We study large n expansions for the partition function of a Coulomb gas <Equation ID="Equ142"> <EquationSource Format="TEX">$\begin{aligned}Z_n=\frac{1}{\pi ^n}\int _{\mathbb {C}^n}\prod _{1\le i<j\le n}|z_i-z_j|^2\prod _{i=1}^n e^{-nQ(z_i)}\, d^2 z_i,\end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>Z</mi> <mi>n</mi> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <msup> <mi>π</mi> <mi>n</mi> </msup> </mfrac> <msub> <mo>∫</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </msub> <munder> <mo>∏</mo> <mrow> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo><</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> </mrow> </munder> <msup> <mrow> <mo stretchy="false">|</mo> <msub> <mi>z</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>z</mi> <mi>j</mi> </msub> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <munderover> <mo>∏</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>n</mi> <mi>Q</mi> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <mspace width="0.166667em" /> <msup> <mi>d</mi> <mn>2</mn> </msup> <msub> <mi>z</mi> <mi>i</mi> </msub> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where Q is a radially symmetric confining potential on the complex plane <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation>.The droplet is not assumed to be connected, but may consist of a number of disjoint annuli and possibly a central disk. The boundary condition is “soft edge”, i.e., Q is smooth in a <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation>-neighbourhood of the droplet.We include the following possibilities: (i) existence of “outposts”, i.e., components of the coincidence set which fall outside of the droplet, (ii) a Fisher-Hartwig singularity at the origin, (iii) perturbations <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$Q-\frac{h}{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Q</mi> <mo>-</mo> <mfrac> <mi>h</mi> <mi>n</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation> where h is a smooth radially symmetric test-function.In each case, the free energy <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\log Z_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>log</mo> <msub> <mi>Z</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> admits a large n expansion of the form <Equation ID="Equ143"> <EquationSource Format="TEX">$\begin{aligned}\log Z_n=C_1n^2+C_2n\log n+C_3 n+C_4\log n+C_5+\mathcal {G}_{n}+o(1)\end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mo>log</mo> <msub> <mi>Z</mi> <mi>n</mi> </msub> <mo>=</mo> <msub> <mi>C</mi> <mn>1</mn> </msub> <msup> <mi>n</mi> <mn>2</mn> </msup> <mo>+</mo> <msub> <mi>C</mi> <mn>2</mn> </msub> <mi>n</mi> <mo>log</mo> <mi>n</mi> <mo>+</mo> <msub> <mi>C</mi> <mn>3</mn> </msub> <mi>n</mi> <mo>+</mo> <msub> <mi>C</mi> <mn>4</mn> </msub> <mo>log</mo> <mi>n</mi> <mo>+</mo> <msub> <mi>C</mi> <mn>5</mn> </msub> <mo>+</mo> <msub> <mi mathvariant="script">G</mi> <mi>n</mi> </msub> <mo>+</mo> <mi>o</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$C_1,\ldots ,C_5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>C</mi> <mn>5</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> are certain geometric functionals. The n-dependent term <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {G}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">G</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> is bounded as <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$n\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>; it arises in the presence of spectral gaps.We use the free energy expansions to study the distribution of fluctuations of linear statistics. We prove that the fluctuations are well approximated by the sum of a Gaussian and certain independent terms which provide the displacement of particles from one component to another. This displacement depends on n and is expressed in terms of the Heine distribution. We also prove (under suitable assumptions) that the number of particles which fall near a spectral outpost converges to a Heine distribution.

Free Energy and Fluctuations in the Random Normal Matrix Model with Spectral Gaps

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2025年9月16日

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Yacin Ameur,
Christophe Charlier,
Joakim Cronvall

摘要

We study large n expansions for the partition function of a Coulomb gas $\begin{aligned}Z_n=\frac{1}{\pi ^n}\int _{\mathbb {C}^n}\prod _{1\le i<j\le n}|z_i-z_j|^2\prod _{i=1}^n e^{-nQ(z_i)}\, d^2 z_i,\end{aligned}$ $\begin{matrix} Z_{n} = \frac{1}{π^{n}} \int_{C^{n}} \prod_{1 \leq i < j \leq n} {| z_{i} - z_{j} |}^{2} \prod_{i = 1}^{n} e^{- n Q (z_{i})} d^{2} z_{i}, \end{matrix}$ where Q is a radially symmetric confining potential on the complex plane $\mathbb {C}$ $C$ .

The droplet is not assumed to be connected, but may consist of a number of disjoint annuli and possibly a central disk. The boundary condition is “soft edge”, i.e., Q is smooth in a $\mathbb {C}$ $C$ -neighbourhood of the droplet.

We include the following possibilities: (i) existence of “outposts”, i.e., components of the coincidence set which fall outside of the droplet, (ii) a Fisher-Hartwig singularity at the origin, (iii) perturbations $Q-\frac{h}{n}$ $Q - \frac{h}{n}$ where h is a smooth radially symmetric test-function.

In each case, the free energy $\log Z_n$ $log Z_{n}$ admits a large n expansion of the form $\begin{aligned}\log Z_n=C_1n^2+C_2n\log n+C_3 n+C_4\log n+C_5+\mathcal {G}_{n}+o(1)\end{aligned}$ $\begin{matrix} log Z_{n} = C_{1} n^{2} + C_{2} n log n + C_{3} n + C_{4} log n + C_{5} + G_{n} + o (1) \end{matrix}$ where $C_1,\ldots ,C_5$ $C_{1}, \dots, C_{5}$ are certain geometric functionals. The n-dependent term $\mathcal {G}_n$ $G_{n}$ is bounded as $n\rightarrow \infty $ $n \to \infty$ ; it arises in the presence of spectral gaps.

We use the free energy expansions to study the distribution of fluctuations of linear statistics. We prove that the fluctuations are well approximated by the sum of a Gaussian and certain independent terms which provide the displacement of particles from one component to another. This displacement depends on n and is expressed in terms of the Heine distribution. We also prove (under suitable assumptions) that the number of particles which fall near a spectral outpost converges to a Heine distribution.