2.2.2) is given in [BGMR06], in which the positive number $Z_{N}(R,1)/N!$ is recognized as the probability that a random monic polynomial in $R[x]$ splits completely in $R$. The more general $\beta>0$ case given in [Sin20] makes explicit use of the partition of $R$ into cosets of $P$ (as in (1.2.1)). In Section 3, we will use the analogous partition of $\mathbb{P}^{1}(K)$ into $q+1$ balls (recall (2.1.4)) to show that

$$\frac{Z_{N}(\mathbb{P}^{1}(K),\beta)}{N!}=\sum_{\begin{subarray}{c}N_{0}+\dots+N_{q}=N\\ N_{0},\dots,N_{q}\geq 0\end{subarray}}\prod_{k=0}^{q}\left(\frac{q}{q+1}\right)^{N_{k}}\frac{Z_{N_{k}}(P,\beta)}{N_{k}!}\qquad\text{for all $\beta>0$ and $N\geq 0$,}$$

(2.2.3)

which immediately implies our second main result:

Like the $q$th Power Law, the $(q+1)$th Power Law roughly states that a log-Coulomb gas in $\mathbb{P}^{1}(K)$ exchanging energy and particles with a heat reservoir “factors” into $q+1$ identical sub-gases in the balls $B_{1}[0:1]$, $B[1:1]$, $B[\xi:1]$, $\dots$, $B[\xi^{q-2}:1]$, $B_{1}[1:0]$ (each isometrically homeomorphic to $P$), with fugacity $\frac{qf}{q+1}$. The $q$th Power Law allows the $(q+1)$th Power Law to be written more crudely as

$$Z(f,\mathbb{P}^{1}(K),\beta)=Z(\tfrac{qf}{q+1},R,\beta)\cdot Z(\tfrac{qf}{q+1},P,\beta)~{},$$

(2.2.4)

which is to say that the gas in $\mathbb{P}^{1}(K)$ “factors” into two sub-gases: one in $\iota(R)$ and one in $B[1:0]$ (which are respectively isometrically homeomorphic to $R$ and $P$), both with fugacity $\frac{qf}{q+1}$.

Although Proposition 1.7 and Theorem 2.1 provide explicit formulas for $Z_{N}(R,\beta)$ and $Z_{N}(\mathbb{P}^{1}(K),\beta)$, they are not efficient for computation because they require a complete list of reduced splitting chains of order $N$. For a practical alternative, we take advantage of both Power Laws and the following ideas from [BGMR06] and [Sin20]: Apply $Z(f,P,\beta)\cdot\frac{\partial}{\partial f}$ to the equation $Z(f,R,\beta)=(Z(f,P,\beta))^{q}$ to get

$$Z(f,P,\beta)\cdot\frac{\partial}{\partial f}Z(f,R,\beta)=q\cdot Z(f,R,\beta)\cdot\frac{\partial}{\partial f}Z(f,P,\beta)~{},$$

then expand both sides as power series in $f$ to obtain the coefficient equations

$$\sum_{k=1}^{N}\frac{Z_{N-k}(P,\beta)}{(N-k)!}\frac{Z_{k}(R,\beta)}{(k-1)!}=q\cdot\sum_{k=1}^{N}\frac{Z_{N-k}(R,\beta)}{(N-k)!}\frac{Z_{k}(P,\beta)}{(k-1)!}\qquad\text{for all }N\geq 1.$$

(2.3.1)

The identities $Z_{j}(P,\beta)=q^{-j-\binom{j}{2}\beta}Z_{j}(R,\beta)$ follow easily from Definition 1.5 and eliminate all instances of $Z_{j}(P,\beta)$ in (2.3.1) while introducing powers of the form $q^{-j-\binom{j}{2}\beta}$. For $N\geq 2$, a careful rearrangement of these powers, the factorials, and the terms in (2.3.1) yields the explicit recurrence

$$\frac{Z_{N}(R,\beta)}{N!q^{\frac{1}{2}\binom{N}{2}\beta}}=\sum_{k=1}^{N-1}\frac{k}{N}\cdot\frac{\sinh\left(\frac{\log(q)}{2}\left[\left(N+\binom{N}{2}\beta\right)\left(1-\frac{2k}{N}\right)+1\right]\right)}{\sinh\left(\frac{\log(q)}{2}\left[\left(N+\binom{N}{2}\beta\right)-1\right]\right)}\cdot\frac{Z_{N-k}(R,\beta)}{(N-k)!q^{\frac{1}{2}\binom{N-k}{2}\beta}}\cdot\frac{Z_{k}(R,\beta)}{k!q^{\frac{1}{2}\binom{k}{2}\beta}}~{}.$$

The expression at left is identically 1 if $N=0$ or $N=1$, so induction confirms that it is polynomial in ratios of hyperbolic sines for all $N\geq 0$. In particular, its dependence on $q$ is carried only by the factor $\log(q)$ appearing inside the hyperbolic sines, which motivates the following lemma: