{"id":"360aee51-49b1-4126-acfc-50a0b67b56ec","arxiv_id":"1908.03694","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For every prime d and any alpha below 1/6, there are infinitely many (d+1)-regular graphs with near-optimal second eigenvalue, logarithmic girth, and many fully localized eigenvectors on sets of size O(m^alpha).","lead":"This paper builds infinite families of high-girth regular graphs that are nearly Ramanujan expanders yet have many eigenvectors confined to tiny sets. It shows that strong expansion and large girth do not force quantum-ergodic delocalization, giving a discrete analogue of scarring.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly identifies the regularity and superharmonicity conditions of Kahale's lemma as the point on which the near-Ramanujan gap rests. My own scrutiny focused there. The proof does not explicitly verify the neighbor-count regularity for all layers, but the construction's large girth and the removed matching edges ensure the relevant balls are tree-like with the required uniform neighbor counts. The sequence s is constructed precisely to satisfy (12) at each layer, with the transition at X_r tuned to match the two-parent and d-1-child structure. The alleged risk from T3 attachments is avoided because the removed matching edges push the T3-attached vertices beyond the radius r+t considered. Therefore no load-bearing concern remains, and the verdict should stand.","tokens_in":181,"tokens_out":39740,"duration_ms":743249,"concrete_test":"Verify explicitly in the Section 3 construction that for X={u,u'}, every vertex in X_{r+1} has exactly one neighbor in X_r and every vertex in X_r has exactly d-1 neighbors in X_{r+1}, and that no T3 vertex appears in the ball of radius r+t around X. This would formally certify the hypotheses of Kahale's Lemma 4.2 for all layers used in Proposition 4.1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After careful review, the central claim appears sound. The most delicate point is the application of Kahale's Lemma 4.2 in Proposition 4.1, specifically whether the distance layers around X={u,u'} and X={v'} satisfy the regularity condition (all vertices in X_i have the same number of neighbors in X_j) up to radius r+t. I checked the local structure: the matching edges u_i v_i are removed, so the vertices v_i (leaves of T3) are not adjacent to their former parents u_i; hence the T3 branches attach to H only at vertices that are at distance at least g-r-1 >= 3r-1 from X, which lies beyond the radius r+t < g/2 used in the proof. Thus, within the ball of radius r+t, the graph is a regular layered tree: X_r has d-1 forward neighbors, X_{r+1} has one parent and d children, and deeper layers follow the same pattern. The superharmonicity check in (12) correctly accounts for these counts, including the two-parent structure at X_r. The exponential growth argument for |X_j|s_j^2 is consistent with the recurrence. I found no gap that would invalidate the near-Ramanujan bound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, for every prime d and every alpha in (0,1/6), an infinite family of (d+1)-regular graphs G_m with girth at least 2 alpha log_d(m)(1-o_d(1)), all nontrivial adjacency eigenvalues bounded in absolute value by (3/sqrt(2))sqrt(d), and at least floor(alpha log_d m) eigenvalues whose eigenvectors are supported on a set of size O(m^alpha). The construction combines a deterministic Erdős–Sachs-style pairing of two d-ary trees (Lemma 2.1) with a high-girth non-bipartite Ramanujan graph H of Lubotzky–Phillips–Sarnak, attaching two additional trees at the leaves of a ball in H. The spectral analysis uses Kahale's growth lemma for eigenfunctions to show that the mass of any large-eigenvalue eigenvector on the interface between the trees and H is negligible, yielding the near-Ramanujan bound. A variant with several well-separated attachments produces many localized eigenvectors with large multiplicity, and the set of localized eigenvalues over the family is dense in (-2 sqrt(d), 2 sqrt(d)).","tokens_in":12100,"tokens_out":46955,"duration_ms":449386,"significance":"If the proof is correct, the paper gives the first construction of high-girth expanders with strongly localized eigenvectors, simultaneously achieving near-optimal spectral gap, logarithmic girth, and small support for many eigenvectors. This strengthens the earlier non-expanding examples of Ganguly–Srivastava and provides a discrete analogue of scarring phenomena relevant to quantum ergodicity on graphs, since the constructed graphs satisfy the Benjamini–Schramm and expansion hypotheses of Anantharaman–Le Masson yet have localized eigenfunctions. The proof is largely self-contained and the main spectral estimate rests on a careful application of a lemma of Kahale; the combinatorial tree-pairing lemma is proved in detail. The paper also includes a high-multiplicity variant and a density statement for the localized eigenvalues, both of which are natural strengthenings of the main theorem.","major_comments":[],"minor_comments":[{"comment":"The verification of inequality (12) for v in X_{r+1} is terse: the text says it suffices to check that sqrt(d) s_{r+1}/x_1 + sqrt(d) x_2 s_{r+1} <= |mu| s_{r+1}, and that this follows from x_2 <= b + epsilon - 1/x_1. Since x_1 = 1/c and b = c + 1/c, this yields x_2 <= 1/c + epsilon, which is exactly what the displayed inequality needs; spelling out this substitution would make the argument easier to follow.","section":"Section 4, proof of Proposition 4.1"},{"comment":"The printed definition of b as `b = 3d-1/√d(2d-1)` is ambiguous. It should read b = (3d-1)/sqrt(d(2d-1)), which equals c + 1/c for c = sqrt((2d-1)/d); the current typography could mislead a reader about the algebra that follows.","section":"Section 4, definition of b"},{"comment":"After the matching edges u_i v_i are removed, each v_i is disconnected from its former parent u_i within the local ball of radius r+t, so the branches of T3 attach to H only deep beyond the radii used in the proof of Proposition 4.1. Adding a short remark to this effect would preempt the natural concern that the glued trees T2 and T3 interfere with the layered regularity required by Lemma 4.2.","section":"Section 3 and Section 4"},{"comment":"In the proof of Lemma 2.1, the notation m(r,s) for the number of paths conflicts with the use of m for the number of vertices elsewhere in the paper; renaming the path count (for example, to N(r,s)) would improve clarity.","section":"Lemma 2.1"}],"recommendation":"minor_revision","confidential_remarks":"The paper appears technically sound; the delicate application of Kahale's lemma is legitimate because the removed matching edges disconnect the T3 branches from the rest of the graph within the relevant local balls. The main issue is presentation, especially the abbreviated verification of (12) and the ambiguous definition of b. I recommend minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The central result is new and worth knowing: for every prime d and alpha in (0,1/6), there are infinitely many (d+1)-regular graphs that are simultaneously near-Ramanujan, have logarithmic girth, and admit many eigenvectors fully supported on sublinear sets. This strengthens the earlier Ganguly–Srivastava construction by adding a spectral gap, and it gives a clean counterexample to the strong form of quantum ergodicity on graphs. The denseness of the localized eigenvalues is a nice bonus.\n\nThe proof is careful and mostly convincing. The key spectral step is Proposition 4.1, which uses Kahale's lemma to bound the mass of any eigenvector on the interface sets. The delicate point is the regularity of the distance layers around X={u,u'}, and the removed matching edges are exactly what make it work: they push the T3 attachments far from X, so within the ball of radius r+t the graph looks like a regular layered tree. I checked that the superharmonicity inequalities in (12) line up with the degree counts, and they do. The dependence on prior work, including the authors' own GS18, is clearly disclosed and not circular.\n\nSoft spots are minor. The claim that the construction is 'explicit' is a bit generous: the matching in Lemma 2.1 comes from an Erdős–Sachs style extremal argument, so it is deterministic but not constructively specified. The spectral gap is (3/sqrt(2)) sqrt(d) rather than the optimal 2 sqrt(d), and the girth is at most about (1/3) log m, so the title's 'high-girth' should be read as logarithmic, not optimal. The multi-tree extension in Theorem 1.3 is plausible but terse; it says the same arguments apply without fully checking the layer structure for multiple interfaces. None of these threaten the main result.\n\nOverall, this is a solid paper that deserves a serious referee. I would send it out and expect acceptance after minor revision, perhaps with a more measured word than 'explicit' and a short appendix on the multi-tree case.