@article{lidman_pinzon-caicedo_zentner_2023, title={Toroidal integer homology three-spheres have irreducible SU(2)$SU(2)$-representations}, volume={16}, ISSN={["1753-8424"]}, DOI={10.1112/topo.12275}, number={1}, journal={JOURNAL OF TOPOLOGY}, author={Lidman, Tye and Pinzon-Caicedo, Juanita and Zentner, Raphael}, year={2023}, month={Mar}, pages={344–367} }
@article{gorsky_lidman_liu_moore_2023, title={Triple Linking Numbers and Heegaard Floer Homology}, volume={2023}, ISSN={["1687-0247"]}, DOI={10.1093/imrn/rnab368}, abstractNote={Abstract We establish some new relationships between Milnor invariants and Heegaard Floer homology. This includes a formula for the Milnor triple linking number from the link Floer complex, detection results for the Whitehead link and Borromean rings, and a structural property of the $d$-invariants of surgeries on certain algebraically split links.}, number={6}, journal={INTERNATIONAL MATHEMATICS RESEARCH NOTICES}, author={Gorsky, Eugene and Lidman, Tye and Liu, Beibei and Moore, Allison H.}, year={2023}, month={Mar}, pages={4501–4554} }
@article{hom_levine_lidman_2022, title={KNOT CONCORDANCE IN HOMOLOGY COBORDISMS}, volume={171}, ISSN={["1547-7398"]}, DOI={10.1215/00127094-2021-0110}, abstractNote={Let CˆZ denote the group of knots in homology spheres that bound homology balls, modulo smooth concordance in homology cobordisms. Answering a question of Matsumoto, the second author previously showed that the natural map from the smooth knot concordance group C to CˆZ is not surjective. Using tools from Heegaard Floer homology, we show that the cokernel of this map, which can be understood as the non-locally-flat piecewise-linear concordance group, is infinitely generated and contains elements of infinite order. In the appendix, we provide a careful proof that any piecewise-linear surface in a smooth 4-manifold can be isotoped to be smooth away from cone points.}, number={15}, journal={DUKE MATHEMATICAL JOURNAL}, author={Hom, Jenniffer and Levine, Adam Simon and Lidman, Tye}, year={2022}, month={Oct}, pages={3089–3131} }
@article{baldwin_lidman_wong_2022, title={Lagrangian Cobordisms and Legendrian Invariants in Knot Floer Homology}, volume={71}, ISSN={["1945-2365"]}, DOI={10.1307/mmj/20195786}, abstractNote={We prove that the LOSS and GRID invariants of Legendrian links in knot Floer homology behave in certain functorial ways with respect to decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on R3. Our results give new, computable, and effective obstructions to the existence of such cobordisms.}, number={1}, journal={MICHIGAN MATHEMATICAL JOURNAL}, author={Baldwin, John A. and Lidman, Tye and Wong, C-M Michael}, year={2022}, month={Mar}, pages={145–175} }
@article{daemi_lidman_vela-vick_wong_2022, title={Ribbon homology cobordisms}, volume={408}, ISSN={["1090-2082"]}, DOI={10.1016/j.aim.2022.108580}, abstractNote={We study 4-dimensional homology cobordisms without 3-handles, showing that they interact nicely with Thurston geometries, character varieties, and instanton and Heegaard Floer homologies. Using these, we derive obstructions to such cobordisms. As one example of these obstructions, we generalize other recent results on the behavior of knot Floer homology under ribbon concordances. Finally, we provide topological applications, including to Dehn surgery problems.}, journal={ADVANCES IN MATHEMATICS}, author={Daemi, Aliakbar and Lidman, Tye and Vela-Vick, David Shea and Wong, C. -M. Michael}, year={2022}, month={Oct} }
@article{baldwin_dowlin_levine_lidman_sazdanovic_2021, title={Khovanov homology detects the figure‐eight knot}, volume={53}, ISSN={0024-6093 1469-2120}, url={http://dx.doi.org/10.1112/blms.12467}, DOI={10.1112/blms.12467}, abstractNote={We use Dowlin's spectral sequence from Khovanov homology to knot Floer homology to prove that reduced Khovanov homology with rational coefficients detects the figure-eight knot.}, number={3}, journal={Bulletin of the London Mathematical Society}, publisher={Wiley}, author={Baldwin, John A. and Dowlin, Nathan and Levine, Adam Simon and Lidman, Tye and Sazdanovic, Radmila}, year={2021}, month={Jan}, pages={871–876} }
