Journal of the London Mathematical Society-Second Series最新文献

{"title":"Spectral large deviations of sparse random matrices","authors":"Shirshendu Ganguly,&nbsp;Ella Hiesmayr,&nbsp;Kyeongsik Nam","doi":"10.1112/jlms.12954","DOIUrl":"https://doi.org/10.1112/jlms.12954","url":null,"abstract":"<p>Eigenvalues of Wigner matrices has been a major topic of investigation. A particularly important subclass of such random matrices, useful in many applications, are what are known as sparse or diluted random matrices, where each entry in a Wigner matrix is multiplied by an independent Bernoulli random variable with mean <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>. Alternatively, such a matrix can be viewed as the adjacency matrix of an Erdős–Rényi graph <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>G</mi>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>,</mo>\u0000 <mi>p</mi>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$mathcal {G}_{n,p}$</annotation>\u0000 </semantics></math> equipped with independent and identically distributed (i.i.d.) edge-weights. An observable of particular interest is the largest eigenvalue. In this paper, we study the large deviations behavior of the largest eigenvalue of such matrices, a topic that has received considerable attention over the years. While certain techniques have been devised for the case when <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math> is fixed or perhaps going to zero not too fast with the matrix size, we focus on the case <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>=</mo>\u0000 <mfrac>\u0000 <mi>d</mi>\u0000 <mi>n</mi>\u0000 </mfrac>\u0000 </mrow>\u0000 <annotation>$p = frac{d}{n}$</annotation>\u0000 </semantics></math>, that is, constant average degree regime of sparsity, which is a central example due to its connections to many models in statistical mechanics and other applications. Most known techniques break down in this regime and even the typical behavior of the spectrum of such random matrices is not very well understood. So far, results were known only for the Erdős–Rényi graph <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>G</mi>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>,</mo>\u0000 <mfrac>\u0000 <mi>d</mi>\u0000 <mi>n</mi>\u0000 </mfrac>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$mathcal {G}_{n,frac{d}{n}}$</annotation>\u0000 </semantics></math> <i>without</i> edge-weights and with <i>Gaussian</i> edge-weights. In the present article, we consider the effect of general weight distributions. More specifically, we consider entry distributions whose tail probabilities decay at rate <span></span><math>\u0000 <semantics>\u0000 ","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141489020","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Discontinuous homomorphisms on C(X) with the negation of CH and a weak forcing axiom","authors":"Yushiro Aoki","doi":"10.1112/jlms.12956","DOIUrl":"https://doi.org/10.1112/jlms.12956","url":null,"abstract":"<p>In this paper, I introduce the properties <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>EPC</mi>\u0000 <msub>\u0000 <mi>ℵ</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 </msub>\u0000 <annotation>$mathrm{EPC}_{aleph _1}$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ProjCes</mi>\u0000 <mo>(</mo>\u0000 <mi>E</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathrm{ProjCes}(E)$</annotation>\u0000 </semantics></math> for forcing notions and show that it is consistent that the forcing axiom for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>EPC</mi>\u0000 <msub>\u0000 <mi>ℵ</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 </msub>\u0000 <mo>+</mo>\u0000 <mi>ProjCes</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>E</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$mathrm{EPC}_{aleph _1}+ mathrm{ProjCes}(E)$</annotation>\u0000 </semantics></math> forcing notions holds, the continuum hypothesis fails, and an ultrapower of the field of reals has the property <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>β</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <annotation>$beta _1$</annotation>\u0000 </semantics></math>. This provides a partial solution to H. Woodin's question concerning the existence of discontinuous homomorphisms on the Banach algebra of all complex-valued continuous functions on a compact space. Furthermore, we prove that the uniformization of a coloring of a ladder system on a stationary–costationary set <span></span><math>\u0000 <semantics>\u0000 <mi>E</mi>\u0000 <annotation>$E$</annotation>\u0000 </semantics></math> is an example of an <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>EPC</mi>\u0000 <msub>\u0000 <mi>ℵ</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 </msub>\u0000 <mo>+</mo>\u0000 <mi>ProjCes</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>ω</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>∖</mo>\u0000 <mi>E</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$mathrm{EPC}_{aleph _1}+ mathrm{ProjCes}(omega _1 setminus E)$</annotation>\u0000 </se","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141488323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Full Souslin trees at small cardinals","authors":"Assaf Rinot,&nbsp;Shira Yadai,&nbsp;Zhixing You","doi":"10.1112/jlms.12957","DOIUrl":"https://doi.org/10.1112/jlms.12957","url":null,"abstract":"<p>A <span></span><math>\u0000 <semantics>\u0000 <mi>κ</mi>\u0000 <annotation>$kappa$</annotation>\u0000 </semantics></math>-tree