\n\nRecommendation: accept, with minor comments.","headline":"A sound and interesting construction: high-girth expanders with localized eigenvectors; the proof's delicate layer-regularity check survives scrutiny.","tokens_in":12614,"tokens_out":13987,"would_cite":true,"duration_ms":133753,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C50","05C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"High-girth expanders exist whose many eigenvectors live entirely on tiny sets.","keywords":["high-girth graphs","Ramanujan graphs","eigenvector localization","spectral gap","quantum ergodicity on graphs","scarring","d-ary trees","vertex expansion"],"falsifier":"Compute the full adjacency spectrum of the explicit graph for a small prime $d$ (say $d=2$) and a moderate $m$ with $\\alpha=0.1$, and check whether any nontrivial eigenvalue exceeds $(3/\\sqrt{2})\\sqrt{d}+\\epsilon$ or whether the claimed tree-supported eigenvectors have support larger than $O(m^\\alpha)$; a single violation refutes the theorem. A simpler verification is to check the inequality $As(v)\\le|\\mu|s(v)$ for the explicit test function $s$ at every layer of the constructed graph.","tokens_in":11709,"feed_emoji":"🌲","tokens_out":14174,"duration_ms":126500,"temperature":0.7,"pith_summary":"This paper proves that for every prime $d$ and every $\\alpha\\in(0,1/6)$, there are infinitely many $(d+1)$-regular graphs whose nontrivial eigenvalues are all at most $(3/\\sqrt{2})\\sqrt{d}$, whose shortest cycles grow like $2\\alpha\\log_d m$, and which have at least $\\lfloor\\alpha\\log_d m\\rfloor$ eigenvalues whose eigenvectors are supported entirely on a set of size $O(m^\\alpha)$. Earlier work produced such localized eigenvectors in high-girth graphs without expansion; this construction adds a near-Ramanujan spectral gap, so expansion and high girth do not force delocalization. The result is a discrete analogue of the 'scarring' phenomenon in quantum ergodic systems, and the graphs are constructed explicitly. The eigenvalues carried by the localized eigenvectors are dense in the whole bulk interval $(-2\\sqrt{d},2\\sqrt{d})$.","feed_headline":"High-girth expanders can hide eigenvectors in tiny sets","feed_subtitle":"Many eigenvectors sit entirely on O(m^alpha) vertices, even inside highly expanding high-girth graphs.","key_machinery":"The construction starts from an explicit family of $(d+1)$-regular Ramanujan graphs and selects a vertex $u$; the ball of radius $r=\\lfloor\\alpha\\log_d m\\rfloor$ around $u$ is a $d$-ary tree $T_1$. Two isomorphic copies $T_2,T_3$ are glued respectively to the leaves of $T_1$ and to vertices at distance $r+1$ from $u$ in the base graph, using a deterministic leaf-pairing that preserves high girth (the pair-swapping method for high-girth regular graphs). The spectral control is carried by a growth-rate lemma: a positive radial test function $s$ is defined on the distance layers from the interface set $X=\\{u,u'\\}$ so that $As(v)\\le|\\mu|s(v)$ holds up to half the girth; the lemma then forces the mass of any eigenvector of eigenvalue $\\mu$ on the interface to be negligible, yielding the bound $|\\mu|\\le \\frac{3d-1}{\\sqrt{d}(2d-1)}\\sqrt{d}<\\frac{3}{\\sqrt2}\\sqrt{d}$ for large $d$. Localized eigenvectors come from radial eigenvectors of the finite $d$-ary trees, whose eigenvalues are dense in $(-2\\sqrt{d},2\\sqrt{d})$; the glued trees carry these vectors with zero extension to the rest of the graph.","core_discovery":"The central claim is that high girth and a near-optimal spectral gap are compatible with extreme eigenvector localization. For every prime $d$ and $\\alpha\\in(0,1/6)$, the paper constructs infinitely many $(d+1)$-regular graphs $G_m$ on $m$ vertices such that (i) every nontrivial eigenvalue of the adjacency matrix has absolute value at most $(3/\\sqrt{2})\\sqrt{d}$; (ii) the girth is at least $2\\alpha\\log_d m\\,(1-o_d(1))$; and (iii) there are at least $\\lfloor\\alpha\\log_d m\\rfloor$ eigenvalues in $(-2\\sqrt{d},2\\sqrt{d})$ whose eigenvectors vanish outside a set $S_m$ of size $O(m^\\alpha)$. Moreover, the eigenvalues realized by these localized eigenvectors are dense in $(-2\\sqrt{d},2\\sqrt{d})$. Because the graphs satisfy the tree-convergence and spectral-gap hypotheses used in quantum ergodicity theorems on graphs, this shows that those hypotheses cannot imply unique ergodicity in the strong sense of every subsequence of eigenvectors equidistributing.","pith_inferences":["If the spectral analysis is correct, the same gluing recipe should work with any good spectral expander as the base, not only Ramanujan graphs; the superharmonic test function only uses the regularity of distance layers, so the $3/\\sqrt2$ bound may persist for a wider class of bases after adjusting constants.","The density of localized eigenvalues suggests that, for these graphs, the average over a spectral window of width about $1/\\log m$ would likely fail to equidistribute; the paper leaves this question open (Remark 1.2), and a finite computation could test it directly.","Because the construction is explicit and deterministic, it provides a concrete testbed for numerical studies of eigenvector statistics on expanding high-girth graphs, complementing random regular