@article{hom_lidman_2019, title={A note on positive-definite, symplectic four-manifolds}, volume={21}, ISSN={["1435-9855"]}, DOI={10.4171/JEMS/835}, abstractNote={We prove that a positive definite smooth four-manifold with $b_2^+ \geq 2$ and having either no 1-handles or no 3-handles cannot admit a symplectic structure.}, number={1}, journal={JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY}, author={Hom, Jennifer and Lidman, Tye}, year={2019}, pages={257–270} }
@article{hanselman_kutluhan_lidman_2019, title={A remark on the geography problem in Heegaard Floer homology}, volume={102}, ISBN={["978-1-4704-4249-1"]}, ISSN={["2324-707X"]}, DOI={10.1090/pspum/102/01810}, journal={BREADTH IN CONTEMPORARY TOPOLOGY}, author={Hanselman, Jonathan and Kutluhan, Cagatay and Lidman, Tye}, year={2019}, pages={103–111} }
@article{hendricks_hom_lidman_2019, title={APPLICATIONS OF INVOLUTIVE HEEGAARD FLOER HOMOLOGY}, volume={20}, ISSN={1474-7480 1475-3030}, url={http://dx.doi.org/10.1017/S147474801900015X}, DOI={10.1017/S147474801900015X}, abstractNote={Abstract We use Heegaard Floer homology to define an invariant of homology cobordism. This invariant is isomorphic to a summand of the reduced Heegaard Floer homology of a rational homology sphere equipped with a spin structure and is analogous to Stoffregen’s connected Seiberg–Witten Floer homology. We use this invariant to study the structure of the homology cobordism group and, along the way, compute the involutive correction terms $\bar{d}$ and $\text{}\underline{d}$ for certain families of three-manifolds.}, number={1}, journal={Journal of the Institute of Mathematics of Jussieu}, publisher={Cambridge University Press (CUP)}, author={Hendricks, Kristen and Hom, Jennifer and Lidman, Tye}, year={2019}, month={Apr}, pages={1–38} }
@article{lidman_moore_vazquez_2019, title={Distance one lens space fillings and band surgery on the trefoil knot}, volume={19}, ISSN={["1472-2739"]}, DOI={10.2140/agt.2019.19.2439}, abstractNote={We prove that if the lens space $L(n, 1)$ is obtained by a surgery along a knot in the lens space $L(3,1)$ that is distance one from the meridional slope, then $n$ is in $\{-6, \pm 1, \pm 2, 3, 4, 7\}$. This result yields a classification of the coherent and non-coherent band surgeries from the trefoil to $T(2, n)$ torus knots and links. The main result is proved by studying the behavior of the Heegaard Floer $d$-invariants under integral surgery along knots in $L(3,1)$. The classification of band surgeries between the trefoil and torus knots and links is motivated by local reconnection processes in nature, which are modeled as band surgeries. Of particular interest is the study of recombination on circular DNA molecules.}, number={5}, journal={ALGEBRAIC AND GEOMETRIC TOPOLOGY}, author={Lidman, Tye and Moore, Allison H. and Vazquez, Mariel}, year={2019}, pages={2439–2484} }
@article{levine_lidman_2019, title={SIMPLY CONNECTED, SPINELESS 4-MANIFOLDS}, volume={7}, ISSN={["2050-5094"]}, DOI={10.1017/fms.2019.11}, abstractNote={We construct infinitely many compact, smooth 4-manifolds which are homotopy equivalent to $S^{2}$ but do not admit a spine (that is, a piecewise linear embedding of $S^{2}$ that realizes the homotopy equivalence). This is the remaining case in the existence problem for codimension-2 spines in simply connected manifolds. The obstruction comes from the Heegaard Floer $d$ invariants.}, journal={FORUM OF MATHEMATICS SIGMA}, author={Levine, Adam Simon and Lidman, Tye}, year={2019}, month={May} }
@article{hom_lidman_2018, title={A NOTE ON SURGERY OBSTRUCTIONS AND HYPERBOLIC INTEGER HOMOLOGY SPHERES}, volume={146}, ISSN={["1088-6826"]}, DOI={10.1090/proc/13925}, abstractNote={Auckly gave two examples of irreducible integer homology spheres (one toroidal and one hyperbolic) which are not surgery on a knot in the three-sphere. Using Heegaard Floer homology, the authors and Karakurt provided infinitely many small Seifert fibered examples. In this note, we extend those results to give infinitely many hyperbolic examples, as well as infinitely many examples with arbitrary JSJ decomposition.}, number={3}, journal={PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY}, author={Hom, Jennifer and Lidman, Tye}, year={2018}, month={Mar}, pages={1363–1365} }
