[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83218-en":3,"doc-seo-83218-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},83218,962075114765,"Quinn","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Norm Bounds for Sparse Random Tensors and Spectral Gap of Random Hypergraphs","Second eigenvalue and spectral gap notions for hypergraphs generalize the graph adjacency-matrix eigenvalue framework of Friedman and Wigderson. The work proves that for r-uniform Erdős-Rényi hypergraphs, the spectral gap emerges as soon as the expected hyperedge count m is far larger than n^{r/2}. Prior bounds resolved only up to logarithmic factors, which are removed via an explicit decomposition of an associated selector process. The results also yield improved injective norm bounds for sparse random tensors with independent entries and extend Seginer’s random-matrix theorem to tensors under a moment-equivalence condition.","arXiv :2607 .07308v1 [math .PR] 8 Jul 2026  \nNorm Bounds for Sparse Random Tensors and Spectral Gap of Random Hypergraphs  \nKevin Lucca∗ Lucas Pesenti†  \nJuly 9, 2026  \nAbstract  \nFriedman and Wigderson (1995) introduced a notion of second eigenvalue for hypergraphs that generalizes the second eigenvalue of the adjacency matrix of a graph. We show that r-uniform Erdős-Rényi hypergraphs on n vertices exhibit a spectral gap as soon as their expected number of hyperedges m satisfies m ≫ nr/2 . Prior work identified this scale only up to logarithmic factors; removing these factors is the main technical challenge.  \nOur proof overcomes this obstacle through an explicit decomposition of an associated selector process, inspired by a generic decomposition theorem of Talagrand (2021) . As a consequence of our techniques, we obtain improved norm bounds for sparse random tensors with independent entries. Finally, under a mild moment equivalence assumption, we extend to tensors a seminal result of Seginer (2000) for random matrices with i.i.d. entries.  \n1 Introduction  \nCan we develop a spectral theory for hypergraphs that matches the power of the well-developed spectral theory of graphs? This long-standing question motivates the development of new tools for studying matrices that extend to tensors.  \nIn this line of work, Friedman and Wigderson [FW95] introduced a notion of second eigenvalue for hypergraphs that generalizes the second eigenvalue of the adjacency matrix of a graph. They also asked for the value of this second eigenvalue for random hypergraphs.  \nDefinition 1.1 (Erdős-Rényi model) . Let n, r ∈ N and p = p (r, n) ∈ [0 , 1] . Consider the distribution over r-uniform hypergraphs on the vertex set [n] := {1,..., n} obtained by independently including each subset of vertices e ⊆ [n] of size |e| = r as a hyperedge with probability p. We represent a hypergraph by its adjacency tensor T ∈ (Rn )⊗r, defined by Ti1 , ...,ir = 1 if {i1 ,..., ir} is a hyperedge, and Ti1 , ...,ir = 0 otherwise. Let Hr(n,p) denote the distribution of the adjacency tensors of Erdős-Rényi hypergraphs.  \nDefinition 1.2 (Injective tensor norm) . The injective norm of T ∈ (Rn )⊗r is  \n∥T∥ inj :=  |⟨T, X⟩| ,  \nwhere X := {x1 ⊗ ... ⊗ xr : ∥x1 ∥2 = .. . = ∥xr∥2 = 1} is the set of unit rank-1 tensors.  \n∗ ETH Zürich. kevin .lucca@ifor .math .ethz .ch †ETH Zürich. lpesenti@ethz .ch.  \nWhen T is invariant under permutations of its indices, the injective norm can equivalently be written as the coupled maximization problem over {x⊗r : ∥x∥2 = 1} [Kel28] .  \nFollowing the terminology of [FW95], when T ∼ Hr(n,p), we call ∥T − ET∥ inj the second eigenvalue1 of T , and ∥T∥ inj − ∥T − ET∥ inj the spectral gap of T. More explicitly, the second eigenvalue of an Erdős-Rényi hypergraph is:  \nn  \n∥T − ET∥ inj = ∥x1∥2 =m... xr∥2=1 i1 , . .X,ir=1 (Ti1,...,ir − p)x 1,i1 . . . xr,ir .  \ndistinct  \nIn the graph case (r = 2), ∥T∥ inj and ∥T − ET∥ inj are, respectively, the spectral norm of the uncentered and centered adjacency matrices of an Erdős-Rényi random graph. These quantities have been central in the development of spectral graph theory. In sparse random graphs, the emergence of a spectral gap coincides with the existence of efficient algorithms certifying (nearly) tight bounds on large cliques and independent sets; see Section 1.2. The emergence of a spectral gap also marks the region in which the expander mixing lemma holds; see Example 3.3.  \n1.1 Emergence of a spectral gap in Erdős-Rényi hypergraphs  \nMotivated by this analogy, we determine the sparsity scale at which the spectral gap of Erdős-Rényi hypergraphs diverges. The upper bound  \n∥T∥ inj − ∥T − ET∥ inj ⩽ ∥ET∥ inj ⩽ pnr/2  \nshows that p ≫ n−r/2 is a necessary condition for the spectral gap to diverge. Our first main result shows that, for r ⩾ 3, this condition is also sufficient.  \nTheorem 1.3 (Spectral gap of Erdős-Rényi hypergraphs; consequence of Theorem 3.1) . Let r ⩾ 3 be fixed, and let p = p (n) be such tha","cbCait8ZWZX7ddTa","https://ap.wps.com/l/cbCait8ZWZX7ddTa","pdf",620387,1,25,"English","en",105,"# Abstract\n# Introduction\n## Emergence of a spectral gap in Erdős-Rényi hypergraphs\n## Motivation: refutation of random constraint satisfaction problems","[{\"question\":\"What concept of eigenvalues is used for hypergraphs in this work?\",\"answer\":\"It uses Friedman and Wigderson’s notion of the second eigenvalue for hypergraphs, defined through the injective tensor norm of the centered adjacency tensor T−ET, generalizing the second eigenvalue for graphs.\"},{\"question\":\"At what sparsity level does the spectral gap for r-uniform Erdős-Rényi hypergraphs appear?\",\"answer\":\"For fixed r≥3, the spectral gap diverges once p n^{r/2}→∞, equivalently when the expected number of hyperedges satisfies m≫n^{r/2}.\"},{\"question\":\"How does the paper overcome the main technical challenge from prior work?\",\"answer\":\"It removes unavoidable polylogarithmic losses by proving bounds via an explicit decomposition of an associated selector process, inspired by Talagrand’s generic decomposition theorem (2021).\"}]",1784186015,63,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"norm-bounds-for-sparse-random-tensors-and-spectral-gap-of-random-hypergraphs","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/norm-bounds-for-sparse-random-tensors-and-spectral-gap-of-random-hypergraphs/83218/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What concept of eigenvalues is used for hypergraphs in this work?","Question",{"text":75,"@type":76},"It uses Friedman and Wigderson’s notion of the second eigenvalue for hypergraphs, defined through the injective tensor norm of the centered adjacency tensor T−ET, generalizing the second eigenvalue for graphs.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"At what sparsity level does the spectral gap for r-uniform Erdős-Rényi hypergraphs appear?",{"text":80,"@type":76},"For fixed r≥3, the spectral gap diverges once p n^{r/2}→∞, equivalently when the expected number of hyperedges satisfies m≫n^{r/2}.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the paper overcome the main technical challenge from prior work?",{"text":84,"@type":76},"It removes unavoidable polylogarithmic losses by proving bounds via an explicit decomposition of an associated selector process, inspired by Talagrand’s generic decomposition theorem 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