Abstract
We study the energy landscape of a model of a single particle on a random potential, that is, we investigate the topology of level sets of smooth random fields on \({\mathbb {R}}^{N}\) of the form \(X_N(x) +\frac{\mu }{2} \Vert x\Vert ^2,\) where \(X_{N}\) is a Gaussian process with isotropic increments. We derive asymptotic formulas for the mean number of critical points with critical values in an open set as the dimension N goes to infinity. In a companion paper, we provide the same analysis for the number of critical points with a given index.
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Acknowledgements
We would like to thank Yan Fyodorov for suggesting the study of fields with isotropic increments and providing several references. We are also grateful to the anonymous referees for many suggestions which have significantly improved the presentation of the paper.
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Antonio Auffinger: research partially supported by NSF Grant CAREER DMS-1653552 and NSF Grant DMS-1517894. Qiang Zeng: research partially supported by SRG 2020-00029-FST and FDCT 0132/2020/A3.
Appendices
Covariance Function and Its Derivatives
Let \(D_N(r)=D(r/N)\). For \(x,y\in {\mathbb {R}}^N\), let \(\varphi (x,y) =\frac{1}{2}(D_N(\Vert x\Vert ^2)+D_N(\Vert y\Vert ^2)-D_N(\Vert x-y\Vert ^2))\). Under \(X_N(0)=0\), isotropic increments imply that \({\mathbb {E}}X_N(x)=0\); see [Yag87, p.439]. We have
Lemma A.1
Assume Assumptions I and II. Then for \(x\in {\mathbb {R}}^N\),
where \(\delta _{ij}\) are the Kronecker delta function.
Proof
By [AT07, Theorem 1.4.2], \(X_N(x)\) is smooth. We can differentiate inside expectation as in [AT07, (5.5.4)] to find the expectations and
Substituting \(x=y\),
Then we note that \(D'_N(r)=D'(r/N)/N\) and \(D_N''(r)=D''(r/N)/N^2\). \(\quad \square \)
Auxiliary Lemmas
For the integral \({\mathbb {E}}\int _{\mathbb {R}}\exp \big (-\frac{1}{2}(N+1)x^2 -\frac{\sqrt{N(N+1)}\mu x}{\sqrt{-D''(0)}}\big ) L_{N+1}(\textrm{d}x)\), we have the following elementary fact which is used in Sect. 2.
Lemma B.1
Let \(\nu _N\) be probability measures on \({\mathbb {R}}\) and \(\mu \ne 0\). Suppose
Then we have
Proof
Let
We claim \(\lim _{N\rightarrow \infty } \frac{1}{N} \log \frac{ a_N}{c_{N}}=0\). Indeed, note that
By Jensen’s inequality,
Then the claim follows from the assumption that \(\lim _{N\rightarrow \infty } \frac{1}{N}\log a_N >-\infty \). From the elementary inequality \(a\wedge b \le (a+b)/2 \le a\vee b\), we have \(\lim _{N\rightarrow \infty } \frac{1}{N} (\log ( a_{N} + c_{N})-\log a_N )=0\). It remains to prove that
Note that
Let t be a large constant (independent of N) such that
and that
It follows that
and since \(\frac{1}{N} \log \frac{a_N}{c_N}\rightarrow 0\) as \(N\rightarrow \infty \),
Note that
Since
we have \(\lim _{N\rightarrow \infty } \frac{1}{N} (\log ( a_{N} + c_{N}) - \log b_N)=0\). \(\quad \square \)
The following discussion is about Assumption IV.
Proof of Lemma 3.2
1. Since \(y\mapsto D'(y)\) is a strictly decreasing convex function and \(D'''(y)>0\) for any \(y>0\), \(|D''(y)|< \frac{D'(0)-D'(y)}{y} \). By assumption,
It follows that
2. We verify (3.10). If (3.11) holds, then \(y\mapsto \beta (y)^2\) is a decreasing function and (3.10) follows from Lemma 3.1.
3. By item 1, it suffices to check (3.10). Consider the function
Condition (3.10) is equivalent to \(f(y)\ge 0\). Note that \(f(0)=0\) and that
By convexity, \( \frac{D'(y)-D'(0)}{y} \le D''(y)\le 0\). If (3.12) holds, \(D''(0)D'(y)y -D'(0)[D'(y)-D'(0)]\le 0\) and
Then (3.10) follows from here since \(D'(0)\ge D'(y)\) and we have \(f'(y)\ge 0\).
4. By Cauchy’s mean value theorem, condition (3.12) is equivalent to (3.13).
5. Direct calculation yields
6. By the representation (3.7) of Thorin–Bernstein functions, we have
By the Cauchy–Schwarz inequality, we have
It follows that \(\frac{\textrm{d}}{\textrm{d}y} \frac{D'(y)}{- D''(y)} \ge 1\) and (3.14) holds. \(\quad \square \)
If \(A=0\) in the representation (1.2), using the Cauchy–Schwarz inequality, we can see
compared with (3.14). It is easy to check that for any \({\varepsilon }>0, 0<\gamma <1\), our major examples \(D(r)=\log (1+r/{\varepsilon })\) and \(D(r)=(r+{\varepsilon })^\gamma -{\varepsilon }^{\gamma }\) satisfy (3.13). With more work, one can check that these functions satisfy (3.11).
On the other hand, according to [SSV12, p. 332],
is a complete Bernstein function which is not Thorin–Bernstein. One can check (at least numerically) that it violates (3.13) but still verifies (3.10). We suspect that (3.8) and (3.9) always hold for any structure function D. The following shows that this is the case at least in a neighborhood of 0.
Lemma B.2
Assume \(A=0\) in (1.2). We have
Consequently, there exists \(\delta >0\) such that \(-2D''(0)>[\alpha (y) y+\beta (y)]\beta (y)\) and \(-4D''(0)>[\alpha (y) y+\beta (y)]\alpha (y)y\) for \(y\in (0,\delta )\).
Proof
We only prove the first inequality as the second is similar. Write
Since \([(\alpha y+\beta )\beta ]'=\frac{T'B-B'T}{B^2}\) and \(\lim _{y\rightarrow 0+} B=-\frac{3}{2} D''(0) \ne 0\), it suffices to show that \(\lim _{y\rightarrow 0+}T'B-B'T<0\). By calculation, we have \(\lim _{y\rightarrow 0+} T= 3D''(0)^2\) and
After some tedious computation, we find \(\lim _{y\rightarrow 0+} T'=4D'''(0) D''(0)\) and \(\lim _{y\rightarrow 0+} B'=-\frac{5}{6} D'''(0)-\frac{D''(0)^2}{D'(0)}\). Then
By the Cauchy–Schwarz inequality,
From here the conclusion follows. \(\quad \square \)
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Auffinger, A., Zeng, Q. Complexity of Gaussian Random Fields with Isotropic Increments. Commun. Math. Phys. 402, 951–993 (2023). https://doi.org/10.1007/s00220-023-04739-0
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DOI: https://doi.org/10.1007/s00220-023-04739-0