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<title>Free convolution</title>

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Free convolution

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">Free convolution is the <a href="Free_probability" title="Free probability">free probability</a> analog of the classical notion of <a href="Convolution" title="Convolution">convolution</a> of probability measures. Due to the non-commutative nature of free probability theory, one has to talk separately about additive and multiplicative free convolution, which arise from addition and multiplication of free random variables (see below; in the classical case, what would be the analog of free multiplicative convolution can be reduced to additive convolution by passing to logarithms of random variables). These operations have some interpretations in terms of <a href="Empirical_spectral_measures" class="mw-redirect" title="Empirical spectral measures">empirical spectral measures</a> of <a href="Random_matrices" class="mw-redirect" title="Random matrices">random matrices</a>.<a href="#cite_note-1">[1]</a>

The notion of free convolution was introduced by <a href="Dan-Virgil_Voiculescu" title="Dan-Virgil Voiculescu">Dan-Virgil Voiculescu</a>.<a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="Free_additive_convolution">Free additive convolution</h2></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

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</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

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<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

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</math><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> be two probability measures on the real line, and assume that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a random variable in a non commutative probability space with law <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

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</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> is a random variable in the same non commutative probability space with law <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

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<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

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</math><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy">. Assume finally that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> are <a href="Free_independence" title="Free independence">freely independent</a>. Then the free additive convolution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \boxplus \nu }">

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<annotation encoding="application/x-tex">{\displaystyle \mu \boxplus \nu }</annotation>

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</math><img src="./d4bcb208591f7c7900047372dfe5fcbf3884af92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.474ex; height:2.676ex;" alt="{\displaystyle \mu \boxplus \nu }" loading="lazy"> is the law of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X+Y}">

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<annotation encoding="application/x-tex">{\displaystyle X+Y}</annotation>

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</math><img src="./191744cf9cddeff3ab2e750e22bcfce7766d355e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.594ex; height:2.343ex;" alt="{\displaystyle X+Y}" loading="lazy">. <a href="Random_matrices" class="mw-redirect" title="Random matrices">Random matrices</a> interpretation: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> are some independent <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> Hermitian (resp. real symmetric) random matrices such that at least one of them is invariant, in law, under conjugation by any unitary (resp. orthogonal) matrix and such that the <a href="Empirical_spectral_measures" class="mw-redirect" title="Empirical spectral measures">empirical spectral measures</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> tend respectively to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

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</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

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</math><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> tends to infinity, then the empirical spectral measure of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A+B}">

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<annotation encoding="application/x-tex">{\displaystyle A+B}</annotation>

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</math><img src="./4279cdbd3cb8ec4c3423065d9a7d83a82cfc89e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A+B}" loading="lazy"> tends to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \boxplus \nu }">

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<annotation encoding="application/x-tex">{\displaystyle \mu \boxplus \nu }</annotation>

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</math><img src="./d4bcb208591f7c7900047372dfe5fcbf3884af92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.474ex; height:2.676ex;" alt="{\displaystyle \mu \boxplus \nu }" loading="lazy">.<a href="#cite_note-4">[4]</a>

In many cases, it is possible to compute the probability measure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \boxplus \nu }">

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<annotation encoding="application/x-tex">{\displaystyle \mu \boxplus \nu }</annotation>

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</math><img src="./d4bcb208591f7c7900047372dfe5fcbf3884af92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.474ex; height:2.676ex;" alt="{\displaystyle \mu \boxplus \nu }" loading="lazy"> explicitly by using complex-analytic techniques and the R-transform of the measures <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

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</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

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</math><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Rectangular_free_additive_convolution">Rectangular free additive convolution</h2></div>

The rectangular free additive convolution (with ratio <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

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<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

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</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy">) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \boxplus _{c}}">

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<annotation encoding="application/x-tex">{\displaystyle \boxplus _{c}}</annotation>

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</math><img src="./b9d2dbb4a38021d7de5d504e14e837024973479c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.752ex; height:2.509ex;" alt="{\displaystyle \boxplus _{c}}" loading="lazy"> has also been defined in the non commutative probability framework by Benaych-Georges<a href="#cite_note-5">[5]</a> and admits the following <a href="Random_matrices" class="mw-redirect" title="Random matrices">random matrices</a> interpretation. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in [0,1]}">

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<annotation encoding="application/x-tex">{\displaystyle c\in [0,1]}</annotation>