@article{lidman_tweedy_2018, title={A note on concordance properties of fibers in Seifert homology spheres}, volume={61}, ISSN={["1496-4287"]}, DOI={10.4153/CMB-2017-081-9}, abstractNote={Abstract In this note, we collect various properties of Seifert homology spheres from the viewpoint of Dehn surgery along a Seifert fiber. We expect that many of these are known to various experts, but include them in one place, which we hope will be useful in the study of concordance and homology cobordism.}, number={4}, journal={CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES}, author={Lidman, Tye and Tweedy, Eamonn}, year={2018}, month={Dec}, pages={754–767} }
@article{lidman_manolescu_2018, title={Floer homology and covering spaces}, volume={22}, ISSN={["1364-0380"]}, DOI={10.2140/gt.2018.22.2817}, abstractNote={We prove a Smith-type inequality for regular covering spaces in monopole Floer homology. Using the monopole Floer / Heegaard Floer correspondence, we deduce that if a 3-manifold Y admits a p^n-sheeted regular cover that is a Z/pZ-L-space (for p prime), then Y is a Z/pZ-L-space. Further, we obtain constraints on surgeries on a knot being regular covers over other surgeries on the same knot, and over surgeries on other knots.}, number={5}, journal={GEOMETRY & TOPOLOGY}, author={Lidman, Tye and Manolescu, Ciprian}, year={2018}, pages={2817–2838} }
@article{lidman_manolescu_2018, title={The equivalence of two Seiberg-Witten Floer homologies}, volume={399}, journal={Astérisque}, author={Lidman, T. and Manolescu, C.}, year={2018} }
@article{lidman_moore_2017, title={COSMETIC SURGERY IN L-SPACES AND NUGATORY CROSSINGS}, volume={369}, ISSN={["1088-6850"]}, DOI={10.1090/tran/6839}, abstractNote={The cosmetic crossing conjecture (also known as the ânugatory crossing conjectureâ) asserts that the only crossing changes that preserve the oriented isotopy class of a knot in the 3-sphere are nugatory. We use the Dehn surgery characterization of the unknot to prove this conjecture for knots in integer homology spheres whose branched double covers are L-spaces satisfying a homological condition. This includes as a special case all alternating and quasi-alternating knots with square-free determinant. As an application, we prove the cosmetic crossing conjecture holds for all knots with at most nine crossings and provide new examples of knots, including pretzel knots, non-arborescent knots and symmetric unions for which the conjecture holds.}, number={5}, journal={TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY}, author={Lidman, Tye and Moore, Allison H.}, year={2017}, month={May}, pages={3639–3654} }
@article{gordon_lidman_2017, title={Corrigendum to “Taut Foliations, Left-Orderability, and Cyclic Branched Covers”}, volume={42}, ISSN={0251-4184 2315-4144}, url={http://dx.doi.org/10.1007/s40306-017-0216-1}, DOI={10.1007/s40306-017-0216-1}, number={4}, journal={Acta Mathematica Vietnamica}, publisher={Springer Nature}, author={Gordon, Cameron and Lidman, Tye}, year={2017}, month={Jun}, pages={775–776} }
@article{gordon_lidman_2017, title={Knot contact homology detects cabled, composite, and torus knots}, volume={145}, ISSN={0002-9939 1088-6826}, url={http://dx.doi.org/10.1090/proc/13643}, DOI={10.1090/proc/13643}, abstractNote={Knot contact homology is an invariant of knots derived from Legendrian contact homology which has numerous connections to the knot group. We use basic properties of knot groups to prove that knot contact homology detects every torus knot. Further, if the knot contact homology of a knot is isomorphic to that of a cable (respectively composite) knot, then the knot is a cable (respectively composite).}, number={12}, journal={Proceedings of the American Mathematical Society}, publisher={American Mathematical Society (AMS)}, author={Gordon, Cameron and Lidman, Tye}, year={2017}, month={Jun}, pages={5405–5412} }
@article{lidman_sivek_2017, title={Quasi-alternating links with small determinant}, volume={162}, ISSN={["1469-8064"]}, DOI={10.1017/s0305004116000578}, abstractNote={Quasi-alternating links of determinant 1, 2, 3, and 5 were previously classified by Greene and Teragaito, who showed that the only such links are two-bridge. In this paper, we extend this result by showing that all quasi-alternating links of determinant at most 7 are connected sums of two-bridge links, which is optimal since there are quasi-alternating links not of this form for all larger determinants. We achieve this by studying their branched double covers and characterizing distance-one surgeries between lens spaces of small order, leading to a classification of formal L-spaces with order at most 7.}, number={2}, journal={MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY}, author={Lidman, Tye and Sivek, Steven}, year={2017}, month={Mar}, pages={319–336} }