graphs where such localization is much weaker.","One could try to push the localization radius exponent $\\alpha$ beyond $1/6$ by using base graphs with girth larger than $\\frac23\\log_d m$; the paper's restriction $\\alpha<1/6$ comes from the currently available girth of explicit Ramanujan families, so a new family with girth $c\\log m$ for $c>2/3$ would improve the theorem."],"forward_implications":["The graphs satisfy the Benjamini–Schramm tree-convergence condition and a spectral gap, yet they carry eigenvectors completely localized on small sets; therefore the two hypotheses used in the graph quantum ergodicity theorem cannot imply unique ergodicity in the sense of every subsequence of eigenvectors becoming equidistributed.","The localized eigenvalues are dense in $(-2\\sqrt{d},2\\sqrt{d})$, so localization occurs at essentially every frequency in the bulk of the spectrum, not only near the spectral edges.","A modified construction (Theorem 1.3) produces $\\lfloor\\alpha\\log_d m\\rfloor$ eigenvalues each with multiplicity $\\Omega(m^{1-4\\alpha})$, with an orthogonal basis of localized eigenvectors for each eigenspace; this gives many localized modes in a narrow spectral window.","The spectral bound $\\frac{3}{\\sqrt2}\\sqrt{d}\\approx 2.121\\sqrt{d}$ lies within a few percent of the optimal $2\\sqrt{d}$ for an infinite family, and the paper notes this construction cannot be improved to $2\\sqrt{d}$ because a certain small set has vertex expansion below $(d+1)/2$."],"supporting_citations":[{"why":"It supplies the starting construction of high-girth graphs with localized tree eigenvectors and the radial tree eigenvalue lemmas used here.","marker":"[GS18]"},{"why":"It provides the explicit Ramanujan graphs used as the base graph and as the degree-correcting high-girth regular gadget.","marker":"[LPS88]"},{"why":"It supplies the growth-rate lemma that bounds eigenfunction mass on the interface, and the expansion theorem used to note the construction is not exactly Ramanujan.","marker":"[Kah95]"},{"why":"It provides the deterministic leaf-pairing swap method that Lemma 2.1 adapts for high girth.","marker":"[ES63]"},{"why":"It gives the lower bound showing $2\\sqrt{d}$ is optimal for an infinite family, used to contextualize the $3/\\sqrt{2}$ bound.","marker":"[Nil91]"},{"why":"It establishes the delocalization lower bound that this construction shows is nearly sharp.","marker":"[BL13]"},{"why":"It is the graph quantum ergodicity theorem used to derive the scarring implication.","marker":"[ALM15]"},{"why":"It is the refined quantum ergodicity over spectral windows referenced for the open problem of small non-equidistributed windows.","marker":"[BLML15]"}],"fun_headline_variants":["Expander graphs with high girth host localized eigenvectors","High girth and strong expansion still allow eigenvector localization","Hidden eigenvectors: high-girth expanders with scarred modes","Tight spectral gap meets localized modes in high-girth graphs","High-girth expanders: eigenvectors confined to tiny sets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The spectral argument would collapse if the explicitly built radial test function fails to satisfy $As(v)\\le|\\mu|s(v)$ at any vertex within half the girth, which is why the parameters are restricted to $\\alpha<1/6$ so the base Ramanujan graph's girth stays above $4r$.","fun_headline_variants_meta":{"raw":{"variants":["Expander graphs with high girth host localized eigenvectors","High girth and strong expansion still allow eigenvector localization","Hidden eigenvectors: high-girth expanders with scarred modes","Tight spectral gap meets localized modes in high-girth graphs","High-girth expanders: eigenvectors confined to tiny sets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000584,"raw_usage":{"total_tokens":2746,"prompt_tokens":946,"completion_tokens":1800,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":562,"completion_tokens_details":{"reasoning_tokens":1718}},"tokens_in":562,"tokens_out":1800,"duration_ms":13372,"temperature":1.0,"reasoning_tokens":1718,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:10:35.520199+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the full adjacency spectrum of the explicit graph for a small prime $d$ (say $d=2$) and a moderate $m$ with $\\alpha=0.1$, and check whether any nontrivial eigenvalue exceeds $(3/\\sqrt{2})\\sqrt{d}+\\epsilon$ or whether the claimed tree-supported eigenvectors have support larger than $O(m^\\alpha)$; a single violation refutes the theorem. A simpler verification is to check the inequality $As(v)\\le|\\mu|s(v)$ for the explicit test function $s$ at every layer of the constructed graph.","supporting_citations":[],"review_version":1}