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</math><img src="./02b2796dc5e3cb6527d1ac6e766dc1da2ef1e120.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.5ex; height:2.843ex;" alt="{\displaystyle c\in [0,1]}" loading="lazy">, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> are some independent <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> complex (resp. real) random matrices such that at least one of them is invariant, in law, under multiplication on the left and on the right by any unitary (resp. orthogonal) matrix and such that the <a href="Empirical_singular_values_distribution" class="mw-redirect" title="Empirical singular values distribution">empirical singular values distribution</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> tend respectively to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

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</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

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</math><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> tend to infinity in such a way that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n/p}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle n/p}</annotation>

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</math><img src="./4b40bd0a1d3c9771117157039602170561e5902f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.727ex; height:2.843ex;" alt="{\displaystyle n/p}" loading="lazy"> tends to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

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<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

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</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy">, then the <a href="Empirical_singular_values_distribution" class="mw-redirect" title="Empirical singular values distribution">empirical singular values distribution</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A+B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A+B}</annotation>

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</math><img src="./4279cdbd3cb8ec4c3423065d9a7d83a82cfc89e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A+B}" loading="lazy"> tends to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \boxplus _{c}\nu }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu \boxplus _{c}\nu }</annotation>

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</math><img src="./c4afc61032d4faef60e4d8975839e61d0c6235d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.418ex; height:2.676ex;" alt="{\displaystyle \mu \boxplus _{c}\nu }" loading="lazy">.<a href="#cite_note-6">[6]</a>

In many cases, it is possible to compute the probability measure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \boxplus _{c}\nu }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu \boxplus _{c}\nu }</annotation>

</semantics>

</math><img src="./c4afc61032d4faef60e4d8975839e61d0c6235d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.418ex; height:2.676ex;" alt="{\displaystyle \mu \boxplus _{c}\nu }" loading="lazy"> explicitly by using complex-analytic techniques and the rectangular R-transform with ratio <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> of the measures <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

</semantics>

</math><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Free_multiplicative_convolution">Free multiplicative convolution</h2></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

</semantics>

</math><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> be two probability measures on the interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,+\infty )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,+\infty )}</annotation>

</semantics>

</math><img src="./7c0ec7f25cac88c59009a7fe528dc000ec7f58c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle [0,+\infty )}" loading="lazy">, and assume that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a random variable in a non commutative probability space with law <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> is a random variable in the same non commutative probability space with law <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

</semantics>

</math><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy">. Assume finally that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> are <a href="Free_independence" title="Free independence">freely independent</a>. Then the free multiplicative convolution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \boxtimes \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu \boxtimes \nu }</annotation>

</semantics>

</math><img src="./aaaf8e9d3b76a72de3818ef3175b0893da82d429.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.474ex; height:2.676ex;" alt="{\displaystyle \mu \boxtimes \nu }" loading="lazy"> is the law of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{1/2}YX^{1/2}}">

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{1/2}YX^{1/2}}</annotation>

</semantics>

</math><img src="./16f6456132a4a0bb60eaeb1fe9b28bb5b317315d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.163ex; height:2.843ex;" alt="{\displaystyle X^{1/2}YX^{1/2}}" loading="lazy"> (or, equivalently, the law of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y^{1/2}XY^{1/2}}">

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y^{1/2}XY^{1/2}}</annotation>

</semantics>

</math><img src="./fd6025ac45f9890ea0d8fc6542cf1b6786978dc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.177ex; height:2.843ex;" alt="{\displaystyle Y^{1/2}XY^{1/2}}" loading="lazy">. <a href="Random_matrices" class="mw-redirect" title="Random matrices">Random matrices</a> interpretation: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> are some independent <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> non negative Hermitian (resp. real symmetric) random matrices such that at least one of them is invariant, in law, under conjugation by any unitary (resp. orthogonal) matrix and such that the <a href="Empirical_spectral_measures" class="mw-redirect" title="Empirical spectral measures">empirical spectral measures</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> tend respectively to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

</semantics>

</math><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> tends to infinity, then the empirical spectral measure of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AB}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle AB}</annotation>

</semantics>

</math><img src="./b04153f9681e5b06066357774475c04aaef3a8bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:2.176ex;" alt="{\displaystyle AB}" loading="lazy"> tends to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \boxtimes \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu \boxtimes \nu }</annotation>

</semantics>

</math><img src="./aaaf8e9d3b76a72de3818ef3175b0893da82d429.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.474ex; height:2.676ex;" alt="{\displaystyle \mu \boxtimes \nu }" loading="lazy">.<a href="#cite_note-7">[7]</a>

A similar definition can be made in the case of laws <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ,\nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu ,\nu }</annotation>

</semantics>

</math><img src="./489a4ac1b5d76663fa5328e6f3c1d382655cb496.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.668ex; height:2.176ex;" alt="{\displaystyle \mu ,\nu }" loading="lazy"> supported on the unit circle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{z:|z|=1\}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{z:|z|=1\}}</annotation>

</semantics>

</math><img src="./3fcc48df613ff7186686a8880ea29f7e5943ff09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.993ex; height:2.843ex;" alt="{\displaystyle \{z:|z|=1\}}" loading="lazy">, with an orthogonal or unitary <a href="Random_matrices" class="mw-redirect" title="Random matrices">random matrices</a> interpretation.