@article{gordon_lidman_2017, title={Taut foliations, left-orderability, and cyclic branched covers (vol 39, pg 599, 2014)}, volume={42}, number={4}, journal={Acta Mathematica Vietnamica}, author={Gordon, C. and Lidman, T.}, year={2017}, pages={775–776} }
@article{hom_lidman_watson_2017, title={The Alexander module, Seifert forms, and categorification}, volume={10}, ISSN={["1753-8424"]}, DOI={10.1112/topo.12001}, abstractNote={We show that bordered Floer homology provides a categorification of a topological quantum field theory (TQFT) described by Donaldson [Proceedings of the Kirbyfest, Berkeley, CA, 1998, Geometry & Topology Monographs 2 (Geometry & Topology Publications, Coventry, 1999) 87–102]. This, in turn, leads to a proof that both the Alexander module of a knot and the Seifert form are completely determined by Heegaard Floer theory.}, number={1}, journal={JOURNAL OF TOPOLOGY}, author={Hom, Jennifer and Lidman, Tye and Watson, Liam}, year={2017}, pages={22–100} }
@article{lidman_moore_2016, title={Pretzel knots with L-space surgeries}, volume={65}, ISSN={0026-2285}, url={http://dx.doi.org/10.1307/mmj/1457101813}, DOI={10.1307/mmj/1457101813}, abstractNote={A rational homology sphere whose Heegaard Floer homology is the same as that of a lens space is called an L-space. We classify pretzel knots with any number of tangles which admit L-space surgeries. This rests on Gabai's classification of fibered pretzel links.}, number={1}, journal={The Michigan Mathematical Journal}, publisher={Michigan Mathematical Journal}, author={Lidman, Tye and Moore, Allison H.}, year={2016}, month={Mar}, pages={105–130} }
@article{hom_karakurt_lidman_2016, title={Surgery obstructions and Heegaard Floer homology}, volume={20}, ISSN={["1364-0380"]}, DOI={10.2140/gt.2016.20.2219}, abstractNote={Using Taubes' periodic ends theorem, Auckly gave examples of toroidal and hyperbolic irreducible integer homology spheres which are not surgery on a knot in the three-sphere. We give an obstruction to a homology sphere being surgery on a knot coming from Heegaard Floer homology. This is used to construct infinitely many small Seifert fibered examples.}, number={4}, journal={GEOMETRY & TOPOLOGY}, author={Hom, Jennifer and Karakurt, Cagri and Lidman, Tye}, year={2016}, pages={2219–2251} }
@article{lidman_sivek_2015, title={Contact structures and reducible surgeries}, volume={152}, ISSN={0010-437X 1570-5846}, url={http://dx.doi.org/10.1112/s0010437x15007599}, DOI={10.1112/s0010437x15007599}, abstractNote={We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus $g$ must have slope $2g-1$ , leading to a proof of the cabling conjecture for positive knots of genus 2. Our techniques also produce bounds on the maximum Thurston–Bennequin numbers of cables.}, number={1}, journal={Compositio Mathematica}, publisher={Wiley}, author={Lidman, Tye and Sivek, Steven}, year={2015}, month={Sep}, pages={152–186} }
@article{hom_lidman_zufelt_2015, title={Reducible surgeries and Heegaard Floer homology}, volume={22}, ISSN={1073-2780 1945-001X}, url={http://dx.doi.org/10.4310/mrl.2015.v22.n3.a8}, DOI={10.4310/mrl.2015.v22.n3.a8}, abstractNote={In this paper, we use Heegaard Floer homology to study reducible surgeries. In particular, suppose K is a non-cable knot in the three-sphere with an L-space surgery. If p-surgery on K is reducible, we show that p equals 2g(K)-1. This implies that any knot with an L-space surgery has at most one reducible surgery, a fact that we show additionally for any knot of genus at most two.}, number={3}, journal={Mathematical Research Letters}, publisher={International Press of Boston}, author={Hom, Jennifer and Lidman, Tye and Zufelt, Nicholas}, year={2015}, pages={763–788} }
