<p>We study the landscape complexity of the Hamiltonian <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$X_N(x) +\frac{\mu }{2} \Vert x\Vert ^2,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>X</mi> <mi>N</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mfrac> <mi>μ</mi> <mn>2</mn> </mfrac> <msup> <mrow> <mo stretchy="false">‖</mo> <mi>x</mi> <mo stretchy="false">‖</mo> </mrow> <mn>2</mn> </msup> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mu >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$X_{N}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>X</mi> <mi>N</mi> </msub> </math></EquationSource> </InlineEquation> is an isotropic Gaussian random field on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {R}^{N}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation>. We derive asymptotic formulas for the expected number of critical points of the Hamiltonian, as the dimension <i>N</i> tends to infinity. These results complement the corresponding findings for locally isotropic Gaussian random fields, as well as the work of Auffinger and Zeng (Auffinger and Zeng 402:951–993 2023, 2022) on non-isotropic Gaussian random fields with isotropic increments.</p>

Complexity Function of Isotropic Gaussian Random Fields

期刊论文
发表日期：

2026年3月12日

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Ieng Tak Leong,
Hao Xu

摘要

We study the landscape complexity of the Hamiltonian $X_N(x) +\frac{\mu }{2} \Vert x\Vert ^2,$ $X_{N} (x) + \frac{μ}{2} {‖ x ‖}^{2},$ where $\mu >0$ $μ > 0$ and $X_{N}$ $X_{N}$ is an isotropic Gaussian random field on $\mathbb {R}^{N}$ $R^{N}$ . We derive asymptotic formulas for the expected number of critical points of the Hamiltonian, as the dimension N tends to infinity. These results complement the corresponding findings for locally isotropic Gaussian random fields, as well as the work of Auffinger and Zeng (Auffinger and Zeng 402:951–993 2023, 2022) on non-isotropic Gaussian random fields with isotropic increments.