is <i>full</i> if each of its limit levels omits no more than one potential branch. Kunen asked whether a full <span></span><math>\u0000 <semantics>\u0000 <mi>κ</mi>\u0000 <annotation>$kappa$</annotation>\u0000 </semantics></math>-Souslin tree may consistently exist. Shelah gave an affirmative answer of height a strong limit Mahlo cardinal <span></span><math>\u0000 <semantics>\u0000 <mi>κ</mi>\u0000 <annotation>$kappa $</annotation>\u0000 </semantics></math>. Here, it is shown that these trees may consistently exist at small cardinals. Indeed, there can be <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ℵ</mi>\u0000 <mn>3</mn>\u0000 </msub>\u0000 <annotation>$aleph _3$</annotation>\u0000 </semantics></math> many full <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ℵ</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$aleph _2$</annotation>\u0000 </semantics></math>-trees such that the product of any countably many of them is an <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ℵ</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$aleph _2$</annotation>\u0000 </semantics></math>-Souslin tree.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12957","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141488977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sarnak's conjecture in quantum computing, cyclotomic unitary group coranks, and Shimura curves","authors":"Colin Ingalls,&nbsp;Bruce W. Jordan,&nbsp;Allan Keeton,&nbsp;Adam Logan,&nbsp;Yevgeny Zaytman","doi":"10.1112/jlms.12952","DOIUrl":"https://doi.org/10.1112/jlms.12952","url":null,"abstract":"<p>Sarnak's conjecture in quantum computing concerns when the groups <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mo>PU</mo>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$operatorname{PU}_{2}$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mo>PSU</mo>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$operatorname{PSU}_{2}$</annotation>\u0000 </semantics></math> over cyclotomic rings <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Z</mi>\u0000 <mo>[</mo>\u0000 <msub>\u0000 <mi>ζ</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <mn>1</mn>\u0000 <mo>/</mo>\u0000 <mn>2</mn>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <annotation>${mathbb {Z}}[zeta _{n}, 1/2]$</annotation>\u0000 </semantics></math> with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>ζ</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mo>=</mo>\u0000 <msup>\u0000 <mi>e</mi>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>π</mi>\u0000 <mi>i</mi>\u0000 <mo>/</mo>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$zeta _n=e^{2pi i/n}$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>4</mn>\u0000 <mo>|</mo>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation>$4|n$</annotation>\u0000 </semantics></math>, are generated by the Clifford-cyclotomic gate set. We previously settled this using Euler–Poincaré characteristics. A generalization of Sarnak's conjecture is to ask when these groups are generated by torsion elements. An obstruction to this is provided by the corank: a group <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> has <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>corank</mo>\u0000 <mi>G</mi>\u0000 <mo>&gt;</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$operatorname{corank}G&amp;gt;0$</annotation>\u0000 </semantics></math> only if <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> is not generated by torsion elements. In this paper, we study the corank of these cyclotomic unitary grou","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141488427","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hofer–Zehnder capacity of disc tangent bundles of projective spaces","authors":"Johanna Bimmermann","doi":"10.1112/jlms.12948","DOIUrl":"https://doi.org/10.1112/jlms.12948","url":null,"abstract":"<p>We compute the Hofer–Zehnder capacity of disc tangent bundles of the complex and real projective spaces of any dimension. The disc bundle is taken with respect to the Fubini–Study resp. round metric, but we can obtain explicit bounds for any other metric. In the case of the complex projective space, we also compute the Hofer–Zehnder capacity for the magnetically twisted case, where the twist is proportional to the Fubini–Study form. For arbitrary twists, we can still give explicit upper bounds.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12948","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141488421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Nonfree almost finite actions for locally finite-by-virtually \u0000 \u0000 Z\u0000 ${mathbb {Z}}$\u0000 groups","authors":"Kang Li,&nbsp;Xin Ma","doi":"10.1112/jlms.12959","DOIUrl":"https://doi.org/10.1112/jlms.12959","url":null,"abstract":"<p>In this paper, we study almost finiteness and almost finiteness in measure of nonfree actions. Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>α</mi>\u0000 <mo>:</mo>\u0000 <mi>G</mi>\u0000 <mi>↷</mi>\u0000 <mi>X</mi>\u0000 </mrow>\u0000 <annotation>$alpha:Gcurvearrowright X$</annotation>\u0000 </semantics></math> be a minimal action of a locally finite-by-virtually <span></span><math>\u0000 <semantics>\u0000 <mi>Z</mi>\u0000 <annotation>${mathbb {Z}}$</annotation>\u0000 </semantics></math> group <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> on the Cantor set <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math>. We prove that under certain assumptions, the action <span></span><math>\u0000 <semantics>\u0000 <mi>α</mi>\u0000 <annotation>$alpha$</annotation>\u0000 </semantics></math> is almost finite in measure if and only if <span></span><math>\u0000 <semantics>\u0000 <mi>α</mi>\u0000 <annotation>$alpha$</annotation>\u0000 </semantics></math> is essentially free. As an application, we obtain that any minimal topologically free action of a virtually <span></span><math>\u0000 <semantics>\u0000 <mi>Z</mi>\u0000 <annotation>${mathbb {Z}}$</annotation>\u0000 </semantics></math> group on an infinite compact metrizable space with the small boundary property is almost finite. This is the first general result, assuming only topological freeness, in this direction, and these lead to new results on uniform property <span></span><math>\u0000 <semantics>\u0000 <mi>Γ</mi>\u0000 <annotation>$Gamma$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mi>Z</mi>\u0000 <annotation>$mathcal {Z}$</annotation>\u0000 </semantics></math>-stability for their crossed product <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>C</mi>\u0000 <mo>∗</mo>\u0000 </msup>\u0000 <annotation>$C^*$</annotation>\u0000 </semantics></math>-algebras. Some concrete examples of minimal topological free (but nonfree) subshifts are provided.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141488426","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On K-stability of \u0000 \u0000 \u0000 P\u0000 3\u0000 \u0000 $mathbb {P}^3$\u0000 blown up along a (2,3) complete intersection","authors":"Tiago Duarte Guerreiro,&nbsp;Luca Giovenzana,&nbsp;Nivedita Viswanathan","doi":"10.1112/jlms.12961","DOIUrl":"https://doi.org/10.1112/jlms.12961","url":null,"abstract":"<p>We prove K-stability of every smooth member of the Fano 3-fold family 2.15 of the Mori, Mukai and Iskovskikh classification.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12961","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141488151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hahn series and Mahler equations: Algorithmic aspects","authors":"C. Faverjon,&nbsp;J. Roques","doi":"10.1112/jlms.12945","DOIUrl":"https://doi.org/10.1112/jlms.12945","url":null,"abstract":"<p>Many articles have recently been devoted to Mahler equations, partly because of their links with other branches of mathematics such as automata theory. Hahn series (a generalization of the Puiseux series allowing arbitrary exponents of the indeterminate as long as the set that supports them is well ordered) play a central role in the theory of Mahler equations. In this paper, we address the following fundamental question: is there an algorithm to calculate the Hahn series solutions of a given linear Mahler equation? What makes this question interesting is the fact that the Hahn series appearing in this context can have complicated supports with infinitely many accumulation points. Our (positive) answer to the above question involves among other things the construction of a computable well-ordered receptacle for the supports of the potential Hahn series solutions.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141430228","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Corrigendum: Transcendental Brauer groups of products of CM elliptic curves","authors":"Rachel Newton","doi":"10.1112/jlms.12953","DOIUrl":"https://doi.org/10.1112/jlms.12953","url":null,"abstract":"<p>This is a corrigendum to the paper <i>Transcendental Brauer groups of products of CM elliptic curves</i>, published in (J. Lond. Math. Soc. <b>93</b> (2016), 397–419). In this note, we point out a mistake affecting the statements of Theorems 1.3, 5.2, 5.3 and Proposition 5.1 in <i>op. cit</i>., and provide a correction.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12953","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141425026","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Coalescence and total-variation distance of semi-infinite inverse-gamma polymers","authors":"Firas Rassoul-Agha,&nbsp;Timo Seppäläinen,&nbsp;Xiao Shen","doi":"10.1112/jlms.12955","DOIUrl":"https://doi.org/10.1112/jlms.12955","url":null,"abstract":"<p>We show that two semi-infinite positive temperature polymers coalesce on the scale predicted by KPZ (Kardar–Parisi–Zhang) universality. The two polymer paths have the same asymptotic direction and evolve in the same environment, independently until coalescence. If they start at distance <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math> apart, their coalescence occurs on the scale <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>k</mi>\u0000 <mrow>\u0000 <mn>3</mn>\u0000 <mo>/</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msup>\u0000 <annotation>$k^{3/2}$</annotation>\u0000 </semantics></math>. It follows that the total variation distance of two semi-infinite polymer measures decays on this same scale. Our results are upper and lower bounds on probabilities and expectations that match, up to constant factors and occasional logarithmic corrections. Our proofs are done in the context of the solvable inverse-gamma polymer model, but without appeal to integrable probability. With minor modifications, our proofs give also bounds on transversal fluctuations of the polymer path. As the free energy of a directed polymer is a discretization of a stochastically forced viscous Hamilton–Jacobi equation, our results suggest that the hyperbolicity phenomenon of such equations obeys the KPZ exponent.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141326625","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}