@article{hom_lidman_vafaee_2014, title={Berge–Gabai knots and L–space satellite operations}, volume={14}, ISSN={1472-2739 1472-2747}, url={http://dx.doi.org/10.2140/agt.2014.14.3745}, DOI={10.2140/agt.2014.14.3745}, abstractNote={Let $P(K)$ be a satellite knot where the pattern, $P$, is a Berge-Gabai knot (i.e., a knot in the solid torus with a non-trivial solid torus Dehn surgery), and the companion, $K$, is a non-trivial knot in $S^3$. We prove that $P(K)$ is an L-space knot if and only if $K$ is an L-space knot and $P$ is sufficiently positively twisted relative to the genus of $K$. This generalizes the result for cables due to Hedden and the first author.}, number={6}, journal={Algebraic & Geometric Topology}, publisher={Mathematical Sciences Publishers}, author={Hom, Jennifer and Lidman, Tye and Vafaee, Faramarz}, year={2014}, month={Dec}, pages={3745–3763} }
@article{lidman_watson_2014, title={Nonfibered L-space knots}, volume={267}, ISSN={0030-8730 0030-8730}, url={http://dx.doi.org/10.2140/pjm.2014.267.423}, DOI={10.2140/pjm.2014.267.423}, abstractNote={We construct an infinite family of knots in rational homology spheres with irreducible, non-fibered complements, for which every non-longitudinal filling is an L-space.}, number={2}, journal={Pacific Journal of Mathematics}, publisher={Mathematical Sciences Publishers}, author={Lidman, Tye and Watson, Liam}, year={2014}, month={May}, pages={423–429} }
@article{karakurt_lidman_2014, title={Rank inequalities for the Heegaard Floer homology of Seifert homology spheres}, volume={367}, ISSN={0002-9947 1088-6850}, url={http://dx.doi.org/10.1090/s0002-9947-2014-06451-9}, DOI={10.1090/s0002-9947-2014-06451-9}, abstractNote={We establish three rank inequalities for the reduced flavor of Heegaard Floer homology of Seifert fibered integer homology spheres. Combining these inequalities with the known classifications of non-zero degree maps between Seifert fibered spaces, we prove that a map f : Y ′ → Y f:Y’ \to Y between Seifert homology spheres yields the inequality | deg f | r a n k H F r e d ( Y ) ≤ r a n k H F r e d ( Y ′ ) |\deg f|\mathrm {rank} HF_{\mathrm {red}}(Y) \leq \mathrm {rank} HF_{\mathrm {red}}(Y’) . These inequalities are also applied in conjunction with an algorithm of Némethi to give a method to solve the botany problem for the Heegaard Floer homology of these manifolds.}, number={10}, journal={Transactions of the American Mathematical Society}, publisher={American Mathematical Society (AMS)}, author={Karakurt, Çağrı and Lidman, Tye}, year={2014}, month={Dec}, pages={7291–7322} }
@article{gordon_lidman_2014, title={Taut Foliations, Left-Orderability, and Cyclic Branched Covers}, volume={39}, ISSN={0251-4184 2315-4144}, url={http://dx.doi.org/10.1007/s40306-014-0091-y}, DOI={10.1007/s40306-014-0091-y}, abstractNote={We study the question of when cyclic branched covers of knots admit taut foliations, have left-orderable fundamental groups, and are not L-spaces.}, number={4}, journal={Acta Mathematica Vietnamica}, publisher={Springer Science and Business Media LLC}, author={Gordon, Cameron and Lidman, Tye}, year={2014}, month={Dec}, pages={599–635} }
@article{clay_lidman_watson_2013, title={Graph manifolds, left-orderability and amalgamation}, volume={13}, ISSN={1472-2739 1472-2747}, url={http://dx.doi.org/10.2140/agt.2013.13.2347}, DOI={10.2140/agt.2013.13.2347}, abstractNote={We show that every irreducible toroidal integer homology sphere graph manifold has a left-orderable fundamental group. This is established by way of a specialization of a result due to Bludov and Glass for the almagamated products that arise, and in this setting work of Boyer, Rolfsen and Wiest may be applied. Our result then depends on input from 3-manifold topology and Heegaard Floer homology.}, number={4}, journal={Algebraic & Geometric Topology}, publisher={Mathematical Sciences Publishers}, author={Clay, Adam and Lidman, Tye and Watson, Liam}, year={2013}, month={Jul}, pages={2347–2368} }
