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Abstract
Symmetric matrices with zero row sums occur in many theoretical settings and in reallife applications. When the offdiagonal elements of such matrices are i.i.d. random variables and the matrices are large, the eigenvalue distributions converge to a peculiar universal curve $p_{\mathrm{zrs}}(\lambda)$ that looks like a cross between the Wigner semicircle and a Gaussian distribution. An analytic theory for this curve, originally due to Fyodorov, can be developed using supersymmetrybased techniques. We extend these derivations to the case of sparse matrices, including the important case of graph Laplacians for large random graphs with $N$ vertices of mean degree $c$. In the regime $1\ll c\ll N$, the eigenvalue distribution of the ordinary graph Laplacian (diffusion with a fixed transition rate per edge) tends to a shifted and scaled version of $p_{\mathrm{zrs}}(\lambda)$, centered at $c$ with width $\sim\sqrt{c}$. At smaller $c$, this curve receives corrections in powers of $1/\sqrt{c}$ accurately captured by our theory. For the normalized graph Laplacian (diffusion with a fixed transition rate per vertex), the large $c$ limit is a shifted and scaled Wigner semicircle, again with corrections captured by our analysis.
Original language  English 

Article number  295001 
Number of pages  29 
Journal  Journal of Physics A: Mathematical and Theoretical 
Volume  56 
Issue number  29 
DOIs  
Publication status  Published  26 Jun 2023 
Bibliographical note
v3: clarifications and references added, accepted for publication in J. Phys. AKeywords
 condmat.disnn
 condmat.statmech
 hepth
 mathph
 math.MP
 math.PR
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Dive into the research topics of 'Random matrices with row constraints and eigenvalue distributions of graph Laplacians'. Together they form a unique fingerprint.Projects
 2 Active

SRP72: SRPOnderzoekszwaartepunt: Highenergy physics (HEP@VUB).
D'Hondt, J., Buitink, S., Craps, B., De Vries, K., Lowette, S. & Mariotti, A.
1/11/22 → 31/10/27
Project: Fundamental

SRP8: Strategic Research Programme: HighEnergy Physics at the VUB
D'Hondt, J., Van Eijndhoven, N., Craps, B. & Buitink, S.
1/11/12 → 31/10/24
Project: Fundamental