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完整後設資料紀錄
DC 欄位 | 值 | 語言 |
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dc.contributor.advisor | 林偉傑 | zh_TW |
dc.contributor.advisor | Wai-Kit Lam | en |
dc.contributor.author | 張雋烺 | zh_TW |
dc.contributor.author | Chun-Long Cheung | en |
dc.date.accessioned | 2024-07-01T16:12:09Z | - |
dc.date.available | 2024-07-02 | - |
dc.date.copyright | 2024-07-01 | - |
dc.date.issued | 2024 | - |
dc.date.submitted | 2024-04-29 | - |
dc.identifier.citation | [1] L. Addario-Berry and B. Reed. Minima in branching random walks. The Annals of Probability, 37, 05 2009. doi:10.1214/08-aop428.
[2] L. Addario-Berry and B. A. Reed. Ballot theorems for random walks with finite variance, 02 2008. arXiv:0802.2491. [3] L. Addario‐Berry and B. Reed. Ballot theorems, old and new. Bolyai Society math- ematical studies, pages 9–35, 10 2008. doi:10.1007/978-3-540-77200-2_1. [4] L.-P. Arguin, D. Belius, and P. Bourgade. Maximum of the characteristic polynomial of random unitary matrices. Communications in Mathematical Physics, 349:703–751, 09 2016. doi:10.1007/s00220-016-2740-6. [5] L.-P. Arguin, D. Belius, and A. J. Harper. Maxima of a randomized Riemann zeta function, and branching random walks. arXiv, 06 2015. doi:10.48550/arxiv.1506.00629. [6] L.-P. Arguin, P. Bourgade, and M. Radziwiłł. The Fyodorov–Hiary–Keating conjecture. i. arXiv, 07 2020. arXiv:2007.00988. [7] L.-P. Arguin, P. Bourgade, and M. Radziwiłł. The Fyodorov–Hiary–Keating conjecture. ii. arXiv, 07 2023. doi:10.48550/arxiv.2307.00982 [8] R. Arratia and S. Tavare. The cycle structure of random permutations. The Annals of Probability, 20, 07 1992. doi:10.1214/aop/1176989707. [9] R. R. Bahadur and R. R. Rao. On deviations of the sample mean. Annals of Mathematical Statistics, 31:1015–1027, 12 1960. doi:10.1214/aoms/1177705674. [10] G. Ben Arous and P. Bourgade. Extreme gaps between eigenvalues of random matrices. The Annals of Probability, 41, 07 2013. doi:10.1214/11-aop710. [11] M. Biskup. Random Graphs, Phase Transitions, and the Gaussian Free Field. Springer International Publishing, 01 2020. doi:10.1007/978-3-030-32011-9. [12] P. Bourgade, P. Lopatto, and O. Zeitouni. Optimal rigidity and maximum of the characteristic polynomial of wigner matrices. arXiv, 12 2023. doi:10.48550/arxiv.2312.13335. [13] P. Bourgade, K. Mody, and M. Pain. Optimal local law and central limit theorem for β-ensembles. Communications in Mathematical Physics, 390:1017–1079, 01 2022. doi:10.1007/s00220-022-04311-2. [14] M. D. Bramson, J. Ding, and O. Zeitouni. Convergence in law of the maximum of the two‐dimensional discrete Gaussian free field. Communications on Pure and Applied Mathematics, 69:62–123, 10 2015. doi:10.1002/cpa.21621. [15] N. Cook and O. Zeitouni. Maximum of the characteristic polynomial for a random permutation matrix. Communications on Pure and Applied Mathematics, 73:1660–1731, 05 2020. doi:10.1002/cpa.21899. [16] P. Deift, A. Its, and I. Krasovsky. On the asymptotics of a Toeplitz determinant with singularities, 2012. arXiv:1206.1292. [17] A. Dembo and O. Zeitouni. Large deviations techniques and applications. Springer, 2010. [18] P. Diaconis and M. Shahshahani. On the eigenvalues of random matrices. Journal of Applied Probability, 31:49–62, 01 1994. [19] Y. V. Fyodorov, G. A. Hiary, and J. P. Keating. Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta function. Physical Review Letters, 108, 04 2012. doi:10.1103/physrevlett.108.170601. [20] A. J. Harper. A note on the maximum of the Riemann zeta function, and log-correlated random variables, 04 2013. arXiv:1304.0677. [21] G. J. O. Jameson. The Prime number theorem. Cambridge University Press, 2003. [22] A. Kroó. On optimal polynomial meshes. Journal of Approximation Theory, 163:1107–1124, 09 2011. doi:10.1016/j.jat.2011.03.007. [23] G. Lambert. Maximum of the characteristic polynomial of the Ginibre ensemble. Communications in Mathematical Physics, 378:943–985, 07 2020. doi:10.1007/s00220-020-03813-1. [24] G. F. Lawler. Intersections of Random Walks. New York, NY Springer, Imprint: Birkhäuser, 2013. [25] B. Rider and B. Virag. The noise in the circular law and the Gaussian free field. International Mathematics Research Notices, 07 2010. doi:10.1093/imrn/rnm006. [26] D. Slepian. The one-sided barrier problem for Gaussian noise. Bell System Technical Journal, 41:463–501, 03 1962. doi:10.1002/j.1538-7305.1962.tb02419.x. [27] C. Stone. A local limit theorem for nonlattice multi-dimensional distribution func- tions. The Annals of Mathematical Statistics, 36:546–551, 04 1965. doi:10.1214/ aoms/1177700165. | - |
dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/92807 | - |
dc.description.abstract | 這篇論文是探討對數相關場的最大值的行為。隨機場中最大值的行為在物理學、機率論和數論等各個領域都具有重要意義。在本論文中,我們將重點放在分支隨機遊走(BRW)和二維離散高斯自由場(GFF)。我們將證明 BRW 最大值的緊密性和收斂性,並建立 GFF 和 BRW 之間的連結。然後我們會給兩個對數相關場的例子:臨界線上的黎曼 zeta 函數和隨機酉矩陣的特徵多項式。 | zh_TW |
dc.description.abstract | This thesis explores the behavior of the maximum of log-correlated fields. The be- havior of maxima in random fields holds significant importance in various fields, includ- ing physics, probability theory, and number theory. In this thesis, we focus on branching random walks (BRW) and two dimensional discrete Gaussian free fields (GFF). We will prove the tightness and convergence in law of the centered maximum for BRW and estab- lish a connection between GFF and BRW. Then we present two examples of log-correlated fields, namely the Riemann zeta function on the critical line and the characteristic poly- nomial of a random unitary matrix. | en |
dc.description.provenance | Submitted by admin ntu (admin@lib.ntu.edu.tw) on 2024-07-01T16:12:09Z No. of bitstreams: 0 | en |
dc.description.provenance | Made available in DSpace on 2024-07-01T16:12:09Z (GMT). No. of bitstreams: 0 | en |
dc.description.tableofcontents | Verification Letter from the Oral Examination Committee i
摘要 iii Abstract v Contents vii Chapter 1 Introduction 1 Chapter 2 Branching Random Walks 3 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Definitions and Assumptions . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Tightness of the centered maximum . . . . . . . . . . . . . . . . . . 4 2.4 Convergence in law of the maximum . . . . . . . . . . . . . . . . . 12 2.5 Relaxing assumption (I) . . . . . . . . . . . . . . . . . . . . . . . . 13 Chapter 3 2D Gaussian Free Field 17 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Comparison with BRW . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Limit of the maximum . . . . . . . . . . . . . . . . . . . . . . . . . 22 Chapter 4 Two examples of log-correlated fields 25 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 Riemann zeta function on the critical line . . . . . . . . . . . . . . . 25 4.3 Characteristic polynomial of random matrices . . . . . . . . . . . . . 30 References 37 | - |
dc.language.iso | en | - |
dc.title | 分支隨機遊走和對數相關場的極值 | zh_TW |
dc.title | Extrema of Branching Random Walks and Log-Correlated Fields | en |
dc.type | Thesis | - |
dc.date.schoolyear | 112-2 | - |
dc.description.degree | 碩士 | - |
dc.contributor.oralexamcommittee | 李志煌;何政衞 | zh_TW |
dc.contributor.oralexamcommittee | Jhih-Huang Li;Ching-Wei Ho | en |
dc.subject.keyword | 分支隨機遊走,高斯自由場,對數相關場, | zh_TW |
dc.subject.keyword | Branching random walks,Gaussian free fields,Log-Correlated Fields, | en |
dc.relation.page | 40 | - |
dc.identifier.doi | 10.6342/NTU202400910 | - |
dc.rights.note | 同意授權(限校園內公開) | - |
dc.date.accepted | 2024-04-30 | - |
dc.contributor.author-college | 理學院 | - |
dc.contributor.author-dept | 應用數學科學研究所 | - |
顯示於系所單位: | 應用數學科學研究所 |
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