@article{lidman_2013, title={On the infinity flavor of Heegaard Floer homology and the integral cohomology ring}, volume={88}, ISSN={0010-2571}, url={http://dx.doi.org/10.4171/cmh/306}, DOI={10.4171/cmh/306}, abstractNote={For a three-manifold $Y$ and torsion $\mathrm{Spin}^c$ structure $\mathfrak{s}$, Ozsváth and Szabóconstruct a spectral sequence with $E^2$ term an exterior algebra over $H^1(Y;\mathbb{Z})$ converging to $H F^\infty(Y,\mathfrak{s})$. They conjecture that the differentials are completely determined by the integral triple cup product form. In this paper, we prove that $H\hskip-2pt F^\infty(Y,\mathfrak{s})$ is in fact determined by the cohomology ring when $\mathfrak{s}$ is torsion. Furthermore, we give a complete calculation of such $HF^\infty(Y,\mathfrak{s})$, with mod 2 coefficients, in the case where $b\_1(Y)$ is 3 or 4.}, number={4}, journal={Commentarii Mathematici Helvetici}, publisher={European Mathematical Publishing House}, author={Lidman, Tye}, year={2013}, pages={875–898} }
@article{evans_lidman_2007, title={Asymptotic Evolution of Acyclic Random Mappings}, volume={12}, ISSN={1083-6489}, url={http://dx.doi.org/10.1214/ejp.v12-437}, DOI={10.1214/ejp.v12-437}, abstractNote={An acyclic mapping from an $n$ element set into itself is a mapping $\varphi$ such that if $\varphi^k(x) = x$ for some $k$ and $x$, then $\varphi(x) = x$. Equivalently, $\varphi^\ell = \varphi^{\ell+1} = \ldots$ for $\ell$ sufficiently large. We investigate the behavior as $n \rightarrow \infty$ of a sequence of a Markov chain on the collection of such mappings. At each step of the chain, a point in the $n$ element set is chosen uniformly at random and the current mapping is modified by replacing the current image of that point by a new one chosen independently and uniformly at random, conditional on the resulting mapping being again acyclic. We can represent an acyclic mapping as a directed graph (such a graph will be a collection of rooted trees) and think of these directed graphs as metric spaces with some extra structure. Informal calculations indicate that the metric space valued process associated with the Markov chain should, after an appropriate time and ``space'' rescaling, converge as $n \rightarrow \infty$ to a real tree ($R$-tree) valued Markov process that is reversible with respect to a measure induced naturally by the standard reflected Brownian bridge. Although we don't prove such a limit theorem, we use Dirichlet form methods to construct a Markov process that is Hunt with respect to a suitable Gromov-Hausdorff-like metric and evolves according to the dynamics suggested by the heuristic arguments. This process is similar to one that appears in earlier work by Evans and Winter as a similarly informal limit of a Markov chain related to the subtree prune and regraft tree (SPR) rearrangements from phylogenetics.}, number={0}, journal={Electronic Journal of Probability}, publisher={Institute of Mathematical Statistics}, author={Evans, Steven and Lidman, Tye}, year={2007}, pages={1051–1180} }
@article{evans_lidman_2007, title={Expectation, Conditional Expectation and Martingales in Local Fields}, volume={12}, ISSN={1083-6489}, url={http://dx.doi.org/10.1214/ejp.v12-405}, DOI={10.1214/ejp.v12-405}, abstractNote={We investigate a possible definition of expectation and conditional expectation for random variables with values in a local field such as the $p$-adic numbers. We define the expectation by analogy with the observation that for real-valued random variables in $L^2$ the expected value is the orthogonal projection onto the constants. Previous work has shown that the local field version of $L^\infty$ is the appropriate counterpart of $L^2$, and so the expected value of a local field-valued random variable is defined to be its ``projection'' in $L^\infty$ onto the constants. Unlike the real case, the resulting projection is not typically a single constant, but rather a ball in the metric on the local field. However, many properties of this expectation operation and the corresponding conditional expectation mirror those familiar from the real-valued case; for example, conditional expectation is, in a suitable sense, a contraction on $L^\infty$ and the tower property holds. We also define the corresponding notion of martingale, show that several standard examples of martingales (for example, sums or products of suitable independent random variables or ``harmonic'' functions composed with Markov chains) have local field analogues, and obtain versions of the optional sampling and martingale convergence theorems.}, number={0}, journal={Electronic Journal of Probability}, publisher={Institute of Mathematical Statistics}, author={Evans, Steven and Lidman, Tye}, year={2007}, pages={498–515} }