Explicit computations of multiplicative free convolution can be carried out using complex-analytic techniques and the S-transform.

<div class="mw-heading mw-heading2"><h2 id="Applications_of_free_convolution">Applications of free convolution</h2></div>

<ul><li>Free convolution can be used to give a proof of the free central limit theorem.</li>

<li>Free convolution can be used to compute the laws and spectra of sums or products of random variables which are free. Such examples include: <a href="Random_walk" title="Random walk">random walk</a> operators on free groups (Kesten measures); and asymptotic distribution of eigenvalues of sums or products of independent <a href="Random_matrix" title="Random matrix">random matrices</a>.</li></ul>

Through its applications to random matrices, free convolution has some strong connections with other works on G-estimation of Girko.

The applications in <a href="Wireless_communication" class="mw-redirect" title="Wireless communication">wireless communications</a>, <a href="Finance" title="Finance">finance</a> and <a href="Biology" title="Biology">biology</a> have provided a useful framework when the number of observations is of the same order as the dimensions of the system.

<ul><li><a href="Convolution" title="Convolution">Convolution</a></li>

<li><a href="Free_probability" title="Free probability">Free probability</a></li>

<li><a href="Random_matrix" title="Random matrix">Random matrix</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> Anderson, G.W.; Guionnet, A.; Zeitouni, O. (2010). An introduction to random matrices. Cambridge: Cambridge University Press. <style data-mw-deduplicate="TemplateStyles:r1238218222">

/* start https://en.wikipedia.org/ */

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/* end https://en.wikipedia.org/ */

</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-19452-5</bdi>.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Voiculescu, D., Addition of certain non-commuting random variables, J. Funct. Anal. 66 (1986), 323–346

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> Voiculescu, D., Multiplication of certain noncommuting random variables, J. Operator Theory 18 (1987), 2223–2235

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> Anderson, G.W.; Guionnet, A.; Zeitouni, O. (2010). An introduction to random matrices. Cambridge: Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-19452-5</bdi>.

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> Benaych-Georges, F., Rectangular random matrices, related convolution, Probab. Theory Related Fields Vol. 144, no. 3 (2009) 471-515.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> Benaych-Georges, F., Rectangular random matrices, related convolution, Probab. Theory Related Fields Vol. 144, no. 3 (2009) 471-515.

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> Anderson, G.W.; Guionnet, A.; Zeitouni, O. (2010). An introduction to random matrices. Cambridge: Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-19452-5</bdi>.

</li>

</ol></div></div>

<ul><li>"Free Deconvolution for Signal Processing Applications", O. Ryan and M. Debbah, ISIT 2007, pp. 1846–1850</li>

<li>James A. Mingo, Roland Speicher: <a rel="nofollow" class="external text" href="https://www.springer.com/us/book/9781493969418">Free Probability and Random Matrices</a>. Fields Institute Monographs, Vol. 35, Springer, New York, 2017.</li>

<li>D.-V. Voiculescu, N. Stammeier, M. Weber (eds.): <a rel="nofollow" class="external text" href="http://www.ems-ph.org/books/book.php?proj_nr=208">Free Probability and Operator Algebras</a>, Münster Lectures in Mathematics, EMS, 2016</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.supelec.fr/d2ri/flexibleradio/Welcome.html">Alcatel Lucent Chair on Flexible Radio</a></li>

<li><a rel="nofollow" class="external text" href="http://www.cmapx.polytechnique.fr/~benaych">http://www.cmapx.polytechnique.fr/~benaych</a></li>

<li><a rel="nofollow" class="external text" href="http://folk.uio.no/oyvindry">http://folk.uio.no/oyvindry</a></li>

<li><a rel="nofollow" class="external text" href="https://www.math.uni-sb.de/ag/speicher/speicher_publikationenE.html">survey articles</a> of Roland Speicher on free probability.</li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2023-06-21" href="https://en.wikipedia.org/wiki/?title=Free_convolution&oldid=1161197513">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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